Deriving Gradient in Spherical Coordinates (For Physics Majors)

Disclaimer
I skipped over some of the more tedious algebra parts. I'm assuming that since you're watching a multivariable calculus video that the algebra isn't the thing you need help with.

[@gauravdeepsingh4478]

There's an error of having a + sign instead of - , when we collect the phi terms in the end. But other than that, you are an amazing teacher

[@pauladriaanse]

ah i was confused why mine was different. thanks

[@WayneL-gh6hd]

And should the phi hat term of x hat be -sin(phi) times phi hat and that of y be cos(phi) times phi hat?

[@dbarcene]

11:50 You have become the very thing you swore to destroy

[@Frostbitecgi]

when you said " i leave that as an exercise " that moment was price less bro :V ... loved it .. it made my day.

[@followwhereyourhearttakesy9315]

Thank you for the derivation. It is a kind of proof that one does not dare to go through all the process but wants to know how it can be shown. I've been looking for this for a few months now. Really appreciate it as a mathematiclal major student in Korea.

[@WayneL-gh6hd]

Using inner product to convert between these two unit vectors is very very brilliant! Thanks for this genius deriving. It helps me so much!

[@meghnaroy7782]

Did he use inner pdt to convert polar to cartesian? Can you please tell how can it be done?

[@volodymyr3169]

Hahaha, you just helped an engineer * Uses numerical solution and drives away on expensive sport scar

[@jdragon8184]

i cant believe he stoped such educational content we engineers who have a crush on physics need such people help

[@4NDr3W_f]

My professor never did this deriving and I couldn't do it myself. Thank you.

[@Flaming_Hammer]

Explanation was excellent. I used a slightly different method though. I directly calculated d/dx, d/dy and d/dz instead of calculating d/dr, d/dϕ and d/dθ. It had far lesser calculations. Excellent video though. Than you so much.

[@CsGOismyreligion]

how?

[@mayankbharadwaj3643]

@Ujwal Reddy can you please help me with that.

[@dawitkone7070]

it's very nice.also very clear to see what you have written on the board.keep it up

[@PearlPrime]

Ikr.

[@Dusk-MTG]

Well that's a way of doing it, but probably the one requiring most calculations. If you use the metric and use the definition grad(f)=#(df) you get the same solution in less than a blackboard of calculations.

[@georgem2430]

This really helped me see what's actually going on! Did you get around to deriving the Laplacian in spherical coordinates? I can't find a video for it on your channel.

[@dansehnal1680]

Thanks!

[@SoumyajitGuin-je2hd]

I really find this solution everywhere. Thank you, sir..

[@fancsaliboglarka2017]

If anyone struggleing, there is an error: in d/dx , when expressed, the d/d(phi) member should be with a negative sign (it appears around 4:03)

[@AndrewDotsonvideos]

Sorry for the mistake! Thank you for time stamping it

[@Jordan-s1l]

Just wanted to point out that you can find the unit vectors in Cartesian coordinates by taking the transpose of the transformation matrix since they are orthogonal. It’s a little easier that way

[@user-dp9yn7zf4l]

Crazy þo think that now I'm actually get to learn from this channel, back in higschool, I watched for the meme, now I'm actually learning and physics major

[@paulboard8221]

1:30 bruh it's just dance dance revolution it's not that deep

[@MaxwellsWitch]

0:48 You mean the correct convention? *OFFENDED MATH MAJORS INTENSIFIES*

[@diogorodrigues5527]

How is it the right one when you use theta if you're on a plane? Theta should always be the one ranging from 0 to 2pi. I'm in physics, but I dont understand why we use such unnatural convention.

[@sherllymentalism4756]

Physics be like: Everything converges uniformly Phi is x to vector r Delta function is a function

[@dbss206]

@@diogorodrigues5527 in elrctro magnetism text book by Sadiku, r, theta n phi convention is there. Radius, colattitude and azimuthal angle. Why is it like that? Gotta Google it

[@bilalhussein9730]

Let's start the real fight: (+,-,-,-) or (-,+,+,+)?

[@howardlam6181]

The funny part is on wolfram mathworld spherical coord page, they use math convention but on the spherical harmonic and associated legendre differential eq. pages, they use the physics convention.

