Need help with derivative notation

[orion]

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.

 

[fresh_42]

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)

 

[orion]

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.

 

[Mark44]

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.

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.

 

[orion]

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.