Coordinate Transformations and Curvilinear Coordinates | Tensor Calculus

[@glassfish7207]

that's the best english writing I've ever seen in my life, thanks btw

[@eulefranz944]

Ahhh was just about to go to bed, this was the missing piece... to wake me up again! Now I have unlimited power again!!! (Until Quantum mechanics forbids it....)

[@HarperChisari]

My multivariate calc class never touched on any of this stuff at all. Thank you very much for making these videos.

[@stoneflower8751]

about to take the multivar calc exam, we also didnt go into any detail. we were just handed the formula for the jacobian and then we used it in examples, "the american way" my prof calls it lol. good thing we have sources other than the official textbook- videos like this help keep math fun.

[@KSM94K]

I'm still learning this though they will not teach this in class, don't know when I need it, unpredictable

[@mohibulhoque2145]

I was literally wondering whether understanding Tensor is in my capability or not. I am pretty sure that by the end of this playlist answer will be Yes. Thanks, those notes are of great help.

[@catchaser3433]

Had no idea what I just watched for 12 minutes, but your hand writing is neat and it sounded well explained :)

[@FacultyofKhan]

Haha thanks! If you want a better idea, I'd recommend watching my Tensor playlist (see description) to get a better understanding!

[@ishansheikh3058]

What u saw is the basics of a painting that went through Einstein's mind when he drew the remarkable piece of art called Relativity.

[@PunmasterSTP]

These videos provide wonderful explanations, and if I can understand them at least partially, I think I may be...ahead of the curve!

[@xephyr417]

I love your handwriting

[@jowlenejones]

Wow!!!! I am so glad I came across this video :)) Thanks!

[@markkennedy9767]

At 10:30 do you mean the determinant of the Jacobian is non-zero (as opposed to the Jacobian itself is non-zero)

[@rodrigoappendino]

2:24 Is it necessary to use the Kronecker delta? If you write just x super I twice, there would be repeated indices already.

[@FacultyofKhan]

Now that I look back, probably not necessary lol as long as I write delta x^i * delta x^i in the square root. The silver lining is that the notation helps solidify the meaning of the Kronecker delta from previous videos, I guess?

[@philp4684]

Near the end of the earlier video in this series entitled "Einstein Notation: Proofs, Examples, and Kronecker Delta", δ_ij x_i y_j is simplified to x_i y_i, so why, in this video at 2:05, is the distance formula given as sqrt(δ_ij x^i y^j), rather than the simpler form sqrt(x^i y^i)?

[@FacultyofKhan]

This video is a bit different from what I had in the previous video you mention. Here, we've got δ_ij Delta x^i Delta x^j, where Delta x^i = x^i - y^i. It's not the same as δ_ij x^i y^j as you mention. You can simplify the expression I have in this video to something like δ_ij Delta x^i Delta x^j = (Delta x^i)(Delta x^i) = Sum (x^i-y^i)^2, which basically gives you the distance formula at 2:05. Hope that clarifies things!

[@philp4684]

Thanks for the reply. I think what happened there was that I intended to go back and fill in the capital deltas after looking up the right key combination to produce them. I must've forgotten to do that before hitting the send button! But my question was really relating to the seemingly unnecessary use of a Kronecker delta. Many years back, I tried reading a book on General Relativity, but it quickly got too hard for me to understand! But I did remember about the Einstein summation convention. I vaguely remember that book saying that the convention only applies if one index is a superscript and the other is a subscript. Your earlier videos in this series, though, contradicted that, so I assumed I had remembered incorrectly. Then, when I watched this video, I saw that you used the Kronecker delta in the distance formula, and wondered if there was something to that vague memory after all. That's why I asked about it. I've now watched your later videos about how the superscripts & subscripts relate to contravariance & covariance, so I'm sure it'll all become clearer as the series continues.

[@FacultyofKhan]

Yea the δ_ij bit is just some convention my textbook ended up using; you could have equivalently used sqrt(delta x^i * delta x^i) without changing anything as you said. Just goes to show that a lot of books on Tensors are hot garbage haha (seriously, self-studying this stuff required some serious work lol).

[@TheBigWazowski]

I think in the past you’ve said that you make these videos, in part, as motivation to learn the material. So I was wondering what your process is for learning the material then organizing it into a video. Are you reading through a textbook then sort of rewriting it in your own way, etc?

[@FacultyofKhan]

Yea pretty much. I go through a textbook that I think is good for the subject and explain the areas which I feel are lacking explanation in the text. I usually also supplement my teaching with other texts and videos. For instance, I'm using Schaum's outline for my tensor series as the main text, but then I'm also using my Mathematical Physics texts (e.g. the one by Boas) to supplement what I'm teaching and provide better explanation.