[@unknown360ful]

THANK YOU SO MUCH! I NEEDED THIS! I was recently looking over Schrodinger in 3D and I was confused about the gradient part!

[@AndrewDotsonvideos]

unknown360ful are you referring to the laplacian part of the schrodinger equation? That corresponds to the divergence of the gradient. I’ll get to that!

[@unknown360ful]

YES! The separation into Radial and Angular Equations and then the use of Legendre Polynomials...

[@diamonddust8840]

From 2:50pm to 3:42pm, i was trying to find an expression for the gradient of a scalar point function in spherical coordinate system, but answer was not coming as everytime i coughs up. Your video saved me from further headache. Thanks 😊

[@aravindsuresh6202]

You my friend deserve a gold medal

[@pauladriaanse]

4:24 When i try it, i get a minus before the sin(phi)/rsin(theta) part of d/dx . . . Is his sign a + or a -?

[@clintdbennett]

It should be a minus.

[@akshathadevilolla5415]

Thanks. I thought it was a mistake on my part

[@k1rv0lak]

I'm taking an engineering course on simulating physical processes and this video was incredibly useful for deriving the gradient of temperature in spherical coordinates for one of my problems. I used this analytical derivation to support the results I got when I ran the simulation on COMSOL. Definitely hairy math, but cool stuff nonetheless - thank you so much!

[@pixb3695]

Thanks for these lectures I've found them very helpful so far. The only comment I'd make is when you do skip over stuff if could just stand back from the board for a moment pls so people can see what you've just written. It makes it harder to follow the flow if you're standing right in front of the work you've just done. Great stuff cheers

[@alejandroduque772]

How does the chain rule work with those operators at 1:37 , I understand the idea but how can we formalize that, any source where i can check that out?

[@satyampriyadarshi27]

It felt like mathematical error with 1=3 . I want to know that chain rule too.

[@fromis____9]

thanks for nice explanation, but I can't understand 7:00. The inner product of x hat and phi hat equals '-sin phi', but in terms of x hat, why is the coefficient of phi hat '-sin theta'?

[@frede1905]

If you imagine setting up a Cartesian coordinate system (x, y, z) with the origin at the location where you wanna calculate the gradient of the function, and then let your x, y and z axes point in the direction of the local unit vectors of the radial coordinate, polar angle and azimuthal angle, then the expression for the gradient in terms of spherical coordinates can be found very straightforwardly.

[@richarichi]

is there a video about this? I still can't figure it out

[@frede1905]

@@richarichi No, it's just a faster, more intuitive approach. You are free to choose your Cartesian coordinate system from which you transform into a spherical coordinate system to derive the gradient. He picked a Cartesian coordinate system with its origin at the center of the spherical coordinate system. I noted that if you want to, say, find the gradient at a point A, you can create a Cartesian coordinate system with its origin at that point and with the various orthogonal x, y, z unit vectors pointing in the direction of the various orthogonal r, theta, phi unit vectors. Then the conversion of the gradient at that point from the Cartesian to the spherical system will be fairly trivial. Alternatively, simply note that the spherical coordinate system is orthogonal, and so all you need are various factors in front of the d/dr, d/d(theta) and d/d(phi) terms to properly normalize them.

[@hikmatullahpakhtoon3694]

Our teacher has given us an assignment to proof gradient, divergence and curl in spherical and cylindrical coordinate. For last 6 hours i haven't solve the gradient in spherical coordinate

[@AnkitYadav-zg5zd]

At 4:09 there is a slight mistake in the d/dx expression.

[@minahiljaveed3723]

Why we use x unit vector,y unit vector,z unit vector instead of r unit vector,theta unit vector and phi unit vector?

[@cyancoyote7366]

Awesome video! Could you do the Laplacian next? :)

[@thamalijayawickrnlinama3007]

This is a very nice explanation. If you could say first what is gradient which you says in 9.29min initially in the video and then show how to find the terms make more sense to me. Just an idea. Thanks a lot for the video.

