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### Course: Multivariable calculus>Unit 4

Lesson 10: Surface integral preliminaries

# Partial derivatives of vector-valued functions

Partial Derivatives of Vector-Valued Functions. Created by Sal Khan.

## Want to join the conversation?

• Is there a difference between ∂s and ds ? If dy=dx *2, is it legal to say (∂r/∂s)*ds =∂r ?
• Edited:
- ∂s is specially reserved for and specifically refers to partial derivatives while ds is used to indicate a regular derivative being taken!

- Since ds and ∂s are two separate things, it would not be legal to do what you mentioned in the second question!

Note: I know this question was asked two years ago! I am simply posting the answers for any who may read your post and also wonder the same things!
. . . I also want to try and get one of those question badges!
• I hear Sal & Grant talk about how "Mathematicians cringe" when we treat differentials like variables/tiny-changes-in-a-direction.

While I've heard some people say that, I've also heard many others say that it is perfectly fine to do.

Also all of these differential math seems to beautifully work out to me, so why would mathematicians not like that? Is there some rigorous reason some people think that mathematicians cringe?

Who are the "official" mathematicians out there who get to overrule others, and why does everyone think that they are cringing so often (lol)?

• Good question... I'll preface by saying I've never entered this debate myself. As someone who knows a decent amount of math though, I'd say that there's no reason not to treat the differentials as tiny changes in a direction as long as we keep clear what we're talking about. Our notation df/dx implies a certain limit, while \delta f / \delta x implies a small step. In some ways it's abuse of notation to say "Consider df/dx as a small step in the x direction," but really all that matters is that your audience is on the same page as you. There are many different contexts in which you would want the exact derivative (think actual calculus/physics), others where a finite difference works great (think numerical approximation). As long as you personally have the two concepts straight in your head, and you make sure your audience is following, then you're all set. Just my take on the issue :)
• Can u make the vidoes smaller and not so long?/
• Thankfully Sal has recently said that he plans to make videos shorter (4-8 minutes long).

lol I feel ya! I just watched a 30 minute long video (!). I was watching it while I ate lunch at least, and at double speed which makes it more manageable. xD
• On the left side of each of the two boxed equations, what is the difference between the dt (or ds) and the dt with the curly d (and ds with the curly s) ? Thanks
• curly d indicates that it is a partial derivative, i.e. more than 1 dependent variable.
The normal d is for regular derivatives.
• So what is the real difference between dx and delta x?
• any videos about derivative of algebraic functions? :(
• A little before , where Sal represents the limits in terms of partial differentials, wouldn't there be a partial r w.r.t partial x time i (r/y for j and r/z for k) or is that redundant? Because in my understanding, r is a function of x,y and z which are then functions of s and t.

Thank you.
• It is the same. You are right in saying that r is a function of x, y and z, and they are functions of s and t. Therefore, you can replace x, y and z with s and t and get r as a function of s and t. Hope this helps.
• I feel like ∂s doesn’t give us much information about which variable is being varied to cause ∂s. Is there some alternate notation that we can use to represent the differential of s while also telling us with respect to which variable we are talking about?
• I think you may be confused. When calculating partial derivatives of a function f(x, y) then we would say something like this;
∂f/∂x = derivative of f(x, y) with respect to x
∂f/∂y = derivative of f(x, y) with respect to y

So the above denotation does tell us what variable we are taking the partial derivative of.

Hope this helps,
- Convenient Colleague