Jacobian Determinant

Overview

The Jacobian determinant is a concept from vector calculus that helps in transforming between coordinate systems. It represents the scaling factor of the transformation at a given point and is especially useful when changing variables in multivariable integrals.

Definition of the Jacobian Determinant

The Jacobian determinant is defined for a transformation $\mathbf{F}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ where $\mathbf{F}$ maps coordinates $(x_1,x_2,\dots ,x_n)$ to $(u_1,u_2,\dots ,u_n)$. It’s represented by the determinant of the Jacobian matrix $J$, which is given by:

$$J=\left|\begin{array}{cccc}\frac{\partial u_1}{\partial x_1}& \frac{\partial u_1}{\partial x_2}& \dots & \frac{\partial u_1}{\partial x_n}\\ \frac{\partial u_2}{\partial x_1}& \frac{\partial u_2}{\partial x_2}& \dots & \frac{\partial u_2}{\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial u_n}{\partial x_1}& \frac{\partial u_n}{\partial x_2}& \dots & \frac{\partial u_n}{\partial x_n}\end{array}\right|$$

Key Points

Interpretation

The Jacobian determinant measures the rate at which area, volume, or hypervolume changes under the transformation $\mathbf{F}$. If $|\det (J)|>1$, the transformation stretches space; if $|\det (J)|<1$, it compresses space. When $\det (J)=0$, the transformation is singular at that point, meaning it compresses the space completely along some dimension.

Applications

Change of Variables in Integrals

For a transformation from coordinates $(x,y)$ to $(u,v)$, the area element $dxdy$ is transformed to $|\det (J)|dudv$. Thus, when changing variables in a double integral, we use the Jacobian determinant as a correction factor:

$${\iint }_{R}f(x,y)dxdy={\iint }_{S}f(u,v)\left|\det (J)\right|dudv$$

Example

Consider the transformation $u=x^2-y^2$ and $v=2xy$. To compute the Jacobian determinant $J$ for this transformation, first find the partial derivatives:

$$\frac{\partial u}{\partial x}=2x,\frac{\partial u}{\partial y}=-2y$$ $$\frac{\partial v}{\partial x}=2y,\frac{\partial v}{\partial y}=2x$$

The Jacobian matrix is:

$$J=\left(\begin{array}{cc}2x& -2y\\ 2y& 2x\end{array}\right)$$

The determinant of $J$ is:

$$\det (J)=(2x)(2x)-(-2y)(2y)=4x^2+4y^2=4(x^2+y^2)$$

Thus, $\left|\det (J)\right|=4(x^2+y^2)$.

Properties of the Jacobian Determinant

Key Properties

• Positive or Negative Values: The sign of the determinant indicates orientation.
• Zero Determinant: If $\det (J)=0$ at a point, the transformation is locally non-invertible there.