[@dopship6257]

It's insane that in Mexico undergraduate computer science students have to demonstrate this as part of the course

[@김민재-x7h3i]

Thanks a lot !! I think there is some mistakes when ^x dot ^phi, ^y dot ^phi 5:10~8:30

[@borfenglin6150]

yeah regarding the last term of xhat and yhat. but most importantly thanks andrew for leading us with the thought process :)

[@debendragurung3033]

good one bro, please keep them coming... @2:30 It looks like Transpose of Jacobian from cartesian to spherical, expressed in spherical cordinates.

[@chrismaudsley127]

Could you go straight to partial d by partial x etc? The differentiations are a little more awkward but there is less algebra I think. Thanks for the gear lectures.

[@ikrishna06]

You are a great genius brother 😍

[@mahimasri1271]

Sir ur way of teaching is marvellous.

[@iamback2415]

For those that have trouble solving the algebra for d/dx, d/dy, and d/dz you can use the chain rule for d/dx to get a value of d/dx= (d/dr)*(dr/dx)+ (d/d(theta))*(d(theta)/dx)+(d/d(phi))*(d(phi))/dx). Then solve dr/dx, d(theta)/dx, and d(phi)/dx and plug in. Solve d/dy and d/dz similarly.

[@Chrisk92-z1h]

Hi Andrew, love this video! But isn't there a better way described in Appendix A of Griffiths Electrodynamics? Would love to see ur take in that one !

[@markkennedy9767]

Nice video. Any easy way of explaining how you get the rho hat, phi hat and theta hat unit vectors. I can visually do the rho hat and phi hat unit vectors in my head but not the theta hat unit vector. Although it should just be orthogonal to the others. But in which direction? Is it a right handed set up for these unit vectors. If so, it would just be the cross product of the other two I suppose? Thanks

[@abuabdullah9878]

0:47 isn't phi the angle between the x-axis and the *projection* of r onto the x-y plane (and not simply the r vector)?

[@akshathadevilolla5415]

Yes.

[@RahulKumar-gc7lc]

Hello dear A lot of thanks for your hard work. Please replace + with - from third term of d/dx at 4:09 and 10:26 ...😊

[@akivas2034]

At 2:00 why is this step obvious?

[@waqaruddin5243]

sir which book you are following?. This is a best lecture

[@archimedes7182]

I think we can derive it in three to four lines, geometrically.

[@alendominic8320]

I got the third term of d/dx as a negative.Is that so

[@46pi26]

I have a mathematician friend, and he always uses rho, phi, and theta for r, theta, and phi, respectively. At least he doesn't use vartheta lol

[@Dusk-MTG]

Definitely not the best way. Just using the metric tensor gives the result in two steps.

[@anonye8996]

Not everyone can do that :”( (second sem level mathematical physics)

[@lihaozheng5291]

Hey man, thx for the great video, but for 5:10, I think the sign of d/dphi is wrong for d/dx

[@aneeqaahmad5034]

Thank you very much Andrew. Can you tell what this practically means ? Or what can be said for spherical coordinates after this in simple words? Thank you for the answer.

[@yimoawanardo]

I don't get why δ/δρ = δx/δρ * δ/δx + δy/δρ * δ/δy + δz/δρ * δ/δz. Can someone explain to me?

[@4001Jester]

it’s the multivariable chain rule

[@satyampriyadarshi27]

I have the same doubt. It felt like mathematical error(1=3).🤯

[@sollinw]

At 1:13, gradient of x to theta is wrong

[@白毛鼹鼠]

helps a lot for my se course，thanks❤

[@vincentkhang5264]

We need more derivation videos!!

[@AlchemistOfNirnroot]

"This channel has no videos"

[@146fallon]

love that inner product to express x,y,z hat. such a genius.

[@bilalhussein9730]

Just use the covariant divergence. Way fewer calculations. You probably do that in one of your tensor videos.

[@rock801]

andrew skipped over the last part completely and this is where I lost him

[@chandakumari-wy2nq]

how did you find out the unit vector of phi and theta i could not find a way to derive it at least you can give a clue how to calculate that please please respond

[@AndrewDotsonvideos]

chanda kumari e_phi is just dr/dphi divided by the madnitude. The same process goes for the theta unit vector

[@chandakumari-wy2nq]

thanks Andrew.... keep it up and wish you good luck

[@KevinS47]

Ok Andrew, I used about 3 hours trying to derive this, but the more deep I get into the algebra the more complicated it gets; this to the point that I get 1 page long equations and it is extremely hard to keep track of everything. Is there a trick to use? a specific way to do it perhaps so that it doesn't get way too messy? I have written 8 pages of algebra (basically) on my notebook, and I did every step cousciously in order not to make stupid sign mistakes or similiar ones... at this point I am giving up on deriving this. If you people have any tips, I am open to suggestions!

