16.1 The Jacobian and its properties

Jacobian Matrix and Determinant

The Jacobian captures how a multivariable function distorts space locally. When you change variables in a multiple integral, the Jacobian tells you exactly how infinitesimal areas or volumes scale under the transformation. Without it, you can't correctly convert $dx\,dy$ into $dr\,d\theta$ (or any other coordinate system).

This section covers the Jacobian matrix, its determinant, and the key properties that make change-of-variables work. We'll also connect it to coordinate transformations and the inverse function theorem.

Definition and Components of the Jacobian Matrix

For a function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$, the Jacobian matrix $J_\mathbf{f}(\mathbf{x})$ is the $m \times n$ matrix of all first-order partial derivatives. Entry $(i,j)$ is $\frac{\partial f_i}{\partial x_j}$, the partial derivative of the $i$-th output component with respect to the $j$-th input variable.

For a transformation $T(u,v) = (x(u,v),\, y(u,v))$ in $\mathbb{R}^2$, the Jacobian matrix looks like:

$J_T = \begin{pmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \$6pt] \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{pmatrix}$

The Jacobian determinant (often just called "the Jacobian") is $\det(J_T)$:

$\frac{\partial x}{\partial u}\frac{\partial y}{\partial v} - \frac{\partial x}{\partial v}\frac{\partial y}{\partial u}$

Two critical facts about this determinant:

• A non-zero determinant at a point means the transformation is locally invertible there.
• The absolute value of the determinant gives the local scaling factor: how much an infinitesimal area (or volume, in higher dimensions) stretches or shrinks under the transformation.

Concrete example. For polar coordinates, $x = r\cos\theta$, $y = r\sin\theta$:

$J = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix}$

$\det(J) = r\cos^2\theta + r\sin^2\theta = r$

That's where the familiar factor of $r$ in $r\,dr\,d\theta$ comes from.

Properties and Interpretation of the Jacobian

The Jacobian matrix is the best linear approximation of a differentiable function near a given point. Think of it as the multivariable generalization of the derivative. It encodes all the local behavior: stretching, rotation, reflection, and shearing.

• If the Jacobian matrix exists at a point, the function is differentiable there. Differentiability is stronger than mere continuity; it guarantees the function behaves smoothly enough for linear approximation to work.
• The rank of the Jacobian matrix determines local injectivity and surjectivity. For a square Jacobian ($n = m$), full rank is equivalent to a non-zero determinant, which means the function is locally one-to-one.
• A positive determinant means the transformation preserves orientation (e.g., it doesn't flip a region into its mirror image). A negative determinant means orientation is reversed. For integration, you take the absolute value either way.

Coordinate Transformations

Concept and Purpose

A coordinate transformation maps points from one coordinate system to another while preserving the underlying geometry. You use them to exploit symmetry or simplify the region of integration.

Common transformations you'll work with:

• Cartesian to polar: $(x,y) \to (r,\theta)$, useful for circular regions
• Cartesian to cylindrical: $(x,y,z) \to (r,\theta,z)$, useful for problems with axial symmetry
• Cartesian to spherical: $(x,y,z) \to (\rho,\phi,\theta)$, useful for spherical regions
• Custom substitutions: any transformation $T(u,v) = (x,y)$ designed to simplify a particular integral

The general change-of-variables formula in two dimensions is:

$\iint_R f(x,y)\,dx\,dy = \iint_S f(x(u,v),\, y(u,v))\, \left|\det(J_T)\right|\,du\,dv$

where $S$ is the region in the $(u,v)$-plane that maps onto $R$ under $T$.

Applying the Chain Rule in Coordinate Transformations

The chain rule connects partial derivatives across coordinate systems. If you have a function $g$ expressed in $(x,y)$ and you switch to $(u,v)$, then:

$\frac{\partial g}{\partial u} = \frac{\partial g}{\partial x}\frac{\partial x}{\partial u} + \frac{\partial g}{\partial y}\frac{\partial y}{\partial u}$

In matrix form, this is exactly multiplication by the Jacobian matrix:

$\begin{pmatrix} \frac{\partial g}{\partial u} \$4pt] \frac{\partial g}{\partial v} \end{pmatrix} = J_T^{\,\top} \begin{pmatrix} \frac{\partial g}{\partial x} \$4pt] \frac{\partial g}{\partial y} \end{pmatrix}$

The Jacobian matrix is the bridge that ensures derivatives transform correctly between coordinate systems.

Properties of Coordinate Transformations

For a change of variables to work properly in an integral, the transformation should be bijective (one-to-one and onto) on the interior of the region, except possibly on a set of measure zero (like boundary curves).

• A bijective transformation has a unique inverse: you can map back from $(x,y)$ to $(u,v)$ without ambiguity.
• Bijective transformations preserve topological properties like connectedness and compactness, so the shape of your integration region stays well-behaved.
• Minor failures of bijectivity on the boundary (e.g., polar coordinates mapping $\theta = 0$ and $\theta = 2\pi$ to the same ray) don't affect the integral because those boundary sets have zero area.

Inverse Function Theorem

Statement and Implications

In plain terms: if the Jacobian determinant isn't zero at a point, the function is locally invertible near that point, and the inverse is smooth. This is the theoretical guarantee that your coordinate transformation actually works in a neighborhood of that point.

The theorem gives a sufficient condition, not a necessary one. A function could still be locally invertible even if the Jacobian determinant is zero, but the theorem won't help you prove it.

Connection to Change of Variables

The inverse function theorem is what justifies the change-of-variables formula. When you write $\iint_R f\,dx\,dy = \iint_S f \circ T\, |\det(J_T)|\,du\,dv$, you need $T$ to be locally invertible so that the mapping between regions is well-defined. The non-vanishing Jacobian determinant guarantees this.

The Jacobian of the Inverse

If $\mathbf{f}$ is locally invertible, the Jacobian of the inverse function satisfies:

$J_{\mathbf{f}^{-1}}(\mathbf{f}(\mathbf{a})) = \left[J_\mathbf{f}(\mathbf{a})\right]^{-1}$

This follows directly from the chain rule applied to $\mathbf{f}^{-1} \circ \mathbf{f} = \text{id}$. A useful consequence for determinants:

$\det\left(J_{\mathbf{f}^{-1}}\right) = \frac{1}{\det(J_\mathbf{f})}$

So if you know the Jacobian going one direction (say, from $(r,\theta)$ to $(x,y)$), you can immediately find the Jacobian going the other direction by taking the reciprocal of the determinant. For polar coordinates, the Jacobian of the inverse transformation $(x,y) \to (r,\theta)$ has determinant $\frac{1}{r}$.