All language subtitles for The Most Important Math Formula For Understanding Physics [HQsZG8Yxb7w].en

math is the language that we use to

math is the language that we use to
describe the laws of nature as

describe the laws of nature as

describe the laws of nature as
physicists and there's no way around it

physicists and there's no way around it

physicists and there's no way around it
if you want to understand physics you're

if you want to understand physics you're

if you want to understand physics you're
going to have to learn a lot of math and

going to have to learn a lot of math and

going to have to learn a lot of math and
if i had to pick one formula that's the

if i had to pick one formula that's the

if i had to pick one formula that's the
most important for understanding physics

most important for understanding physics

most important for understanding physics
it would be this one taylor's formula it

it would be this one taylor's formula it

it would be this one taylor's formula it
shows up in virtually everything we do

shows up in virtually everything we do

shows up in virtually everything we do
in physics and in this video i want to

in physics and in this video i want to

in physics and in this video i want to
explain how it works and give you a few

explain how it works and give you a few

explain how it works and give you a few
examples of its importance in different

examples of its importance in different

examples of its importance in different
corners of physics you probably learned

corners of physics you probably learned

corners of physics you probably learned
this theorem before if you've taken a

this theorem before if you've taken a

this theorem before if you've taken a
calculus class though you might not have

calculus class though you might not have

calculus class though you might not have
written it in this nice and compact

written it in this nice and compact

written it in this nice and compact
notation i'll show you that it's

notation i'll show you that it's

notation i'll show you that it's
equivalent to the taylor series that

equivalent to the taylor series that

equivalent to the taylor series that
lets us expand any smooth function in

lets us expand any smooth function in

lets us expand any smooth function in
powers of x

powers of x

powers of x
in the first half of the video i'm going

in the first half of the video i'm going

in the first half of the video i'm going
to explain where this incredibly

to explain where this incredibly

to explain where this incredibly
important formula comes from and what it

important formula comes from and what it

important formula comes from and what it
means and then in the second half i'll

means and then in the second half i'll

means and then in the second half i'll
tell you about three applications in

tell you about three applications in

tell you about three applications in
physics where it shows up though again

physics where it shows up though again

physics where it shows up though again
you'd be hard-pressed to find a chapter

you'd be hard-pressed to find a chapter

you'd be hard-pressed to find a chapter
of any physics textbook where it's not

of any physics textbook where it's not

of any physics textbook where it's not
applied number one we'll look at how the

applied number one we'll look at how the

applied number one we'll look at how the
taylor series enables us to understand

taylor series enables us to understand

taylor series enables us to understand
the complicated equations we often need

the complicated equations we often need

the complicated equations we often need
to solve in physics by studying a limit

to solve in physics by studying a limit

to solve in physics by studying a limit
where the equations simplify second i'll

where the equations simplify second i'll

where the equations simplify second i'll
show you how einstein's e equals m c

show you how einstein's e equals m c

show you how einstein's e equals m c
squared or actually his more general

squared or actually his more general

squared or actually his more general
formula for the energy of a relativistic

formula for the energy of a relativistic

formula for the energy of a relativistic
particle of mass m and momentum p

particle of mass m and momentum p

particle of mass m and momentum p
correctly reproduces the more familiar

correctly reproduces the more familiar

correctly reproduces the more familiar
kinetic energy one-half mv squared for

kinetic energy one-half mv squared for

kinetic energy one-half mv squared for
particles that aren't moving too close

particles that aren't moving too close

particles that aren't moving too close
to the speed of light and also how that

to the speed of light and also how that

to the speed of light and also how that
same taylor series leads to the fine

same taylor series leads to the fine

same taylor series leads to the fine
structure of the energy levels of the

structure of the energy levels of the

structure of the energy levels of the
hydrogen atom and third we'll look at

hydrogen atom and third we'll look at

hydrogen atom and third we'll look at
how taylor's formula leads directly to

how taylor's formula leads directly to

how taylor's formula leads directly to
the definition of the momentum operator

the definition of the momentum operator

the definition of the momentum operator
in quantum mechanics i'm not assuming

in quantum mechanics i'm not assuming

in quantum mechanics i'm not assuming
you've learned much about relativity or

you've learned much about relativity or

you've learned much about relativity or
quantum mechanics before by the way the

quantum mechanics before by the way the

quantum mechanics before by the way the
point is just to see some of the

point is just to see some of the

point is just to see some of the
different ways that taylor's formula

different ways that taylor's formula

different ways that taylor's formula
shows up in different areas of physics

shows up in different areas of physics

shows up in different areas of physics
so let's start with the math and

so let's start with the math and

so let's start with the math and
understand what this formula is all

understand what this formula is all

understand what this formula is all
about say we have some function f x

about say we have some function f x

about say we have some function f x
here's a random example it looks really

here's a random example it looks really

here's a random example it looks really
complicated but instead of trying to

complicated but instead of trying to

complicated but instead of trying to
understand the whole complicated

understand the whole complicated

understand the whole complicated
function at once let's zoom in and look

function at once let's zoom in and look

function at once let's zoom in and look
at it in a smaller region where it's a

at it in a smaller region where it's a

at it in a smaller region where it's a
lot simpler take this point here for

lot simpler take this point here for

lot simpler take this point here for
example and let's choose our origin so

example and let's choose our origin so

example and let's choose our origin so
that this point is sitting at x equals

that this point is sitting at x equals

that this point is sitting at x equals
zero so the height of the function there

zero so the height of the function there

zero so the height of the function there
is f of zero imagine that this curve is

is f of zero imagine that this curve is

is f of zero imagine that this curve is
the shape of a treacherous mountain and

the shape of a treacherous mountain and

the shape of a treacherous mountain and
you're an intrepid explorer plotting out

you're an intrepid explorer plotting out

you're an intrepid explorer plotting out
its map you're high up in the air and

its map you're high up in the air and

its map you're high up in the air and
it's very foggy so you can't see very

it's very foggy so you can't see very

it's very foggy so you can't see very
far in either direction

far in either direction

far in either direction
you just need to carefully walk along

you just need to carefully walk along

you just need to carefully walk along
the mountain and record its shape using

the mountain and record its shape using

the mountain and record its shape using
an altimeter that measures your height

an altimeter that measures your height

an altimeter that measures your height
above the ground starting from this

above the ground starting from this

above the ground starting from this
point x equals zero initially all you

point x equals zero initially all you

point x equals zero initially all you
can say is that your starting height is

can say is that your starting height is

can say is that your starting height is
f zero for all you know the whole

f zero for all you know the whole

f zero for all you know the whole
mountain might just be a flat horizontal

mountain might just be a flat horizontal

mountain might just be a flat horizontal
line at this height then when you try to

line at this height then when you try to

line at this height then when you try to
write down in your field journal a

write down in your field journal a

write down in your field journal a
function that describes the height of

function that describes the height of

function that describes the height of
the mountain your first guess is just f

the mountain your first guess is just f

the mountain your first guess is just f
of x equals f of zero a horizontal line

of x equals f of zero a horizontal line

of x equals f of zero a horizontal line
at your starting height

at your starting height

at your starting height
but now you take a little hop to the

but now you take a little hop to the

but now you take a little hop to the
right and you discover that the height

right and you discover that the height

right and you discover that the height
of the mountain has changed so it's not

of the mountain has changed so it's not

of the mountain has changed so it's not
actually a horizontal line instead as

actually a horizontal line instead as

actually a horizontal line instead as
far as you can tell now it looks like a

far as you can tell now it looks like a

far as you can tell now it looks like a
line that's sloped at an angle where the

line that's sloped at an angle where the

line that's sloped at an angle where the
slope is given by the first derivative

slope is given by the first derivative

slope is given by the first derivative
of f at x equals 0.

of f at x equals 0.