[@saudmahgoub8702]

@KevinS47 I know my reply is 4 years late but i hope it may help other students who may see it in the future. Anyways, operators in vector calculus are generally multilinear maps which are tensors. In other words, their form depends on the coordinate system you are using. So in Tensor calculus (which precedes vector calculus) all these operators are generalized to arbitrary coordinates in one formula using the metric tensor. But learning everything in tensor calculus just to derive the gradient is an overkill especially if you are a beginner learning multivariable calc. To avoid the nasty algebra one must realize that these partial derivatives that you get from using the chain rule to write the r,ø,theta basis in terms of the x,y,z are just the inverse of the partial derivatives factors that you use to write the x,y,z basis in terms of the r,ø,theta basis. Inverse in the sense that if you put these factors in a linear map and then invert the linear map you get the other factors. That's the main insight from linear algebra. So to find the linear map take the 3 components of each of your three spherical bases and put them as column vectors inside a linear map to form a 3 × 3 matrix. This matrix here is nothing but the infamous jacobian matrix back from multivariable calc. Now we have reduced the nasty algebra to a linear algebra problem of finding the inverse of 3 × 3 matrix. This is much simpler and is done using inv(J)=1/det(J) × C where C is the matrix of cofactors of the Jacobian J. If this is your first time dealing with matrix inversion it's likely going to be a pretty awkward computational problem but once you practice it a couple of times you will have a feel for it. After the inversion process what you are left with is the inverse of the jacobian where the column vectors are the x,y,z bases and the components of each column vector as are the good old partial derivatives that you were originally looking for. I know the process may look pretty intensive but that's a result of treating vectors the classical way as arrows in space, rather than tensors. Once you develop your mathematical skills and grind all the way to tensor calculus you will be able to derive the gradient, divergence, and the scalar and vector laplacians operators for arbitrary coordinates in a one liner.

[@rustysaw3288]

How do you do the tedious algebra? I don't arrive to the expected answer.

[@StealMySpotlight]

Please please please derive the divergence and curl in spherical curvilinear coordinates as well... I am going to cry, very new to these coordinates and my professor took one lecture to finish off entire thing. I am desperately searching for the logics and visualisation of the curl, gradient, divergence ever since. All i know is high school level calc and semester exams never wait for anyone!! 😢😢😢😢😢😭😭😭

[@PHILLYMEDIC69]

ah ok i see... i asked my physics prof for a formula once.. and he was like "why dont you just derive" and told me to fuck off basically. But then im watching this video and it makes sense.

[@anchakaghatge4347]

this video was helpful indeed! thank you!!

[@عبداللهالبطريق]

why do you have to derive it?

[@estevaoalvaro8231]

Someone can explain to me how I can turn "del/del r" in "del/del x" or "del/del y" in "del/ del θ", e.g how I can found "(cosφcosθ/r)del/delθ"? I talk about the part in as Andrew skip the video. Thanks !

[@tomkerruish2982]

2:00 You should use "Yakety Sax" aka "Benny Hill Music" for sections like this.

[@satyampriyadarshi27]

I was looking for the derivation of divergence in sp. co-ordinates😞😞. Should I open youtube channel for viewers physics doubts only?

[@parinita1031]

What knowledge you have its awesome .you just explain each and every thing in detail .thank you

[@abelpalmer552]

What's that formula at the top of the board? Does it not erase?

[@4001Jester]

so where is the laplacian in spherical coordinates hmm?

[@himaniparmar120]

Nice explanation great buddy...👍 tx

[@blackmane1999]

Great vid Andrew

[@shreyaspradhan8546]

I legit thought this was a meme video at the beginning.

[@princessvalles2281]

Thank you! I love you

[@markkennedy9767]

Now what does the gradient in spherical coordinates mean intuitively?!!

[@uuuuuuuulim]

This was great! Thanks!

