Problem 20 - Calculus Early Transcendentals

Question

Compute the Jacobian \(J(u, v)\) for the following transformations. $$T: x=u \cos \pi v, y=u \sin \pi v$$

Step-by-Step Solution

Answer

Answer: The Jacobian \(J(u, v)\) for the given transformation is \(u\pi\).

Step 1: Write down the given transformation functions

We are given the following transformation functions: $$x(u,v) = u \cos(\pi v)$$ $$y(u,v) = u \sin(\pi v)$$

Step 2: Calculate the partial derivatives with respect to u and v

Next, we need to calculate the partial derivatives \(\frac{\partial x}{\partial u}\), \(\frac{\partial x}{\partial v}\), \(\frac{\partial y}{\partial u}\), and \(\frac{\partial y}{\partial v}\). $$\frac{\partial x}{\partial u} = \frac{\partial}{\partial u} (u \cos(\pi v)) = \cos(\pi v)$$ $$\frac{\partial x}{\partial v} = \frac{\partial}{\partial v} (u \cos(\pi v)) = -u \pi \sin(\pi v)$$ $$\frac{\partial y}{\partial u} = \frac{\partial}{\partial u} (u \sin(\pi v)) = \sin(\pi v)$$ $$\frac{\partial y}{\partial v} = \frac{\partial}{\partial v} (u \sin(\pi v)) = u \pi \cos(\pi v)$$

Step 3: Compute the Jacobian matrix and its determinant

Now that we have the partial derivatives, we can form the Jacobian matrix by plugging in the computed values: $$J(u, v) = \begin{bmatrix} \cos(\pi v) & -u\pi\sin(\pi v) \\ \sin(\pi v) & u\pi\cos(\pi v) \end{bmatrix}$$ Finally, we compute the determinant of this 2x2 matrix: $$\det(J(u, v)) = (\cos(\pi v))(u\pi\cos(\pi v)) - (-u\pi\sin(\pi v))(\sin(\pi v))$$ Simplifying the expression, we get: $$\det(J(u, v)) = u\pi\cos^2(\pi v) + u\pi\sin^2(\pi v)$$ We can factor out \(u\pi\) from the expression: $$\det(J(u, v)) = u\pi(\cos^2(\pi v) + \sin^2(\pi v))$$

Step 4: Simplify the determinant expression

Since \(\cos^2(\pi v) + \sin^2(\pi v) = 1\), we can simplify the determinant as: $$\det(J(u, v)) = u\pi$$ Thus, the Jacobian \(J(u, v)\) for the given transformation is: $$J(u, v) = u\pi.$$