of f at x equals 0.
now your new best guess for the height

now your new best guess for the height

now your new best guess for the height
function is the equation of this sloped

function is the equation of this sloped

function is the equation of this sloped
line f of x equals f of 0 plus f prime

line f of x equals f of 0 plus f prime

line f of x equals f of 0 plus f prime
of 0 times x but now you take another

of 0 times x but now you take another

of 0 times x but now you take another
hop and you discover that the curve

hop and you discover that the curve

hop and you discover that the curve
isn't a straight line after all instead

isn't a straight line after all instead

isn't a straight line after all instead
it starts to deflect away from a line

it starts to deflect away from a line

it starts to deflect away from a line
like a parabola so now you expect

like a parabola so now you expect

like a parabola so now you expect
there's a better approximation to the

there's a better approximation to the

there's a better approximation to the
function like this with an x squared

function like this with an x squared

function like this with an x squared
term and some coefficient a in front of

term and some coefficient a in front of

term and some coefficient a in front of
it but at this point you start to think

it but at this point you start to think

it but at this point you start to think
to yourself that you might be able to

to yourself that you might be able to

to yourself that you might be able to
get an even better description of the

get an even better description of the

get an even better description of the
function over a wider range by including

function over a wider range by including

function over a wider range by including
many more powers of x x cubed x to the

many more powers of x x cubed x to the

many more powers of x x cubed x to the
fourth and so on so you stop exploring

fourth and so on so you stop exploring

fourth and so on so you stop exploring
for a moment and sit down to do some

for a moment and sit down to do some

for a moment and sit down to do some
math

math

math
you want to express this function f as a

you want to express this function f as a

you want to express this function f as a
sum of powers of x with some

sum of powers of x with some

sum of powers of x with some
coefficients c0 c1 c2 and so on the

coefficients c0 c1 c2 and so on the

coefficients c0 c1 c2 and so on the
question is how to pick these numbers so

question is how to pick these numbers so

question is how to pick these numbers so
that this series does a good job of

that this series does a good job of

that this series does a good job of
matching up with f

matching up with f

matching up with f
well we've already seen what the first

well we've already seen what the first

well we've already seen what the first
couple of coefficients are when we plug

couple of coefficients are when we plug

couple of coefficients are when we plug
in x equals 0 everything except the

in x equals 0 everything except the

in x equals 0 everything except the
first term disappears and we get f 0

first term disappears and we get f 0

first term disappears and we get f 0
equals c 0. so that first number in the

equals c 0. so that first number in the

equals c 0. so that first number in the
series is just the value of the function

series is just the value of the function

series is just the value of the function
at our starting point x equals zero as

at our starting point x equals zero as

at our starting point x equals zero as
for the next one c one x and everything

for the next one c one x and everything

for the next one c one x and everything
after it disappears when we plug in x

after it disappears when we plug in x

after it disappears when we plug in x
equals zero but what if we take the

equals zero but what if we take the

equals zero but what if we take the
derivative first remembering the rule

derivative first remembering the rule

derivative first remembering the rule
that to take the derivative of x to the

that to take the derivative of x to the

that to take the derivative of x to the
n we bring the power down out front and

n we bring the power down out front and

n we bring the power down out front and
then reduce the exponent by one we get f

then reduce the exponent by one we get f

then reduce the exponent by one we get f
prime of x equals c one plus two c two

prime of x equals c one plus two c two

prime of x equals c one plus two c two
times x plus three c three x squared

times x plus three c three x squared

times x plus three c three x squared
plus dot dot dot now when we plug in x

plus dot dot dot now when we plug in x

plus dot dot dot now when we plug in x
equals zero the c one term survives f

equals zero the c one term survives f

equals zero the c one term survives f
prime of 0 equals c1 so again like we

prime of 0 equals c1 so again like we

prime of 0 equals c1 so again like we
already knew we should set this first

already knew we should set this first

already knew we should set this first
coefficient c1 to be the derivative of f

coefficient c1 to be the derivative of f

coefficient c1 to be the derivative of f
at x equals 0. but now we've got the

at x equals 0. but now we've got the

at x equals 0. but now we've got the
idea if we take the derivative again we

idea if we take the derivative again we

idea if we take the derivative again we
get f double prime of x equals two c two

get f double prime of x equals two c two

get f double prime of x equals two c two
plus three times two c three x and so on

plus three times two c three x and so on

plus three times two c three x and so on
and so when we plug in x equals zero we

and so when we plug in x equals zero we

and so when we plug in x equals zero we
learned that f double prime of zero

learned that f double prime of zero

learned that f double prime of zero
equals two c two then we should choose c

equals two c two then we should choose c

equals two c two then we should choose c
two to be half the second derivative of

two to be half the second derivative of

two to be half the second derivative of
f at x equals zero and on and on it goes

f at x equals zero and on and on it goes

f at x equals zero and on and on it goes
hopefully you see the pattern for the x

hopefully you see the pattern for the x

hopefully you see the pattern for the x
to the n term in the sum we need to take

to the n term in the sum we need to take

to the n term in the sum we need to take
the derivative n times until this is the

the derivative n times until this is the

the derivative n times until this is the
only term that's around when we plug in

only term that's around when we plug in

only term that's around when we plug in
x equals zero each derivative brings

x equals zero each derivative brings

x equals zero each derivative brings
down a power first n then n minus 1 n

down a power first n then n minus 1 n

down a power first n then n minus 1 n
minus 2 and so on all the way down to 3

minus 2 and so on all the way down to 3

minus 2 and so on all the way down to 3
2 1. in other words the nth derivative

2 1. in other words the nth derivative

2 1. in other words the nth derivative
is n factorial times c n so the nth

is n factorial times c n so the nth

is n factorial times c n so the nth
coefficient is 1 over n factorial times

coefficient is 1 over n factorial times

coefficient is 1 over n factorial times
the nth derivative of f evaluated at x

the nth derivative of f evaluated at x

the nth derivative of f evaluated at x
equals 0. for this x cubed term for

equals 0. for this x cubed term for

equals 0. for this x cubed term for
example we get 1 over 3 factorial which

example we get 1 over 3 factorial which

example we get 1 over 3 factorial which
is 1 6 times the third derivative of f

is 1 6 times the third derivative of f

is 1 6 times the third derivative of f
at 0. by the way i go through all this

at 0. by the way i go through all this

at 0. by the way i go through all this
step by step in the notes which you can

step by step in the notes which you can

step by step in the notes which you can
get at the link in the description to

get at the link in the description to

get at the link in the description to
dig through all the details here after

dig through all the details here after

dig through all the details here after
you've watched

you've watched

you've watched
now we're in business we've written an

now we're in business we've written an

now we're in business we've written an
approximation for our function f as a

approximation for our function f as a

approximation for our function f as a
sum of powers of x this is the taylor

sum of powers of x this is the taylor

sum of powers of x this is the taylor
series for f when x is tiny so that

series for f when x is tiny so that

series for f when x is tiny so that
we're close to the starting point here

we're close to the starting point here

we're close to the starting point here
each higher power of x is an even tinier

each higher power of x is an even tinier

each higher power of x is an even tinier
number than the one that came before it

number than the one that came before it

number than the one that came before it
and so we already get a good

and so we already get a good

and so we already get a good
approximation by keeping just the first

approximation by keeping just the first

approximation by keeping just the first
few terms in the series the farther away

few terms in the series the farther away

few terms in the series the farther away
we venture from x equals 0 the more

we venture from x equals 0 the more

we venture from x equals 0 the more
terms we need to include in the series

terms we need to include in the series

terms we need to include in the series
in order to get a good description of

in order to get a good description of

in order to get a good description of
the function but now that we have the

the function but now that we have the

the function but now that we have the
general formula for the coefficients we

general formula for the coefficients we

general formula for the coefficients we
can include as many terms as we like

can include as many terms as we like

can include as many terms as we like
here's what it looks like for this

here's what it looks like for this

here's what it looks like for this
function as we make the total number of

function as we make the total number of

function as we make the total number of
terms bigger and bigger you can see that

terms bigger and bigger you can see that

terms bigger and bigger you can see that
the series approximation matches the

the series approximation matches the

the series approximation matches the
curve better and better as n gets large

curve better and better as n gets large

curve better and better as n gets large
and here's the kicker when we include

and here's the kicker when we include

and here's the kicker when we include
all the terms by summing up the infinite

all the terms by summing up the infinite

all the terms by summing up the infinite
series over all powers of x we actually

series over all powers of x we actually

series over all powers of x we actually
reproduce the exact function that we

reproduce the exact function that we

reproduce the exact function that we
started with as long as it was smooth

started with as long as it was smooth

started with as long as it was smooth
and well behaved and the series

and well behaved and the series

and well behaved and the series
converges

converges

converges
this is truly remarkable it says that if

this is truly remarkable it says that if

this is truly remarkable it says that if
we know all the derivatives of a smooth

we know all the derivatives of a smooth

we know all the derivatives of a smooth
function at a single point we can

function at a single point we can

function at a single point we can
reconstruct the rest of the function

reconstruct the rest of the function

reconstruct the rest of the function
everywhere else let's do some examples

everywhere else let's do some examples

everywhere else let's do some examples
how about f of x equals sine of x let's

how about f of x equals sine of x let's

how about f of x equals sine of x let's
write down its taylor series around x

write down its taylor series around x

write down its taylor series around x
equals 0. we just need to know what its

equals 0. we just need to know what its

equals 0. we just need to know what its
derivatives are at that point so let's

derivatives are at that point so let's

derivatives are at that point so let's
make a little table

make a little table

make a little table
sine of x passes through the origin so

sine of x passes through the origin so

sine of x passes through the origin so
we start off with f of zero equals zero

we start off with f of zero equals zero

we start off with f of zero equals zero
zero factorial is defined to be one by

zero factorial is defined to be one by

zero factorial is defined to be one by
the way and so this first term in the

the way and so this first term in the

the way and so this first term in the
taylor series is just zero next up we

taylor series is just zero next up we

taylor series is just zero next up we
need the derivative of sine of x which

need the derivative of sine of x which

need the derivative of sine of x which
is cosine of x now when we plug in x

is cosine of x now when we plug in x

is cosine of x now when we plug in x
equals zero we get cosine of 0 which is

equals zero we get cosine of 0 which is

equals zero we get cosine of 0 which is
1. and so the first interesting term in

1. and so the first interesting term in

1. and so the first interesting term in
the taylor series is just x a straight

the taylor series is just x a straight

the taylor series is just x a straight
line through the origin with slope 1.

line through the origin with slope 1.