[@chrisallen9509]

How much is calc 3 used in later physics classes? My friend in modern physics said he's barely touched it which I found surprising.

[@thenumberquelve158]

I'm doing upper division Classical Physics and Modern Physics BOTH this semester, currently approaching the end of Week 4 with a mid-term coming up in Classical. So far, Calc 3 has a lot more to do with Classical than it does in Modern. There's lots of derivations between force and energy that use gradients, and some graphs that require spherical coordinates & such. So far in Modern physics, the deepest dive into calculus we've done so far is partial derivatives & Euler's Formula for wave equations. But I'm told that when we get to Schrodinger's Equation, it's all about manipulating Imaginary Numbers, so brace yourself for that.

[@Topspeedcraft]

For example before quantum mechanics you have to pass complex Variable course and differential equations,which will be used a lot in optics and QM. So not too much calc 3 meaning no triple integrals but a fair bit of Diff eqs, Fourier, Leibniz, And complex number integrals and transforms

[@ES-qe1nh]

Use inverse metric tensor

[@hx3319]

I have a little problem I would love if someone could help me figure out. Show that 11,111,1111,.... (in binary) are not an integer to the power of any whole number greather than or equal to 2. We can write the binary numbers as 2^n-1, and it's easy to see that an even number to the power of anything never gets odd. An odd number to the power of an even number is always congruent to 1 mod4 and the binary are alway congruent to 3 mod4, hence they are not odd^even. But I have a problem solving odd^odd as they sometimes are congruent to 3 mod 4. Can somebody help? I tried to prove it by seeing if 2^n-1 is never equal to x^p. Expanding 2^n by the binomial theorem don't get me anywhere.. Great vids btw!! Lovem

[@dineshkumarpradhan9427]

Nice lecture sir

[@shivanipanwar00]

It's like Laplace's equestion in terms of spherical coordinates 🤟

[@MayankGoel447]

This video me a bigger picture of the idea!

[@aarshmehtani4712]

Thanks sir luv u❤️

[@deepaksivala8509]

Very informative

[@johnnyc8669]

I just memorize it by dividing each term by its jacobian in an integral

[@Kano1729]

you are lagend ⭐⭐⭐⭐⭐

[@StuartJuggernaut]

this is so goated

[@user-xt7cs6sh5i]

thanks man my lazy ass was not gonna try to do this lol.

[@ahmedkarrar25]

IDK how to get d/dx guys, any help? EDIT: I found it by myself. 😃 It's every x element from d/dr, d/dø, and d/d theta. So you have to get dr/dx instead of dx/dr and so on, dr/dy instead of dy/dr...etc

[@emergency.jergens]

I have such a big crush on you it makes math fun

[@Dmongreeneyes]

It's been a few years since I've done calc 3 and I attempted to do this in my own... got stuck at the algebra part 😔.

[@AndrewDotsonvideos]

The algebra is disgusting..

[@Dmongreeneyes]

Andrew Dotson I also realized that this isn't calc3 but rather a differential equations problem... I never took diffeq

[@AlchemistOfNirnroot]

Is it? It's definitely in my Vector Analysis book (Shaum's Outlines). I'm only comfortable with ODE's.

[@bonbonpony]

@@AndrewDotsonvideos If you're doing it backwards, then it most certainly is :q Because then it involves a lot of trigonometry which unnecessarily obfuscates it.

[@alejrandom6592]

Ok but why didn't you do it the simple way

[@odasakunosuke5137]

ah I don't know anymore if I'm going to have a future in taking physics 😂

[@දියසෙන්කුමාර]

if write something on board then stay little bit away from board. please

[@viniciusf.linhares1416]

it was very usefell ,since when i take the galactic dynamics book, just starting asking, why the hell the gradient is this shit?

[@hiccup3.14]

666 Total satanic.... Ok I know my exam is on Monday But I need to get these videos.... Where can I find the curl and Divergence? Has he derived them as well?

[@mrgenius781]

Please do it according to Griffths Electrodynamics

[@Ab-ub2ii]

griffith didn't prove it just have written down the formulas, please specify what did you mean by 'according to the book'

[@patriciamcgeorge2575]

2:17 when you write so fast ∂ becomes ə