Jacobian Matrix and Determinant - Examples

Examples

Example 1. The transformation from spherical coordinates (r, θ, φ) to Cartesian coordinates (x1, x2, x3), is given by the function F : R+ × × [0,2π) → R3 with components:

The Jacobian matrix for this coordinate change is

J_F(r,\theta,\phi) =\begin{bmatrix} \dfrac{\partial x_1}{\partial r} & \dfrac{\partial x_1}{\partial \theta} & \dfrac{\partial x_1}{\partial \phi} \\ \dfrac{\partial x_2}{\partial r} & \dfrac{\partial x_2}{\partial \theta} & \dfrac{\partial x_2}{\partial \phi} \\ \dfrac{\partial x_3}{\partial r} & \dfrac{\partial x_3}{\partial \theta} & \dfrac{\partial x_3}{\partial \phi} \\ \end{bmatrix}=\begin{bmatrix} \sin\theta\, \cos\phi & r\, \cos\theta\, \cos\phi & -r\, \sin\theta\, \sin\phi \\ \sin\theta\, \sin\phi & r\, \cos\theta\, \sin\phi & r\, \sin\theta\, \cos\phi \\ \cos\theta & -r\, \sin\theta & 0 \end{bmatrix}.

The determinant is r2 sin θ. As an example, since dV = dx1 dx2 dx3 this determinant implies that the differential volume element dV = r2 sin θ dr dθ dϕ. Nevertheless this determinant varies with coordinates. To avoid any variation the new coordinates can be defined as Now the determinant equals 1 and volume element becomes .

Example 2. The Jacobian matrix of the function F : R3 → R4 with components

is

J_F(x_1,x_2,x_3) =\begin{bmatrix} \dfrac{\partial y_1}{\partial x_1} & \dfrac{\partial y_1}{\partial x_2} & \dfrac{\partial y_1}{\partial x_3} \\ \dfrac{\partial y_2}{\partial x_1} & \dfrac{\partial y_2}{\partial x_2} & \dfrac{\partial y_2}{\partial x_3} \\ \dfrac{\partial y_3}{\partial x_1} & \dfrac{\partial y_3}{\partial x_2} & \dfrac{\partial y_3}{\partial x_3} \\ \dfrac{\partial y_4}{\partial x_1} & \dfrac{\partial y_4}{\partial x_2} & \dfrac{\partial y_4}{\partial x_3} \\ \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 5 \\ 0 & 8x_2 & -2 \\ x_3\cos(x_1) & 0 & \sin(x_1) \end{bmatrix}.

This example shows that the Jacobian need not be a square matrix.

Example 3.

The Jacobian determinant is equal to . This shows how an integral in the Cartesian coordinate system is transformed into an integral in the polar coordinate system:

Example 4. The Jacobian determinant of the function F : R3 → R3 with components

\begin{align} y_1 &= 5x_2 \\ y_2 &= 4x_1^2 - 2 \sin (x_2x_3) \\ y_3 &= x_2 x_3 \end{align}

is

\begin{vmatrix} 0 & 5 & 0 \\ 8 x_1 & -2 x_3 \cos(x_2 x_3) & -2x_2\cos(x_2 x_3) \\ 0 & x_3 & x_2 \end{vmatrix} = -8 x_1 \cdot \begin{vmatrix} 5 & 0 \\ x_3 & x_2 \end{vmatrix} = -40 x_1 x_2.

From this we see that F reverses orientation near those points where x1 and x2 have the same sign; the function is locally invertible everywhere except near points where x1 = 0 or x2 = 0. Intuitively, if you start with a tiny object around the point (1,1,1) and apply F to that object, you will get an object set with approximately 40 times the volume of the original one.

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