Derivation of function with multiple variables

Question

I started studying Functions with multiple variables , and i am little confused about few stuff concerning how does things work in n-dimensions

let s say that i do have a function $ f : {R^n} \mapsto {R^m}$

for m=n=1 , i know how to derivation of this function works and when $\frac{df}{dx} >0$ (or <0) i know how does the variation of the function work

but for $m \geq n \geq 1$, i know that i have to present the Jacobi-matrix and so on , but i can not imagine how would i be able to analyse the variation of this function , because i do not know if the there is a notation that might define how to find out if a matrix postive or negative is .

so can you please give me some related references to read , or maybe someone can explain to me how does things work in n-dimensional function.

Answer 1

In order to understand how to interpret the Jacobi matrix, you must first change how you see the derivative. In $\Bbb R\to \Bbb R$, I assume that you're used to thinking about it as "How steep is the curve?".

That's not the interpretation that the Jacobi matrix offers. The Jacobi matrix uses the interpretation "If I take some small step in a given direction, what happens to the function value?"

That's what the Jacobi matrix is. Take some point $x\in \Bbb R^m$, and calculate the Jacobi matrix $J_x$ of your function at that point. Despite the big formula, it will just become a nice and simple matrix with numbers in it. A small step is given by some $m$-dimensional (column) vector $y$, and we now have that $f(x+y)$ (the function value after taking that small step) is approximately equal to $f(x)+J_x y$, where the latter term uses the regular matrix product (and the Jacobi matrix is the matrix that gives the best such approximation). So the Jacobi matrix records how the function changes from point to point in the immediate neighbourhood of $x$.

There is, of course, a strict way of quantifying this using limits and such, but in my opinion and my experience as a teacher, both on this site and face to face with students, they are nearly meaningless without the geometric intuition to back them up.