Intuitive proof of multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory?

Question

What is an intuitive proof of the multivariable changing of variables formula (Jacobian) without using mapping and/or measure theory?

I think that textbooks overcomplicate the proof.

If possible, use linear algebra and calculus to solve it, since that would be the simplest for me to understand.

Answer 1

Let there be some vector function $f(x) = x'$, which can be interpreted as remapping points or changing coordinates. For example, $f(x) = \sqrt{x \cdot x} e_1 + \arctan \frac{x^2}{x^1} e_2$ remaps the cartesian coordinates $x^1, x^2$ to polar coordinates on the basis vectors $e_1, e_2$.

Now, let $c(\tau)$ be a path parameterized by the scalar parameter $\tau$. Let $f(c) = c'(\tau)$ be the image of this path under the transformation. The chain rule tells us that

$$\frac{dc'}{d\tau} = \Big(\frac{dc}{d\tau} \cdot \nabla \Big) f$$

Define $a \cdot \nabla f \equiv \underline f(a)$ as the Jacobian operator acting on a vector $a$, and the equation can be rewritten as

$$\frac{dc}{d\tau} = \underline f^{-1} \Big(\frac{dc'}{d\tau} \Big)$$

(Note that the primes have switched, so we use the inverse Jacobian.)

This is all we need to show that a line integral in the original coordinates is related to a line integral in the new coordinates by using the Jacobian. For some scalar field $\phi$, if $\phi(x) = \phi'(x')$, then

$$\int_c \phi \, d\ell = \int_{c'} \phi' \, \underline f^{-1}(d\ell')$$

because $d\ell'$ can be converted to $\frac{d\ell'}{d\tau} \, d\tau$.

Edit: didn't see the word intuitive. As far as intuitive explanations go, you can think of a coordinate transformation like so. Imagine the lines of a polar coordinate system being warped and stretched so that they become rectangular instead. This makes working with them easier, but because the shapes of coordinate lines, paths, and areas have changed (and because you don't want them to change the result, since changing coordinates should not change the result), the naive errors introduced must be corrected for with a factor of the Jacobian operator.

Answer 2

A lengthy proof of the change of variables formula for Riemann integrals in $\mathbb R^n$ (that does not use measure theory) is given in Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard. A discussion of the intuition behind it is given on page 493.

Answer 3

The answers here are good but I am tempted to add a part which I think is quite important and something which others haven't talked about from what I see: Namely, why are we allowed to use, in the linearized limit, parallelograms (and hence Jacobian determinants) to approximate areas in the first place.

In fact, whenever you have a general coordinate transformation $(u,v) \to (x(u,v),y(u,v))$ of the plane, you find that you are forced to sum over quadrilaterals instead of parallelograms in general. One can try this by partitioning the uv plane into discrete values (i.e, $(u_i,v_j)$ where $i,j$ run from, say, $1$ to $n$) and seeing the corresponding images $(x(u_i,v_j),y(u_i,v_j))$, connecting these images together by straight lines forces you to sum over quadrilaterals instead of parallelograms (you can try this for yourselves for polar coordinates explicitly!), however one can show that the area of a quadrilateral differs from that of a parallelogram by second order, and hence won't matter in the limit when the partition goes to zero. This leads us directly to the Jacobian determinant and the exterior algebra of differential forms.