[@iqdx]

Your videos are excellent and very helpful. In that context at 7:30 it is less clear than usual your reference to first and fourth quadrants when there are 12 quadrants in 3D space (4 in each of 3 planes). What did I miss?

[@FacultyofKhan]

Ahh yes my apologies, I meant the quadrants in the x1/x2 plane, since the arctan expression is based on the coordinates x^1 and x^2. Hope that helps!

[@ThePhysicist]

Why is it necessary for a coordinate transformation to have a continuous second partial derivative?

[@markkennedy9767]

Why is the existence of continuous second partials required for transformation of coordinates. And how do you get the inverse functions generally (apart from knowing in the polar and cylindrical cases what they would be already).

[@anupyadav1502]

What software are you using? How is your handwriting so good 🧐

[@davidmwakima3027]

Your writing is super clear! The examples help build a lot of intuition about what linear functionals, which I have heard in my introduction to tensors class, are. What textbook or notes are you using?

[@lethukuthulathelibra]

What should be the response to a question that asks which points does the transformation fail

[@password6975]

Fantastic video thank you

[@physiqueirm1586]

The third POINT 3:07 does it make sense ? I don't get the equality that he wrote ? someone help ?

[@sagarmodak989]

Actually it is transformation equation. When we go from one co-ordinate system to another, we can write them as a function of each other. Focus on the examples he gave later, you will understand it.

[@sanjaythorat9914]

It seems there is a small mistake or probably I too didn't understand it quite well. I checked Wiki article: en.wikipedia.org/wiki/Curvilinear_coordinates and it seems to be helping.

[@cameronspalding9792]

@2:27 The formula written for the length of the vector in Einstein notation is problematic because out of context it’s not obvious if you sum the terms then square root or if you just square root then sum

[@nkintc]

Hello! Could you cover Buckingham Pi therm and how this relates? When looking into the math, it seems like it's a coordinate transformation using SI units but I don't understand. Could you maybe enlighten us?

[@francisavenir4039]

Hello, can someone explain why arctan is only valid in first and fourth quadrant? shouldn't it be first and third since it is defined in that region

[@FacultyofKhan]

If we restrict the output of arctan to the interval from (-pi/2, pi,2), that should correspond to angles in the 1st and 4th quadrants. Hope that helps!

[@kartikvarshney9257]

Hello faculty of Khan, can you provide practice problems based on your teachings or you could suggest any book

[@bingusiswatching6335]

Wait so the jacobian is all the first derivatives and the hessian is the second derivatives?

[@bingusiswatching6335]

Jacobian matrix

[@sagarmodak989]

Why any transformation needs to satisfy those conditions to be a co-ordinate transformation? Is there any valid reason for that or is it just an arbitrary idea?

[@FacultyofKhan]

I believe it's the definition of a coordinate transformation.

[@sagarmodak989]

Okay. Thanks for replying.

[@naturematters08]

@@sagarmodak989 Hello

[@sagarmodak989]

@@naturematters08 hello

[@naturematters08]

@@sagarmodak989 Actually i said hi to you because my sir name is also Modak

[@MohdSameer-rx9gj]

11 minutes vedio took me about 40 minutes to complete...can't say how much would it have taken you to make it...!!!

[@danmccarron0]

THis is all badass. You do a really good job. Calling you out though - you model it heavily after the Schaum's outline by David Kay, don't you? Not that this is a bad thing... :)

[@FacultyofKhan]

Aww you got me! Kidding aside, yes, I use Schaum's outline as my source material, with additional explanations added.

[@laykefindley6604]

Ayeee.....fieeeeeeeeen!!! My soul hurts hearing it pronounced this way. XD

[@mrslave41]

3:36 "invertible" was never defined.

[@calvinkhor890]

Do people say 'Ay-feen' or 'af-fine'?

[@jacobvandijk6525]

I vote for the last option (but I'm Dutch ;-))

[@ZohanSyahFatomi]

Dude, can you make series please. Thank you :)

[@dalisabe62]

@ 2:26, you zoomed by too fast at the point where the serious stuff begins. delta x i expressed in tensor notations. I expected more justification and definition of what delta x i is and how it is equal to the right side.

[@jacobrose4805]

I went to like this video and hesitated when I saw the number of likes was at 69. I liked it anyway.

[@maxpercer7119]

I'm a little confused. In the start of the video, how can you talk about vectors and their components without a basis, and thus you imposed a coordinate system? so this seems circular.