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Chapter 13: Problem 37

Evaluate the Jacobians \(J(u, v, w)\) for the following transformations. $$x=v+w, y=u+w, z=u+v$$

Short Answer

Expert verified
Question: Determine the Jacobian for the given transformation: x = v + w, y = u + w, z = u + v. Answer: The Jacobian for the given transformation is J(u,v,w) = 2.

Step by step solution

01

Determine the partial derivatives

First, we'll write down the transformation equations and find the partial derivatives of x, y, and z with respect to u, v, and w. Given transformation equations: $$x = v + w$$ $$y = u + w$$ $$z = u + v$$ Now find the partial derivatives: $$\frac{\partial x}{\partial u} = 0, \frac{\partial x}{\partial v} = 1, \frac{\partial x}{\partial w} = 1$$ $$\frac{\partial y}{\partial u} = 1, \frac{\partial y}{\partial v} = 0, \frac{\partial y}{\partial w} = 1$$ $$\frac{\partial z}{\partial u} = 1, \frac{\partial z}{\partial v} = 1, \frac{\partial z}{\partial w} = 0$$
02

Form the matrix of partial derivatives

Next, we form the matrix of partial derivatives using the values we found in step 1: $$ \begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} & \frac{\partial x}{\partial w} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} & \frac{\partial y}{\partial w} \\ \frac{\partial z}{\partial u} & \frac{\partial z}{\partial v} & \frac{\partial z}{\partial w} \end{bmatrix} = \begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix} $$
03

Evaluate the determinant of the matrix

Now, we'll evaluate the determinant of the matrix, which represents the Jacobian J(u,v,w): $$ J(u, v, w) = \begin{vmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{vmatrix} $$ Expanding along the first row, we get: $$J(u, v, w) = 0(0 - 1) - 1(1(0) - 1(1)) + 1(1 - 1(1))$$ $$J(u,v,w) = - (-1) - 1(-1) + 1(0)$$ $$J(u, v, w) = 1 + 1 = 2$$ Thus, the Jacobian for the given transformation is J(u,v,w) = 2.

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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.)

A thin (one-dimensional) wire of constant density is bent into the shape of a semicircle of radius \(a\). Find the location of its center of mass. (Hint: Treat the wire as a thin halfannulus with width \(\Delta a,\) and then let \(\Delta a \rightarrow 0\).)

Intersecting spheres One sphere is centered at the origin and has a radius of \(R\). Another sphere is centered at \((0,0, r)\) and has a radius of \(r,\) where \(r>R / 2 .\) What is the volume of the region common to the two spheres?

An important integral in statistics associated with the normal distribution is \(I=\int_{-\infty}^{\infty} e^{-x^{2}} d x .\) It is evaluated in the following steps. a. Assume that $$\begin{aligned} I^{2} &=\left(\int_{-\infty}^{\infty} e^{-x^{2}} d x\right)\left(\int_{-\infty}^{\infty} e^{-y^{2}} d y\right) \\ &=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^{2}-y^{2}} d x d y \end{aligned}$$ where we have chosen the variables of integration to be \(x\) and \(y\) and then written the product as an iterated integral. Evaluate this integral in polar coordinates and show that \(I=\sqrt{\pi} .\) Why is the solution \(I=-\sqrt{\pi}\) rejected? b. Evaluate \(\int_{0}^{\infty} e^{-x^{2}} d x, \int_{0}^{\infty} x e^{-x^{2}} d x,\) and \(\int_{0}^{\infty} x^{2} e^{-x^{2}} d x\) (using part (a) if needed).

Let \(D\) be the solid bounded by the ellipsoid \(x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1,\) where \(a>0, b>0,\) and \(c>0\) are real numbers. Let \(T\) be the transformation \(x=\)au, \(y=b v, z=c w\) Find the average square of the distance between points of \(D\) and the origin.
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