line through the origin with slope 1.
that's already a very good approximation

that's already a very good approximation

that's already a very good approximation
to sine of x when x is a small number

to sine of x when x is a small number

to sine of x when x is a small number
and so we use the approximation sine of

and so we use the approximation sine of

and so we use the approximation sine of
x equals x often in physics but as x

x equals x often in physics but as x

x equals x often in physics but as x
gets a little bigger clearly the

gets a little bigger clearly the

gets a little bigger clearly the
straight line isn't going to cut it

straight line isn't going to cut it

straight line isn't going to cut it
anymore so let's keep going for the next

anymore so let's keep going for the next

anymore so let's keep going for the next
term we need the derivative of cosine of

term we need the derivative of cosine of

term we need the derivative of cosine of
x which is minus sine of x but that

x which is minus sine of x but that

x which is minus sine of x but that
vanishes again when x equals zero and so

vanishes again when x equals zero and so

vanishes again when x equals zero and so
the x squared term actually disappears

the x squared term actually disappears

the x squared term actually disappears
that's part of the reason the linear

that's part of the reason the linear

that's part of the reason the linear
approximation was so good to begin with

approximation was so good to begin with

approximation was so good to begin with
now for f triple prime of x we need the

now for f triple prime of x we need the

now for f triple prime of x we need the
derivative of minus sine and we get

derivative of minus sine and we get

derivative of minus sine and we get
minus cosine then f triple prime of zero

minus cosine then f triple prime of zero

minus cosine then f triple prime of zero
equals minus one and so the qubit

equals minus one and so the qubit

equals minus one and so the qubit
coefficient is minus one over three

coefficient is minus one over three

coefficient is minus one over three
factorial

factorial

factorial
let's do one more line we want the

let's do one more line we want the

let's do one more line we want the
derivative of minus cosine which is sine

derivative of minus cosine which is sine

derivative of minus cosine which is sine
this vanishes again when we plug in x

this vanishes again when we plug in x

this vanishes again when we plug in x
equals zero and so there's no x to the

equals zero and so there's no x to the

equals zero and so there's no x to the
four term in the series

four term in the series

four term in the series
we could keep going like this and take

we could keep going like this and take

we could keep going like this and take
more derivatives but notice that with

more derivatives but notice that with

more derivatives but notice that with
the fourth derivative here we've just

the fourth derivative here we've just

the fourth derivative here we've just
gotten back to where we started with

gotten back to where we started with

gotten back to where we started with
sine of x so this same sequence of four

sine of x so this same sequence of four

sine of x so this same sequence of four
derivatives is just going to repeat

derivatives is just going to repeat

derivatives is just going to repeat
itself over and over again then we can

itself over and over again then we can

itself over and over again then we can
just write down the whole series without

just write down the whole series without

just write down the whole series without
any more work the next term is x to the

any more work the next term is x to the

any more work the next term is x to the
fifth over five factorial followed by

fifth over five factorial followed by

fifth over five factorial followed by
minus one over seven factorial x to the

minus one over seven factorial x to the

minus one over seven factorial x to the
seventh and plus one over nine factorial

seventh and plus one over nine factorial

seventh and plus one over nine factorial
x to the nine and so on here's what it

x to the nine and so on here's what it

x to the nine and so on here's what it
looks like going up to x to the 29 you

looks like going up to x to the 29 you

looks like going up to x to the 29 you
can see that it does a good job

can see that it does a good job

can see that it does a good job
reproducing the sine curve right up to

reproducing the sine curve right up to

reproducing the sine curve right up to
the edges of these four periods

the edges of these four periods

the edges of these four periods
notice that only odd powers of x show up

notice that only odd powers of x show up

notice that only odd powers of x show up
here in the taylor series that's because

here in the taylor series that's because

here in the taylor series that's because
sine of x is an odd function when you

sine of x is an odd function when you

sine of x is an odd function when you
compare it on the right and left sides

compare it on the right and left sides

compare it on the right and left sides
of the y axis it looks the same except

of the y axis it looks the same except

of the y axis it looks the same except
that it's been flipped over in other

that it's been flipped over in other

that it's been flipped over in other
words sine of minus x equals minus sine

words sine of minus x equals minus sine

words sine of minus x equals minus sine
of x odd powers of x share that property

of x odd powers of x share that property

of x odd powers of x share that property
but even powers don't and that's why

but even powers don't and that's why

but even powers don't and that's why
there are no even powers of x in the

there are no even powers of x in the

there are no even powers of x in the
taylor series for sine like i mentioned

taylor series for sine like i mentioned

taylor series for sine like i mentioned
before oftentimes in physics we're not

before oftentimes in physics we're not

before oftentimes in physics we're not
actually interested in the whole taylor

actually interested in the whole taylor

actually interested in the whole taylor
series what we really want is a good

series what we really want is a good

series what we really want is a good
approximation to a complicated function

approximation to a complicated function

approximation to a complicated function
that makes a problem simpler to solve

that makes a problem simpler to solve

that makes a problem simpler to solve
i'll show you examples of what i mean in

i'll show you examples of what i mean in

i'll show you examples of what i mean in
a minute so in this case we might stop

a minute so in this case we might stop

a minute so in this case we might stop
with the first term and just apply the

with the first term and just apply the

with the first term and just apply the
fact that sine of x is approximately

fact that sine of x is approximately

fact that sine of x is approximately
equal to x when x is small this is

equal to x when x is small this is

equal to x when x is small this is
called the small angle approximation and

called the small angle approximation and

called the small angle approximation and
you may have run into it in your first

you may have run into it in your first

you may have run into it in your first
physics class when you learned about the

physics class when you learned about the

physics class when you learned about the
simple pendulum the key point here is

simple pendulum the key point here is

simple pendulum the key point here is
that when x is small like say 0.1 then

that when x is small like say 0.1 then

that when x is small like say 0.1 then
when we take larger powers of x in the

when we take larger powers of x in the

when we take larger powers of x in the
successive terms in the taylor series

successive terms in the taylor series

successive terms in the taylor series
they get even smaller x cubed equals

they get even smaller x cubed equals

they get even smaller x cubed equals
.001 x to the fifth equals .0001

.001 x to the fifth equals .0001

.001 x to the fifth equals .0001
and so on not to mention the effect of

and so on not to mention the effect of

and so on not to mention the effect of
the huge factorials in the denominators

the huge factorials in the denominators

the huge factorials in the denominators
that's why we can ignore the higher

that's why we can ignore the higher

that's why we can ignore the higher
order terms for small x and get a good

order terms for small x and get a good

order terms for small x and get a good
approximation to our function just by

approximation to our function just by

approximation to our function just by
keeping the leading term let's do one

keeping the leading term let's do one

keeping the leading term let's do one
more quick example f of x equals e to

more quick example f of x equals e to

more quick example f of x equals e to
the x this will be important in a moment

the x this will be important in a moment

the x this will be important in a moment
for seeing the slickest way to write

for seeing the slickest way to write

for seeing the slickest way to write
down taylor's formula this one's easy

down taylor's formula this one's easy

down taylor's formula this one's easy
because the derivative of e to the x is

because the derivative of e to the x is

because the derivative of e to the x is
just e to the x again and when we plug

just e to the x again and when we plug

just e to the x again and when we plug
in x equals zero we get one and the

in x equals zero we get one and the

in x equals zero we get one and the
coefficients are one over n factorial so

coefficients are one over n factorial so

coefficients are one over n factorial so
we can just jump right to the taylor

we can just jump right to the taylor

we can just jump right to the taylor
series e to the x equals one plus x plus

series e to the x equals one plus x plus

series e to the x equals one plus x plus
one over two factorial x squared plus

one over two factorial x squared plus

one over two factorial x squared plus
one over three factorial x cubed and so

one over three factorial x cubed and so

one over three factorial x cubed and so
on and once again if x is tiny then we

on and once again if x is tiny then we

on and once again if x is tiny then we
can get a good approximation by stopping

can get a good approximation by stopping

can get a good approximation by stopping
at the linear term it's not quite as

at the linear term it's not quite as

at the linear term it's not quite as
good as we had for sine of x though

good as we had for sine of x though

good as we had for sine of x though
because in that case the x squared

because in that case the x squared

because in that case the x squared
correction vanished now before we get to

correction vanished now before we get to

correction vanished now before we get to
the physics examples the last thing i

the physics examples the last thing i

the physics examples the last thing i
want to do is show you a few convenient

want to do is show you a few convenient

want to do is show you a few convenient
ways of writing taylor's formula

ways of writing taylor's formula

ways of writing taylor's formula
spelling out the whole sum like this

spelling out the whole sum like this

spelling out the whole sum like this
obviously isn't very concise but we can

obviously isn't very concise but we can

obviously isn't very concise but we can
write the same thing much more compactly

write the same thing much more compactly

write the same thing much more compactly
using some notation the sum over n of

using some notation the sum over n of

using some notation the sum over n of
one over n factorial the nth derivative

one over n factorial the nth derivative

one over n factorial the nth derivative
of f evaluated at zero times x to the n

of f evaluated at zero times x to the n

of f evaluated at zero times x to the n
this is the taylor series for f x

this is the taylor series for f x

this is the taylor series for f x
expanded around x equals zero but come

expanded around x equals zero but come

expanded around x equals zero but come
to think of it there was nothing special

to think of it there was nothing special

to think of it there was nothing special
about x equals zero here that's just

about x equals zero here that's just

about x equals zero here that's just
where we happened to put the origin when

where we happened to put the origin when

where we happened to put the origin when
we drew the graph of f of x we could

we drew the graph of f of x we could

we drew the graph of f of x we could
just as well write an expansion around

just as well write an expansion around

just as well write an expansion around
any other point call it x zero say then

any other point call it x zero say then

any other point call it x zero say then
the taylor expansion of f around x0 is

the taylor expansion of f around x0 is

the taylor expansion of f around x0 is
given by the sum of one over n factorial

given by the sum of one over n factorial

given by the sum of one over n factorial
times the derivatives of f evaluated at

times the derivatives of f evaluated at

times the derivatives of f evaluated at
that starting point x0 times the powers

that starting point x0 times the powers

that starting point x0 times the powers
of the distance from there x minus x0

of the distance from there x minus x0

of the distance from there x minus x0
for example here's what we get with the

for example here's what we get with the

for example here's what we get with the
first few terms of the taylor series for

first few terms of the taylor series for

first few terms of the taylor series for
this function expanded around this given

