Question

In Exercises \(1-8,\) find the Jacobian \(\partial(x, y) / \partial(u, v)\) for the indicated change of variables. \(x=u \cos \theta-v \sin \theta, y=u \sin \theta+v \cos \theta\)

Short answer

The Jacobian \(\partial(x, y) / \partial(u, v)\) is equal to 1.

Step-by-step solution

Compute derivatives

First, we need to calculate the four derivatives: \[ \frac{\partial x}{\partial u} = \cos\theta, \frac{\partial x}{\partial v} = -\sin\theta, \frac{\partial y}{\partial u} = \sin\theta, \frac{\partial y}{\partial v} = \cos\theta.\]

Compute Jacobian determinant

The Jacobian determinant is: \[ J(x,y|u,v) = \det\left(\begin{array}{cc}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{array} \right)\] Now compute the determinant by multiplying the elements on the leading diagonal, and then subtracting the product of the elements on the other diagonal. Hence, the determinant is \(\cos^2 \theta + \sin^2 \theta.\)

Use Pythagorean Identity

From the Pythagorean Identity in trigonometry, we have \(\sin^2 \theta + \cos^2 \theta = 1.\) Thus, the Jacobian is equal to 1.