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Chapter 13: Problem 20

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

Short Answer

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

Step by step solution

01

Write down the given transformation functions

We are given the following transformation functions: $$x(u,v) = u \cos(\pi v)$$ $$y(u,v) = u \sin(\pi v)$$
02

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 \(\frac{\partial y}{\partial v}\). $$\frac{\partial x}{\partial u} = \frac{\partial}{\partial u} (u \cos(\pi v)) = \cos(\pi v)$$ $$\frac{\partial x}{\partial v} = \frac{\partial}{\partial v} (u \cos(\pi v)) = -u \pi \sin(\pi v)$$ $$\frac{\partial y}{\partial u} = \frac{\partial}{\partial u} (u \sin(\pi v)) = \sin(\pi v)$$ $$\frac{\partial y}{\partial v} = \frac{\partial}{\partial v} (u \sin(\pi v)) = u \pi \cos(\pi v)$$
03

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: $$\det(J(u, v)) = u\pi(\cos^2(\pi v) + \sin^2(\pi v))$$
04

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: $$J(u, v) = u\pi.$$

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Most popular questions from this chapter

Linear transformations Consider the linear transformation \(T\) in \(\mathbb{R}^{2}\) given by \(x=a u+b v, y=c u+d v,\) where \(a, b, c,\) and \(d\) are real numbers, with \(a d \neq b c\) a. Find the Jacobian of \(T\) b. Let \(S\) be the square in the \(u v\) -plane with vertices (0,0) \((1,0),(0,1),\) and \((1,1),\) and let \(R=T(S) .\) Show that \(\operatorname{area}(R)=|J(u, v)|\) c. Let \(\ell\) be the line segment joining the points \(P\) and \(Q\) in the uv- plane. Show that \(T(\ell)\) (the image of \(\ell\) under \(T\) ) is the line segment joining \(T(P)\) and \(T(Q)\) in the \(x y\) -plane. (Hint: Use vectors.) d. Show that if \(S\) is a parallelogram in the \(u v\) -plane and \(R=T(S),\) then \(\operatorname{area}(R)=|J(u, v)| \operatorname{area}(S) .\) (Hint: Without loss of generality, assume the vertices of \(S\) are \((0,0),(A, 0)\) \((B, C),\) and \((A+B, C),\) where \(A, B,\) and \(C\) are positive, and use vectors.)

Changing order of integration If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. The solid above the cone \(z=r\) and below the sphere \(\rho=2,\) for \(z \geq 0,\) in the orders \(d z d r d \theta, d r d z d \theta,\) and \(d \theta d z d r\)

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