this function expanded around this given

this function expanded around this given
point

point

point
actually there's another way of writing

actually there's another way of writing

actually there's another way of writing
this expression that's often more useful

this expression that's often more useful

this expression that's often more useful
let epsilon denote this quantity in

let epsilon denote this quantity in

let epsilon denote this quantity in
parentheses x minus x0 it measures how

parentheses x minus x0 it measures how

parentheses x minus x0 it measures how
far away you are from the starting point

far away you are from the starting point

far away you are from the starting point
so when epsilon is small you're very

so when epsilon is small you're very

so when epsilon is small you're very
close to x0 and as it gets bigger you

close to x0 and as it gets bigger you

close to x0 and as it gets bigger you
get farther away then x is given by x0

get farther away then x is given by x0

get farther away then x is given by x0
plus epsilon and we can write the same

plus epsilon and we can write the same

plus epsilon and we can write the same
expression like this

expression like this

expression like this
this way of writing things makes it

this way of writing things makes it

this way of writing things makes it
really clear that we can think of the

really clear that we can think of the

really clear that we can think of the
taylor series as starting at the point

taylor series as starting at the point

taylor series as starting at the point
x0 and then expanding out away from

x0 and then expanding out away from

x0 and then expanding out away from
there by evaluating f at x0 plus epsilon

there by evaluating f at x0 plus epsilon

there by evaluating f at x0 plus epsilon
in powers of the displacement but this

in powers of the displacement but this

in powers of the displacement but this
isn't even the slickest way to write the

isn't even the slickest way to write the

isn't even the slickest way to write the
taylor series which is the formula i

taylor series which is the formula i

taylor series which is the formula i
showed you at the very beginning of the

showed you at the very beginning of the

showed you at the very beginning of the
video to see how that works we'll switch

video to see how that works we'll switch

video to see how that works we'll switch
to the other notation for derivatives so

to the other notation for derivatives so

to the other notation for derivatives so
the nth derivative of f is obtained by

the nth derivative of f is obtained by

the nth derivative of f is obtained by
applying d by dx to it n times or in

applying d by dx to it n times or in

applying d by dx to it n times or in
other words d by dx to the power n

other words d by dx to the power n

other words d by dx to the power n
acting on f then we'll plug this into

acting on f then we'll plug this into

acting on f then we'll plug this into
the taylor series which lets us write it

the taylor series which lets us write it

the taylor series which lets us write it
like this i went ahead and dropped the

like this i went ahead and dropped the

like this i went ahead and dropped the
x0 subscript now because that was just a

x0 subscript now because that was just a

x0 subscript now because that was just a
label that we don't need anymore so far

label that we don't need anymore so far

label that we don't need anymore so far
this doesn't look like a huge

this doesn't look like a huge

this doesn't look like a huge
simplification but now let's drag that

simplification but now let's drag that

simplification but now let's drag that
epsilon to the end to the left inside

epsilon to the end to the left inside

epsilon to the end to the left inside
the parentheses

the parentheses

the parentheses
now this looks really interesting it

now this looks really interesting it

now this looks really interesting it
says that if we want to know the value

says that if we want to know the value

says that if we want to know the value
of our function f at a point that's

of our function f at a point that's

of our function f at a point that's
shifted away from x by an amount epsilon

shifted away from x by an amount epsilon

shifted away from x by an amount epsilon
what we should do is take the function

what we should do is take the function

what we should do is take the function
at the original point x and apply this

at the original point x and apply this

at the original point x and apply this
special combination of derivatives to it

special combination of derivatives to it

special combination of derivatives to it
but hang on a second that might look

but hang on a second that might look

but hang on a second that might look
familiar remember from a minute ago that

familiar remember from a minute ago that

familiar remember from a minute ago that
the taylor series we found for e to the

the taylor series we found for e to the

the taylor series we found for e to the
z was one plus z plus one over two

z was one plus z plus one over two

z was one plus z plus one over two
factorial z squared plus one over three

factorial z squared plus one over three

factorial z squared plus one over three
factorial z cubed plus dot dot

factorial z cubed plus dot dot

factorial z cubed plus dot dot
or in some notation the sum of one over

or in some notation the sum of one over

or in some notation the sum of one over
n factorial z to the n but that's

n factorial z to the n but that's

n factorial z to the n but that's
exactly what this differential operator

exactly what this differential operator

exactly what this differential operator
looks like where z is this thing in

looks like where z is this thing in

looks like where z is this thing in
parentheses epsilon d by dx then this

parentheses epsilon d by dx then this

parentheses epsilon d by dx then this
big sum of derivatives is nothing but e

big sum of derivatives is nothing but e

big sum of derivatives is nothing but e
to the epsilon d by d x and so at least

to the epsilon d by d x and so at least

to the epsilon d by d x and so at least
formally we can write f of x plus

formally we can write f of x plus

formally we can write f of x plus
epsilon equals e to the epsilon d by d x

epsilon equals e to the epsilon d by d x

epsilon equals e to the epsilon d by d x
acting on f x

acting on f x

acting on f x
this is the most compact convenient and

this is the most compact convenient and

this is the most compact convenient and
beautiful way of writing taylor's

beautiful way of writing taylor's

beautiful way of writing taylor's
formula it neatly repackages the whole

formula it neatly repackages the whole

formula it neatly repackages the whole
infinite sum over derivatives of f and

infinite sum over derivatives of f and

infinite sum over derivatives of f and
powers of the displacement into a single

powers of the displacement into a single

powers of the displacement into a single
operator e to the epsilon d by dx acting

operator e to the epsilon d by dx acting

operator e to the epsilon d by dx acting
on the function just to make sure it's

on the function just to make sure it's

on the function just to make sure it's
clear how this works let's try applying

clear how this works let's try applying

clear how this works let's try applying
it to a really simple function f of x

it to a really simple function f of x

it to a really simple function f of x
equals mx plus b obviously the taylor

equals mx plus b obviously the taylor

equals mx plus b obviously the taylor
series for this one is going to be

series for this one is going to be

series for this one is going to be
really boring it already is its own

really boring it already is its own

really boring it already is its own
taylor series we expand out the

taylor series we expand out the

taylor series we expand out the
exponential and get one plus epsilon

exponential and get one plus epsilon

exponential and get one plus epsilon
times the first derivative plus one half

times the first derivative plus one half

times the first derivative plus one half
epsilon squared times the second

epsilon squared times the second

epsilon squared times the second
derivative and so on and then all that

derivative and so on and then all that

derivative and so on and then all that
acts on the function mx plus b when we

acts on the function mx plus b when we

acts on the function mx plus b when we
multiply out the one we just get back mx

multiply out the one we just get back mx

multiply out the one we just get back mx
plus b and when the first derivative

plus b and when the first derivative

plus b and when the first derivative
term x we get epsilon m as for that

term x we get epsilon m as for that

term x we get epsilon m as for that
second derivative and everything else

second derivative and everything else

second derivative and everything else
all that disappears because when you

all that disappears because when you

all that disappears because when you
take more than one derivative of a

take more than one derivative of a

take more than one derivative of a
straight line you get zero so altogether

straight line you get zero so altogether

straight line you get zero so altogether
we've got mx plus b plus epsilon m or

we've got mx plus b plus epsilon m or

we've got mx plus b plus epsilon m or
equivalently m times x plus epsilon plus

equivalently m times x plus epsilon plus

equivalently m times x plus epsilon plus
b which is precisely f of x plus epsilon

b which is precisely f of x plus epsilon

b which is precisely f of x plus epsilon
just as expected

just as expected

just as expected
one last beautiful thing about this way

one last beautiful thing about this way

one last beautiful thing about this way
of writing taylor's formula before we

of writing taylor's formula before we

of writing taylor's formula before we
get to the physics it makes the

get to the physics it makes the

get to the physics it makes the
generalization to the multivariable

generalization to the multivariable

generalization to the multivariable
taylor expansion really straightforward

taylor expansion really straightforward

taylor expansion really straightforward
say we now have a function f of x y z

say we now have a function f of x y z

say we now have a function f of x y z
for example this might be the potential

for example this might be the potential

for example this might be the potential
energy function of a particle moving

energy function of a particle moving

energy function of a particle moving
around in three dimensional space then

around in three dimensional space then

around in three dimensional space then
what's the taylor expansion of this the

what's the taylor expansion of this the

what's the taylor expansion of this the
most direct way to approach it is to

most direct way to approach it is to

most direct way to approach it is to
expand like before with one variable at

expand like before with one variable at

expand like before with one variable at
a time if we apply the taylor expansion

a time if we apply the taylor expansion

a time if we apply the taylor expansion
just in x we get f of x still with y

just in x we get f of x still with y

just in x we get f of x still with y
plus epsilon y z plus epsilon z plus

plus epsilon y z plus epsilon z plus

plus epsilon y z plus epsilon z plus
epsilon x d by dx of that plus half

epsilon x d by dx of that plus half

epsilon x d by dx of that plus half
epsilon x squared times the second

epsilon x squared times the second

epsilon x squared times the second
derivative with respect to x and so on

derivative with respect to x and so on

derivative with respect to x and so on
where these are now partial derivatives

where these are now partial derivatives

where these are now partial derivatives
because f is a function of more than one

because f is a function of more than one

because f is a function of more than one
variable all that means is that we take

variable all that means is that we take

variable all that means is that we take
the derivative of f with respect to x

the derivative of f with respect to x

the derivative of f with respect to x
like we normally would treating the

like we normally would treating the

like we normally would treating the
other variables y and z as constants but

other variables y and z as constants but

other variables y and z as constants but
now we have to do the same expansion

now we have to do the same expansion

now we have to do the same expansion
over again in each of these terms for y

over again in each of these terms for y

over again in each of these terms for y
and then again in each of those terms

and then again in each of those terms

and then again in each of those terms
for z it's a bit of a mess but our

for z it's a bit of a mess but our

for z it's a bit of a mess but our
exponential formula makes the whole

exponential formula makes the whole

exponential formula makes the whole
thing incredibly simple let's write r

thing incredibly simple let's write r

thing incredibly simple let's write r
vector equals x y z for the position

vector equals x y z for the position

vector equals x y z for the position
vector and epsilon vector equals epsilon

vector and epsilon vector equals epsilon

vector and epsilon vector equals epsilon
x epsilon y epsilon z for the

x epsilon y epsilon z for the

x epsilon y epsilon z for the
displacement vector then we're trying to

displacement vector then we're trying to

displacement vector then we're trying to
tailor expand f of r vector plus epsilon

