Understanding Jacobian Matrices - Working Through Examples

[Shamshiel]

I get the idea of Jacobian matrices. I think. Working through different examples, I don't have any problems.

For example,

f1 = x^2 + y^2
f2 = 3x + 4y

would result in

[2x 2y]
[3 4]

Similarly, by my understanding, something like

x^2 + y^2
3y + 4x

would result in

[2x 2y]
[4 3]

But when going up to systems like, say

f1 = x + x^2 + x^3 + y

...I'm baffled. I don't see how that could work. I'm under the impression that the Jacobian matrix should have two columns, but I'd have four there. I've had great difficulty finding any examples or explanations online, and I suspect my problem is I don't really understand them to begin with. And I really want to understand the math behind this, because otherwise I'm not really getting it.

Thanks!

 

[HallsofIvy]

What do you mean by "systems" like that? A single formula is not really a system is it?

In any case, the Jacobean of a function from R^m (m variables) to Rⁿ (n values) is an m by n matrix- there are m columns, one for each variable, and n rows, one for each value. Yes, your first two examples are from R² to R² and so are 2 by 2 matrices as you show.

Your third example is from R² (two variables, x and y) to R¹ (one value, f1) and so would be a "2 by 1" matrix, with two rows and one column:
$$\begin{bmatrix}1+2x+ 3x^2 \\ 1\end{bmatrix}$$

 

[Bacle]

To generalize: for each function f_i (x1,...,xn) that you have, you will have n
partial derivatives; one partial with respect to each variable.

This matrix describes the linear map that approximates the change of the function
in the neighborhood of a point, just like f'(x) does, in a map f:R-->R.

 

[lavinia]

I suggest you understand Jacobians as coordinate representations of the differentials of functions.

 

[Shamshiel]

Thanks guys! That helps a lot. :) Sorry for my lack of clarity with that third example, though.