Need help with derivative notation

  • #1
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If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##.

Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial f}{\partial x^n})##.

But for a proof I need a compact notation like ##\frac{df}{dx}## for this multivariable case. Does such a compact notation exist? I mean, a notation without making explicit reference to components.

Thanks in advance.
 

Answers and Replies

  • #2
13,800
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If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##.

Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial f}{\partial x^n})##.

But for a proof I need a compact notation like ##\frac{df}{dx}## for this multivariable case. Does such a compact notation exist? I mean, a notation without making explicit reference to components.

Thanks in advance.
How about the ##∇f## operator?
(https://en.wikipedia.org/wiki/Gradient)
 
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Likes orion
  • #3
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Thanks, fresh 42. I'm sorry I'm late in responding, but I forgot I wrote this question. It turns out that after I wrote this, I realized a mistake I was making in the proof and you are right, the gradient works. Thanks again.
 
  • #4
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If I have a scalar function of a variable ##x## I can write the derivative as: ##f'(x)=\frac{df}{dx}##.

Now suppose ##x## is no longer a single variable but a vector: ## x=(x^1, x^2, ..., x^n)##. Then of course we have for the derivative ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial f}{\partial x^n})##.
This -- ## x=(x^1, x^2, ..., x^n)## -- should probably be written as ## x=(x_1, x_2, ..., x_n)## to avoid confusion. Although I have seen a few textbooks that use superscripts as indexes, most use superscripts to denote exponents rather than indexes.

Also, this -- ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial f}{\partial x^n})## -- should be written as ##(\frac{\partial f}{\partial x_1}, ..., \frac{\partial f}{\partial x_n})## for the same reason.
orion said:
But for a proof I need a compact notation like ##\frac{df}{dx}## for this multivariable case. Does such a compact notation exist? I mean, a notation without making explicit reference to components.

Thanks in advance.
 
  • #5
93
2
This -- ## x=(x^1, x^2, ..., x^n)## -- should probably be written as ## x=(x_1, x_2, ..., x_n)## to avoid confusion. Although I have seen a few textbooks that use superscripts as indexes, most use superscripts to denote exponents rather than indexes.

Also, this -- ##(\frac{\partial f}{\partial x^1}, ..., \frac{\partial f}{\partial x^n})## -- should be written as ##(\frac{\partial f}{\partial x_1}, ..., \frac{\partial f}{\partial x_n})## for the same reason.
No, it has to be written the way I wrote it. Otherwise, the Einstein summation convention does not work and also there is a need to distinguish contravariant components from covariant components.

I realize that I posted in a calculus forum but that was because I wanted input on a derivative notation from vector calculus. It's actually a proof in differential geometry. But in the end I found out that my notational problem was pointing a way to an error in my proof.
 

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