tailor expand f of r vector plus epsilon

tailor expand f of r vector plus epsilon
all we need to do to generalize our

all we need to do to generalize our

all we need to do to generalize our
original formula is to replace the

original formula is to replace the

original formula is to replace the
epsilon d by dx in the exponent with the

epsilon d by dx in the exponent with the

epsilon d by dx in the exponent with the
dot product between epsilon vector and

dot product between epsilon vector and

dot product between epsilon vector and
the quote-unquote vector of partial

the quote-unquote vector of partial

the quote-unquote vector of partial
derivatives d by d x d by d y d by d z

derivatives d by d x d by d y d by d z

derivatives d by d x d by d y d by d z
which is usually denoted by this upside

which is usually denoted by this upside

which is usually denoted by this upside
down triangle called del and so this dot

down triangle called del and so this dot

down triangle called del and so this dot
product just means epsilon x d by d x

product just means epsilon x d by d x

product just means epsilon x d by d x
plus epsilon y d by d y plus epsilon z d

plus epsilon y d by d y plus epsilon z d

plus epsilon y d by d y plus epsilon z d
by d z then by combining the exponential

by d z then by combining the exponential

by d z then by combining the exponential
formulas for the taylor series in x y

formulas for the taylor series in x y

formulas for the taylor series in x y
and z we get this beautiful compact

and z we get this beautiful compact

and z we get this beautiful compact
formula for the taylor expansion in

formula for the taylor expansion in

formula for the taylor expansion in
three or any number of variables

three or any number of variables

three or any number of variables
okay that's enough math now let's put it

okay that's enough math now let's put it

okay that's enough math now let's put it
to work with part two the physics i

to work with part two the physics i

to work with part two the physics i
promised to show you three applications

promised to show you three applications

promised to show you three applications
number one how to make the complicated

number one how to make the complicated

number one how to make the complicated
equations that we often need to solve in

equations that we often need to solve in

equations that we often need to solve in
physics simpler by studying special

physics simpler by studying special

physics simpler by studying special
linearized limits number two the

linearized limits number two the

linearized limits number two the
non-relativistic limit of einstein's

non-relativistic limit of einstein's

non-relativistic limit of einstein's
energy formula and how it contributes to

energy formula and how it contributes to

energy formula and how it contributes to
the fine structure of the hydrogen atom

the fine structure of the hydrogen atom

the fine structure of the hydrogen atom
and number three the definition of the

and number three the definition of the

and number three the definition of the
momentum operator in quantum mechanics

momentum operator in quantum mechanics

momentum operator in quantum mechanics
let's go one by one again you don't

let's go one by one again you don't

let's go one by one again you don't
necessarily need to know anything going

necessarily need to know anything going

necessarily need to know anything going
in about relativity or quantum mechanics

in about relativity or quantum mechanics

in about relativity or quantum mechanics
the point is just to get a taste of how

the point is just to get a taste of how

the point is just to get a taste of how
taylor's formula appears in several very

taylor's formula appears in several very

taylor's formula appears in several very
different areas of physics starting with

different areas of physics starting with

different areas of physics starting with
number one making complicated problems

number one making complicated problems

number one making complicated problems
simple the basic procedure to solve a

simple the basic procedure to solve a

simple the basic procedure to solve a
problem in classical mechanics is to

problem in classical mechanics is to

problem in classical mechanics is to
write down all the forces on a particle

write down all the forces on a particle

write down all the forces on a particle
and then add them up and write f equals

and then add them up and write f equals

and then add them up and write f equals
ma and then solve this equation for the

ma and then solve this equation for the

ma and then solve this equation for the
position of the particle as a function

position of the particle as a function

position of the particle as a function
of time that's easier said than done

of time that's easier said than done

of time that's easier said than done
though especially the last step solving

though especially the last step solving

though especially the last step solving
f equals m a because for all but the

f equals m a because for all but the

f equals m a because for all but the
simplest systems this equation quickly

simplest systems this equation quickly

simplest systems this equation quickly
becomes too hard to solve exactly f

becomes too hard to solve exactly f

becomes too hard to solve exactly f
equals m a is a differential equation

equals m a is a differential equation

equals m a is a differential equation
which just means that it contains

which just means that it contains

which just means that it contains
derivatives of the function that you're

derivatives of the function that you're

derivatives of the function that you're
trying to solve for r of t in this case

trying to solve for r of t in this case

trying to solve for r of t in this case
and differential equations are much

and differential equations are much

and differential equations are much
harder to solve than the algebraic

harder to solve than the algebraic

harder to solve than the algebraic
equations that we all first learn about

equations that we all first learn about

equations that we all first learn about
in middle school and high school a

in middle school and high school a

in middle school and high school a
simple example that i've told you about

simple example that i've told you about

simple example that i've told you about
in a few past videos is the pendulum

in a few past videos is the pendulum

in a few past videos is the pendulum
when solving for the motion of a

when solving for the motion of a

when solving for the motion of a
pendulum the main force we're interested

pendulum the main force we're interested

pendulum the main force we're interested
in is the component of gravity that

in is the component of gravity that

in is the component of gravity that
points along the tangent direction to

points along the tangent direction to

points along the tangent direction to
the circle where the particle is

the circle where the particle is

the circle where the particle is
constrained to move that's given by mg

constrained to move that's given by mg

constrained to move that's given by mg
sine of theta where theta is the angle

sine of theta where theta is the angle

sine of theta where theta is the angle
that the pendulum makes with the

that the pendulum makes with the

that the pendulum makes with the
vertical axis like i showed you in the

vertical axis like i showed you in the

vertical axis like i showed you in the
very first video i posted here on the

very first video i posted here on the

very first video i posted here on the
channel then the f equals ma equation

channel then the f equals ma equation

channel then the f equals ma equation
for theta can be written after a little

for theta can be written after a little

for theta can be written after a little
simplifying as the second derivative of

simplifying as the second derivative of

simplifying as the second derivative of
theta with respect to time equals minus

theta with respect to time equals minus

theta with respect to time equals minus
g over l sine of theta simple as this

g over l sine of theta simple as this

g over l sine of theta simple as this
physical setup looks this equation is

physical setup looks this equation is

physical setup looks this equation is
already very complicated because of this

already very complicated because of this

already very complicated because of this
factor of sine of theta it makes it what

