visualize map $\mathbb{R^n} \to \mathbb{R^n}$ on "orthogonal grid"

Question

I found this image in a chapter on the chain rule for multivariable functions. I have never heard of orthogonal grids and want to know what they are referring to here.

Thanks!

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Answer 1

What it's referring to is to picture what your mapping $f$ does to lines parallel to the $x$-axis and the $y$-axis. In other words, what is the image of lines $x=a$ and $y=b$ under the function $f$?

For example, in your second example, it is easy to see that a line given by $x=a$ is send to a circle of radius $a$. And a line given by $y=b$ is send to a line with the slope given by $\mathrm{atan2}(\sin(b),\cos(b))$ (Appropriate care is needed here. Read about the function $\mathrm{atan2}$ here).

For example, if $x=a$, then for $f(x,y)=(x\cos(y),x\sin(y))$ we get

$$f(a,y)=(a\cos(y),a\sin(y))$$

But these set of points satisfy the equation $x^2+y^2=a^2$. Hence, $f$ sends lines parallel to the $x$-axis to circles of radius $a$. Can you now do the other ones on your own?