factor of sine of theta it makes it what

factor of sine of theta it makes it what
we call a non-linear differential

we call a non-linear differential

we call a non-linear differential
equation which can be very nasty to try

equation which can be very nasty to try

equation which can be very nasty to try
to solve on the other hand when theta is

to solve on the other hand when theta is

to solve on the other hand when theta is
small you can picture a pendulum gently

small you can picture a pendulum gently

small you can picture a pendulum gently
rocking back and forth like a

rocking back and forth like a

rocking back and forth like a
grandfather clock and that motion

grandfather clock and that motion

grandfather clock and that motion
certainly doesn't seem very complicated

certainly doesn't seem very complicated

certainly doesn't seem very complicated
is it possible then that we can simplify

is it possible then that we can simplify

is it possible then that we can simplify
this equation when theta is relatively

this equation when theta is relatively

this equation when theta is relatively
small the taylor series lets us do just

small the taylor series lets us do just

small the taylor series lets us do just
that like we worked out before the

that like we worked out before the

that like we worked out before the
taylor series for sine is theta minus

taylor series for sine is theta minus

taylor series for sine is theta minus
one over three factorial theta cubed

one over three factorial theta cubed

one over three factorial theta cubed
plus one over five factorial theta to

plus one over five factorial theta to

plus one over five factorial theta to
the fifth plus dot dot dot then for tiny

the fifth plus dot dot dot then for tiny

the fifth plus dot dot dot then for tiny
thetas we can apply the small angle

thetas we can apply the small angle

thetas we can apply the small angle
approximation like we saw before then

approximation like we saw before then

approximation like we saw before then
this complicated f equals ma equation

this complicated f equals ma equation

this complicated f equals ma equation
becomes vastly simpler there's no sine

becomes vastly simpler there's no sine

becomes vastly simpler there's no sine
theta factor here anymore making this

theta factor here anymore making this

theta factor here anymore making this
equation complicated and non-linear by

equation complicated and non-linear by

equation complicated and non-linear by
applying the taylor series we've been

applying the taylor series we've been

applying the taylor series we've been
able to linearize the differential

able to linearize the differential

able to linearize the differential
equation to turn it into a problem we

equation to turn it into a problem we

equation to turn it into a problem we
can solve much more easily in the

can solve much more easily in the

can solve much more easily in the
special case when the pendulum isn't too

special case when the pendulum isn't too

special case when the pendulum isn't too
far away from equilibrium this is just

far away from equilibrium this is just

far away from equilibrium this is just
the equation of a simple harmonic

the equation of a simple harmonic

the equation of a simple harmonic
oscillator now like a mass on a spring

oscillator now like a mass on a spring

oscillator now like a mass on a spring
and the general solution is a sum of

and the general solution is a sum of

and the general solution is a sum of
sines and cosines with angular frequency

sines and cosines with angular frequency

sines and cosines with angular frequency
square root g over l so the pendulum

square root g over l so the pendulum

square root g over l so the pendulum
indeed rocks gently back and forth from

indeed rocks gently back and forth from

indeed rocks gently back and forth from
side to side if you've been watching my

side to side if you've been watching my

side to side if you've been watching my
recent videos and all this looks

recent videos and all this looks

recent videos and all this looks
familiar it's no accident i told you a

familiar it's no accident i told you a

familiar it's no accident i told you a
few weeks ago about how the first thing

few weeks ago about how the first thing

few weeks ago about how the first thing
we should do in any physics problem is

we should do in any physics problem is

we should do in any physics problem is
expand the potential energy function

expand the potential energy function

expand the potential energy function
around a stable equilibrium point in a

around a stable equilibrium point in a

around a stable equilibrium point in a
taylor series u of x equals u of zero

taylor series u of x equals u of zero

taylor series u of x equals u of zero
plus u prime of zero times x plus half u

plus u prime of zero times x plus half u

plus u prime of zero times x plus half u
double prime of zero x squared plus dot

double prime of zero x squared plus dot

double prime of zero x squared plus dot
dot where i chose my coordinates here so

dot where i chose my coordinates here so

dot where i chose my coordinates here so
that the equilibrium point is at x

that the equilibrium point is at x

that the equilibrium point is at x
equals zero the first term u of zero is

equals zero the first term u of zero is

equals zero the first term u of zero is
just a constant and that doesn't matter

just a constant and that doesn't matter

just a constant and that doesn't matter
you're always allowed to change what you

you're always allowed to change what you

you're always allowed to change what you
call the ground level of your potential

call the ground level of your potential

call the ground level of your potential
energy function and shift this constant

energy function and shift this constant

energy function and shift this constant
away the second term meanwhile vanishes

away the second term meanwhile vanishes

away the second term meanwhile vanishes
because we've chosen to expand around a

because we've chosen to expand around a

because we've chosen to expand around a
minimum of the potential where u prime

minimum of the potential where u prime

minimum of the potential where u prime
is equal to zero so typically the first

is equal to zero so typically the first

is equal to zero so typically the first
interesting term in the taylor expansion

interesting term in the taylor expansion

interesting term in the taylor expansion
of a potential around equilibrium is the

of a potential around equilibrium is the

of a potential around equilibrium is the
quadratic term which is just like the

quadratic term which is just like the

quadratic term which is just like the
potential energy one half kx squared of

potential energy one half kx squared of

potential energy one half kx squared of
a block on a spring this is why systems

a block on a spring this is why systems

a block on a spring this is why systems
oscillate back and forth around their

oscillate back and forth around their

oscillate back and forth around their
equilibrium position i'll put a link in

equilibrium position i'll put a link in

equilibrium position i'll put a link in
the description to the video where i

the description to the video where i

the description to the video where i
talked all about this as for the force

talked all about this as for the force

talked all about this as for the force
that's related to the potential energy

that's related to the potential energy

that's related to the potential energy
by f equals minus to u by dx and

by f equals minus to u by dx and

by f equals minus to u by dx and
therefore the taylor series for the

therefore the taylor series for the

therefore the taylor series for the
force on a particle near equilibrium

force on a particle near equilibrium

force on a particle near equilibrium
starts with f equals minus u double

starts with f equals minus u double

starts with f equals minus u double
prime of zero times x again just like

prime of zero times x again just like

prime of zero times x again just like
the spring force minus kx

the spring force minus kx

the spring force minus kx
in particular the force is linear so the

in particular the force is linear so the

in particular the force is linear so the
trick i taught you a couple of weeks ago

trick i taught you a couple of weeks ago

trick i taught you a couple of weeks ago
about the simple harmonic motion you

about the simple harmonic motion you

about the simple harmonic motion you
discover when you expand the potential

discover when you expand the potential

discover when you expand the potential
energy around a stable equilibrium is

energy around a stable equilibrium is

energy around a stable equilibrium is
secretly the same thing as linearizing

secretly the same thing as linearizing

secretly the same thing as linearizing
the f equals ma equation

the f equals ma equation

the f equals ma equation
next up let's look at the newtonian

next up let's look at the newtonian

next up let's look at the newtonian
limit of einstein's theory of special

limit of einstein's theory of special

limit of einstein's theory of special
relativity in newtonian mechanics a free

relativity in newtonian mechanics a free

relativity in newtonian mechanics a free
particle has kinetic energy one-half mv

particle has kinetic energy one-half mv

particle has kinetic energy one-half mv
squared alternatively if we plug in the

squared alternatively if we plug in the

squared alternatively if we plug in the
momentum p equals mv we can write the

momentum p equals mv we can write the

momentum p equals mv we can write the
same thing as p squared over 2m

same thing as p squared over 2m

same thing as p squared over 2m
this is the energy of a non-relativistic

this is the energy of a non-relativistic

this is the energy of a non-relativistic
free particle with momentum p

free particle with momentum p

free particle with momentum p
non-relativistic means that the particle

non-relativistic means that the particle

non-relativistic means that the particle
isn't moving very fast compared to the

isn't moving very fast compared to the

isn't moving very fast compared to the
speed of light when particles do

speed of light when particles do

speed of light when particles do
approach the speed of light some weird

approach the speed of light some weird

approach the speed of light some weird
and wild things happen that were

and wild things happen that were

and wild things happen that were
discovered by einstein 100 and some

discovered by einstein 100 and some

discovered by einstein 100 and some
years ago when he wrote down his special

years ago when he wrote down his special

years ago when he wrote down his special
theory of relativity in special

theory of relativity in special

theory of relativity in special
relativity the energy of a free particle

relativity the energy of a free particle

relativity the energy of a free particle
of mass m and momentum p is given by

of mass m and momentum p is given by

of mass m and momentum p is given by
this new formula the square root of m

this new formula the square root of m

this new formula the square root of m
squared c to the four plus p squared c

squared c to the four plus p squared c

squared c to the four plus p squared c
squared where c is the speed of light

squared where c is the speed of light

squared where c is the speed of light
you've seen this before even if you've

you've seen this before even if you've

you've seen this before even if you've
never studied special relativity because

never studied special relativity because

never studied special relativity because
if the particle is at rest so that p

if the particle is at rest so that p

if the particle is at rest so that p
equals zero we get e equals m c squared

equals zero we get e equals m c squared

equals zero we get e equals m c squared
which might be the most famous equation

which might be the most famous equation

which might be the most famous equation
in physics but when the particle is

in physics but when the particle is

in physics but when the particle is
moving we need this more general formula

moving we need this more general formula

moving we need this more general formula
including the contribution from the

including the contribution from the

including the contribution from the
momentum this formula holds even if the

momentum this formula holds even if the

momentum this formula holds even if the
speed of the particle approaches the

speed of the particle approaches the

speed of the particle approaches the
speed of light but on the other hand we

speed of light but on the other hand we

speed of light but on the other hand we
know what the energy is supposed to be

know what the energy is supposed to be

know what the energy is supposed to be
when p is small so how do we see that

when p is small so how do we see that

when p is small so how do we see that
einstein's formula correctly reproduces

einstein's formula correctly reproduces

einstein's formula correctly reproduces
newton's formula for a slow moving

newton's formula for a slow moving

newton's formula for a slow moving
particle the idea is of course to apply

particle the idea is of course to apply

particle the idea is of course to apply
the taylor expansion of einstein's

the taylor expansion of einstein's

the taylor expansion of einstein's
energy when p is small let's first of

energy when p is small let's first of

energy when p is small let's first of
all pull this factor of m squared c to

all pull this factor of m squared c to

all pull this factor of m squared c to
the 4 outside the square root then we

the 4 outside the square root then we

the 4 outside the square root then we
can write the whole thing like this this

can write the whole thing like this this

can write the whole thing like this this
makes it clear that what we want to do

makes it clear that what we want to do

makes it clear that what we want to do
here is compute the taylor series for f

here is compute the taylor series for f

here is compute the taylor series for f
of x equals 1 plus x square root when x

of x equals 1 plus x square root when x

of x equals 1 plus x square root when x
equals p squared over m squared c

equals p squared over m squared c

equals p squared over m squared c
squared is small actually this kind of

squared is small actually this kind of

squared is small actually this kind of
taylor series shows up so often in

taylor series shows up so often in

taylor series shows up so often in
physics that it's worth writing down the

physics that it's worth writing down the

physics that it's worth writing down the
slightly more general case for f of x

slightly more general case for f of x

slightly more general case for f of x
equals one plus x to some power q our

equals one plus x to some power q our

equals one plus x to some power q our
current case with the square root would

current case with the square root would

current case with the square root would
be q equals half i'll let you work out

be q equals half i'll let you work out

be q equals half i'll let you work out
the first few terms of this taylor

the first few terms of this taylor

the first few terms of this taylor
series for yourself for practice i also

series for yourself for practice i also

series for yourself for practice i also
go through the details in the notes i

go through the details in the notes i

go through the details in the notes i
get f of x equals one plus q x plus half

get f of x equals one plus q x plus half

get f of x equals one plus q x plus half
q q minus one times x squared the first

q q minus one times x squared the first

q q minus one times x squared the first
pair of terms here is again a very

pair of terms here is again a very

pair of terms here is again a very
useful approximation that comes up a lot

useful approximation that comes up a lot

useful approximation that comes up a lot
in physics now back to the relativistic

in physics now back to the relativistic

in physics now back to the relativistic
energy we just plug in q equals half and

energy we just plug in q equals half and

energy we just plug in q equals half and
x equals p squared over m squared c

x equals p squared over m squared c

x equals p squared over m squared c
squared then here's what we get and if

squared then here's what we get and if

squared then here's what we get and if
we multiply through by the m c squared

we multiply through by the m c squared

we multiply through by the m c squared
here's where we end up e equals m c

here's where we end up e equals m c

here's where we end up e equals m c
squared plus p squared over two m minus

squared plus p squared over two m minus

squared plus p squared over two m minus
p to the four over eight m cubed c

p to the four over eight m cubed c

p to the four over eight m cubed c
squared plus the higher powers of p the

squared plus the higher powers of p the

squared plus the higher powers of p the
first term is e equals m c squared again

first term is e equals m c squared again

first term is e equals m c squared again
that's what we get by evaluating the

that's what we get by evaluating the

that's what we get by evaluating the
energy of a particle at rest and

energy of a particle at rest and

energy of a particle at rest and
relativity it doesn't have a direct

relativity it doesn't have a direct

relativity it doesn't have a direct
analog in newtonian mechanics but on the

analog in newtonian mechanics but on the

analog in newtonian mechanics but on the
other hand it's just a constant and

other hand it's just a constant and

other hand it's just a constant and
you're always free to add a constant to

you're always free to add a constant to

you're always free to add a constant to
the total energy in newtonian mechanics

the total energy in newtonian mechanics

the total energy in newtonian mechanics
without changing anything as for the

without changing anything as for the

without changing anything as for the
second term there we see how the taylor

second term there we see how the taylor

second term there we see how the taylor
series reproduces precisely the kinetic

series reproduces precisely the kinetic

series reproduces precisely the kinetic
energy that we expect in newtonian

energy that we expect in newtonian

energy that we expect in newtonian
mechanics actually i'm being slightly

mechanics actually i'm being slightly

mechanics actually i'm being slightly
sloppy here because the definition of

sloppy here because the definition of

sloppy here because the definition of
the momentum p actually gets modified in

the momentum p actually gets modified in

the momentum p actually gets modified in
relativity and we should really tailor

relativity and we should really tailor

relativity and we should really tailor
expand that as well but in the

expand that as well but in the

expand that as well but in the
non-relativistic limit we of course get

non-relativistic limit we of course get

non-relativistic limit we of course get
back the newtonian momentum p equals mv

back the newtonian momentum p equals mv

back the newtonian momentum p equals mv
but what about this next term in the

but what about this next term in the

but what about this next term in the
taylor series that goes like p to the

taylor series that goes like p to the

taylor series that goes like p to the
four what are we supposed to make of

four what are we supposed to make of

four what are we supposed to make of
that the point is newtonian mechanics is

that the point is newtonian mechanics is

that the point is newtonian mechanics is
a good description of the world for

a good description of the world for

a good description of the world for
particles that aren't moving anywhere

particles that aren't moving anywhere

particles that aren't moving anywhere
close to the speed of light but it's

close to the speed of light but it's

close to the speed of light but it's
only an approximation this next term in

only an approximation this next term in

only an approximation this next term in
the taylor expansion is the leading

the taylor expansion is the leading

the taylor expansion is the leading
relativistic correction to the newtonian

relativistic correction to the newtonian

relativistic correction to the newtonian
energy when the speed is tiny compared

energy when the speed is tiny compared

energy when the speed is tiny compared
to the speed of light and this

to the speed of light and this

to the speed of light and this
additional term gives a very small

additional term gives a very small

additional term gives a very small
correction to newton's result and we can

correction to newton's result and we can

correction to newton's result and we can
ignore it without losing much accuracy

ignore it without losing much accuracy

ignore it without losing much accuracy
but as the speed gets larger this

but as the speed gets larger this

but as the speed gets larger this
correction becomes increasingly

correction becomes increasingly

correction becomes increasingly
important one place we can see this

important one place we can see this

important one place we can see this
correction in action is in the binding

correction in action is in the binding

correction in action is in the binding
energy of a hydrogen atom that's the

energy of a hydrogen atom that's the

energy of a hydrogen atom that's the
amount of energy you would need to kick

amount of energy you would need to kick

amount of energy you would need to kick
the electron out of its quote-unquote

the electron out of its quote-unquote

the electron out of its quote-unquote
orbit around the proton at the center of

orbit around the proton at the center of

orbit around the proton at the center of
the atom in a video from a couple of

the atom in a video from a couple of

the atom in a video from a couple of
months ago i showed you how we can get

months ago i showed you how we can get

months ago i showed you how we can get
90 percent of the way to the answer for

90 percent of the way to the answer for

90 percent of the way to the answer for
the binding energy just by applying

the binding energy just by applying

the binding energy just by applying
dimensional analysis in other words by

dimensional analysis in other words by

dimensional analysis in other words by
making a list of the parameters we have

making a list of the parameters we have

making a list of the parameters we have
available to play with and their units

available to play with and their units

available to play with and their units
and seeing how we can combine them to

and seeing how we can combine them to

and seeing how we can combine them to
get something with the units that we

get something with the units that we

get something with the units that we
want in this case we saw that we can

want in this case we saw that we can

want in this case we saw that we can
combine the electron mass m in kilograms

combine the electron mass m in kilograms

combine the electron mass m in kilograms
its electric charge e in coulombs

its electric charge e in coulombs

its electric charge e in coulombs
coulomb's constant k which sets the

coulomb's constant k which sets the

coulomb's constant k which sets the
strength of the electric force in

strength of the electric force in

strength of the electric force in
newton's meters squared per coulomb

newton's meters squared per coulomb

newton's meters squared per coulomb
squared and planck's constant h-bar

squared and planck's constant h-bar

squared and planck's constant h-bar
which sets the scale of quantum

which sets the scale of quantum

which sets the scale of quantum
mechanics in kilograms meters squared

mechanics in kilograms meters squared

mechanics in kilograms meters squared
per second to get units of energy like

per second to get units of energy like

per second to get units of energy like
so then the binding energy of the

so then the binding energy of the

so then the binding energy of the
hydrogen atom must be proportional to

hydrogen atom must be proportional to

hydrogen atom must be proportional to
this just by thinking about the units

this just by thinking about the units

this just by thinking about the units
like this gets us almost all the way to

like this gets us almost all the way to

like this gets us almost all the way to
the answer the actual formula for the

the answer the actual formula for the

the answer the actual formula for the
binding energy comes with a factor of

binding energy comes with a factor of

binding energy comes with a factor of
half though which we can't get by only

half though which we can't get by only

half though which we can't get by only
thinking about the units because 2

thinking about the units because 2

thinking about the units because 2
doesn't have any units this is bohr's

doesn't have any units this is bohr's

doesn't have any units this is bohr's
formula for the binding energy of

formula for the binding energy of

formula for the binding energy of
hydrogen and it was one of the first

hydrogen and it was one of the first

hydrogen and it was one of the first
great accomplishments of quantum

great accomplishments of quantum

great accomplishments of quantum
mechanics its numerical value about 13.6

mechanics its numerical value about 13.6

mechanics its numerical value about 13.6
electron volts matches very closely to

electron volts matches very closely to

electron volts matches very closely to
the experimental value of the binding

the experimental value of the binding

the experimental value of the binding
energy and yet bohr's formula is only an

energy and yet bohr's formula is only an

energy and yet bohr's formula is only an
approximation it neglects the small but

approximation it neglects the small but

approximation it neglects the small but
fascinating and experimentally

fascinating and experimentally

fascinating and experimentally
observable effects of special relativity

observable effects of special relativity

observable effects of special relativity
but where do we go wrong in our

but where do we go wrong in our

but where do we go wrong in our
dimensional analysis argument we wrote

dimensional analysis argument we wrote

dimensional analysis argument we wrote
down the only possible way to combine m

down the only possible way to combine m

down the only possible way to combine m
e k and h bar to make units of energy

e k and h bar to make units of energy

e k and h bar to make units of energy
well it's not that we went wrong per se

well it's not that we went wrong per se

well it's not that we went wrong per se
it's that in writing down the

it's that in writing down the

it's that in writing down the
non-relativistic approximation to the

non-relativistic approximation to the

non-relativistic approximation to the
binding energy we omitted the speed of

binding energy we omitted the speed of

binding energy we omitted the speed of
light c from our list of parameters so

light c from our list of parameters so

light c from our list of parameters so
if we want to include the effects of

if we want to include the effects of

if we want to include the effects of
special relativity we need to consider

special relativity we need to consider

special relativity we need to consider
how c can enter the formula for the

how c can enter the formula for the

how c can enter the formula for the
energy but something remarkable happens

energy but something remarkable happens

energy but something remarkable happens
when we add c to the list of parameters

when we add c to the list of parameters

when we add c to the list of parameters
we can form a dimensionless combination

we can form a dimensionless combination

we can form a dimensionless combination
by alpha equals ke squared over h bar c

by alpha equals ke squared over h bar c

by alpha equals ke squared over h bar c
this combination is called the fine

this combination is called the fine

this combination is called the fine
structure constant i'll leave it for you

structure constant i'll leave it for you

structure constant i'll leave it for you
to check that all the dimensions really

to check that all the dimensions really

to check that all the dimensions really
do cancel out here when you plug in the

do cancel out here when you plug in the

do cancel out here when you plug in the
units if you put in the numbers you'll

units if you put in the numbers you'll

units if you put in the numbers you'll
find that alpha is about .0073

find that alpha is about .0073

find that alpha is about .0073
or a little more memorably about one

or a little more memorably about one

or a little more memorably about one
divided by 137

divided by 137

divided by 137
since alpha is unitless dimensional

since alpha is unitless dimensional

since alpha is unitless dimensional
analysis doesn't tell us anything about

analysis doesn't tell us anything about

analysis doesn't tell us anything about
how it appears in the formula for the

how it appears in the formula for the

how it appears in the formula for the
energy no more than it could tell us

energy no more than it could tell us

energy no more than it could tell us
about the factor of 2 in the denominator

about the factor of 2 in the denominator

about the factor of 2 in the denominator
any function of alpha can multiply our

any function of alpha can multiply our

any function of alpha can multiply our
expression for the energy without

expression for the energy without

expression for the energy without
spoiling the units this is how

spoiling the units this is how

spoiling the units this is how
relativity allows small corrections to

relativity allows small corrections to

relativity allows small corrections to
bohr's formula which remember was itself

bohr's formula which remember was itself

bohr's formula which remember was itself
already an excellent approximation to

already an excellent approximation to

already an excellent approximation to
the experimental value of the hydrogen

the experimental value of the hydrogen

the experimental value of the hydrogen
binding energy but we can get an even

binding energy but we can get an even

binding energy but we can get an even
better theoretical prediction by

better theoretical prediction by

better theoretical prediction by
considering the relativistic corrections

considering the relativistic corrections

considering the relativistic corrections
with that leading relativistic

with that leading relativistic

with that leading relativistic
correction that we derived by applying

correction that we derived by applying

correction that we derived by applying
the taylor series to einstein's formula

the taylor series to einstein's formula

the taylor series to einstein's formula
we can determine the small modification

we can determine the small modification

we can determine the small modification
that relativity makes to bohr's formula

that relativity makes to bohr's formula

that relativity makes to bohr's formula
the details require quantum mechanics so

the details require quantum mechanics so

the details require quantum mechanics so
i won't go into that here but the result

i won't go into that here but the result

i won't go into that here but the result
is that this function f is given by 1.

is that this function f is given by 1.

is that this function f is given by 1.
that was for the original bohr answer

that was for the original bohr answer

that was for the original bohr answer
plus 5 4 alpha squared remember that

plus 5 4 alpha squared remember that

plus 5 4 alpha squared remember that
alpha is a tiny number so this

alpha is a tiny number so this

alpha is a tiny number so this
correction that goes like alpha squared

correction that goes like alpha squared

correction that goes like alpha squared
is even tinier still it's therefore

is even tinier still it's therefore

is even tinier still it's therefore
called a fine structure correction to

called a fine structure correction to

called a fine structure correction to
the energy there are in fact further

the energy there are in fact further

the energy there are in fact further
corrections to this formula both at

corrections to this formula both at

corrections to this formula both at
order alpha squared as well as even

order alpha squared as well as even

order alpha squared as well as even
smaller corrections at higher orders in

smaller corrections at higher orders in

smaller corrections at higher orders in
alpha from various physical effects

alpha from various physical effects

alpha from various physical effects
finally while we're on the subject of

finally while we're on the subject of

finally while we're on the subject of
quantum mechanics let's finish by seeing

quantum mechanics let's finish by seeing

quantum mechanics let's finish by seeing
how taylor's formula is related to the

how taylor's formula is related to the

how taylor's formula is related to the
definition of momentum in quantum

definition of momentum in quantum

definition of momentum in quantum
mechanics in classical mechanics the

mechanics in classical mechanics the

mechanics in classical mechanics the
main question is to solve for the

main question is to solve for the

main question is to solve for the
trajectory x of t of a particle as a

trajectory x of t of a particle as a

trajectory x of t of a particle as a
function of time in quantum mechanics on

function of time in quantum mechanics on

function of time in quantum mechanics on
the other hand the goal is to find the

the other hand the goal is to find the

the other hand the goal is to find the
wave function psi of x and how it

wave function psi of x and how it

wave function psi of x and how it
evolves with time wherever the wave

evolves with time wherever the wave

evolves with time wherever the wave
function or rather its square is bigger

function or rather its square is bigger

function or rather its square is bigger
the more likely you are to find the

the more likely you are to find the

the more likely you are to find the
particle at that location when you make

particle at that location when you make

particle at that location when you make
a measurement those things that we

a measurement those things that we

a measurement those things that we
measure about the particle like its

measure about the particle like its

measure about the particle like its
position and momentum are represented by

position and momentum are represented by

position and momentum are represented by
operators that act on the wave function

operators that act on the wave function

operators that act on the wave function
we write x hat for the operator that

we write x hat for the operator that

we write x hat for the operator that
measures the position and p-hat for the

measures the position and p-hat for the

measures the position and p-hat for the
operator that measures the momentum the

operator that measures the momentum the

operator that measures the momentum the
point of this video isn't to learn

point of this video isn't to learn

point of this video isn't to learn
quantum mechanics right now but i gave

quantum mechanics right now but i gave

quantum mechanics right now but i gave
you a bit of a crash course in the video

you a bit of a crash course in the video

you a bit of a crash course in the video
i posted about symmetries in quantum

i posted about symmetries in quantum

i posted about symmetries in quantum
mechanics that i'll link in the

mechanics that i'll link in the

mechanics that i'll link in the
description if you want to see more i

description if you want to see more i

description if you want to see more i
told you there about how the momentum

told you there about how the momentum

told you there about how the momentum
operator is closely related to

operator is closely related to

operator is closely related to
translations in space so let's define an

translations in space so let's define an

translations in space so let's define an
operator call it u of epsilon that

operator call it u of epsilon that

operator call it u of epsilon that
shifts the wave function over by epsilon

shifts the wave function over by epsilon

shifts the wave function over by epsilon
if you haven't seen those earlier videos

if you haven't seen those earlier videos

if you haven't seen those earlier videos
where i explained more about what all

where i explained more about what all

where i explained more about what all
this means don't sweat it right now

this means don't sweat it right now

this means don't sweat it right now
we'll just take this as a definition and

we'll just take this as a definition and

we'll just take this as a definition and
discover how it's related to taylor's

discover how it's related to taylor's

discover how it's related to taylor's
formula and indeed this looks familiar

formula and indeed this looks familiar

formula and indeed this looks familiar
physics aside psi of x is just a

physics aside psi of x is just a

physics aside psi of x is just a
function and this formula tells us that

function and this formula tells us that

function and this formula tells us that
we're looking for an operator that

we're looking for an operator that

we're looking for an operator that
shifts psi of x over to psi of x minus

shifts psi of x over to psi of x minus

shifts psi of x over to psi of x minus
epsilon and that's exactly what taylor's

epsilon and that's exactly what taylor's

epsilon and that's exactly what taylor's
formula does

formula does

formula does
therefore we identify the translation

therefore we identify the translation

therefore we identify the translation
operator u with e to the minus epsilon d

operator u with e to the minus epsilon d

operator u with e to the minus epsilon d
by dx for reasons we won't delve into

by dx for reasons we won't delve into

by dx for reasons we won't delve into
right now this translation operator u is

right now this translation operator u is

right now this translation operator u is
related to the momentum operator by u of

related to the momentum operator by u of

related to the momentum operator by u of
epsilon equals e to the minus i over h

epsilon equals e to the minus i over h

epsilon equals e to the minus i over h
bar epsilon times p and comparing the

bar epsilon times p and comparing the

bar epsilon times p and comparing the
two sides taylor's formula shows us that

two sides taylor's formula shows us that

two sides taylor's formula shows us that
we should identify the momentum operator

we should identify the momentum operator

we should identify the momentum operator
in quantum mechanics with p equals h bar

in quantum mechanics with p equals h bar

in quantum mechanics with p equals h bar
over i d by dx when you do start

over i d by dx when you do start

over i d by dx when you do start
studying quantum mechanics this will be

studying quantum mechanics this will be

studying quantum mechanics this will be
one of the first formulas you'll learn

one of the first formulas you'll learn

one of the first formulas you'll learn
it follows directly from taylor's

it follows directly from taylor's

it follows directly from taylor's
formula

formula

formula
this has been just a small selection of

this has been just a small selection of

this has been just a small selection of
physics applications where taylor's

physics applications where taylor's

physics applications where taylor's
formula shows up but again you'd really

formula shows up but again you'd really

formula shows up but again you'd really
be hard-pressed to find any chapter of

be hard-pressed to find any chapter of

be hard-pressed to find any chapter of
any physics textbook where it isn't

any physics textbook where it isn't

any physics textbook where it isn't
applied keep your eyes open and you'll

applied keep your eyes open and you'll

applied keep your eyes open and you'll
see taylor's formula everywhere you can

see taylor's formula everywhere you can

see taylor's formula everywhere you can
find the notes for this video as well as

find the notes for this video as well as

find the notes for this video as well as
links to all the earlier videos that i

links to all the earlier videos that i

links to all the earlier videos that i
mentioned down in the description if you

mentioned down in the description if you

mentioned down in the description if you
like the video and you want to help

like the video and you want to help

like the video and you want to help
support the channel i'll also put a link

support the channel i'll also put a link

support the channel i'll also put a link
to my patreon page thank you so much for

to my patreon page thank you so much for

to my patreon page thank you so much for
watching and i'll see you back here soon

watching and i'll see you back here soon

watching and i'll see you back here soon
for another physics lesson