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Chapter 13: Q. 52 (page 1080)

The formulas for converting from spherical coordinates to rectangular coordinates are $x = p \sin 蠒 \cos 胃, y = p \sin 蠒 \sin 胃, z = p \cos 蠒$ . Prove that the Jacobianrole="math" $\frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = - p^{2} \sin 蠒$ .

Short Answer

Expert verified

It is proven that $\frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = - p^{2} \sin 蠒$ .

Step by step solution

01

Given information

The transformation equations are,

$x = p \sin 蠒 \cos 胃, y = p \sin 蠒 \sin 胃, z = p \cos 蠒$

02

Proof

The definition of Jacobian of a transformation using partial derivatives is given as

L . H . S = \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = |\begin{matrix} \frac{鈭� x}{鈭� p} & \frac{鈭� y}{鈭� p} & \frac{鈭� z}{鈭� p} \\ \frac{鈭� x}{鈭� 胃} & \frac{鈭� y}{鈭� 胃} & \frac{鈭� z}{鈭� 胃} \\ \frac{鈭� x}{鈭� 蠒} & \frac{鈭� y}{鈭� 蠒} & \frac{鈭� z}{鈭� 蠒} \end{matrix}| \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = |\begin{matrix} \sin 蠒 \cos 胃 & \sin 蠒 \sin 胃 & \cos 蠒 \\ - p \sin 蠒 \sin 胃 & p \sin 蠒 \cos 胃 & 0 \\ p \cos 蠒 \cos 胃 & p \cos 蠒 \sin 胃 & - p \sin 蠒 \end{matrix}| \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = - p^{2} \sin^{3} 蠒 - p^{2} \sin 蠒 \cos^{2} 蠒 \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = - p^{2} \sin 蠒 (\sin^{2} 蠒 + \cos^{2} 蠒) \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = - p^{2} \sin 蠒 (1) \frac{鈭� (x, y, z)}{鈭� (p, 胃, 蠒)} = - p^{2} \sin 蠒 = R . H . S

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Most popular questions from this chapter

Let $f (x, y, z)$ be a continuous function of three variables, let $惟_{x y} = {(x, y) | a 鈮� x 鈮� b a n d h_{1} (x) 鈮� y 鈮� h_{2} (x)}$ be a set of points in the $x y$ -plane, and let $惟 = {(x, y, z) | (x, y) 鈭� 惟_{x y} a n d g_{1} (x, y) 鈮� z 鈮� g_{2} (x, y)}$ be a set of points in 3-space. Find an iterated triple integral equal to the the triple integral ${鈭�}_{惟} f (x, y, z) d V$ . How would your answer change if $惟_{x y} = {(x, y) | a 鈮� y 鈮� b a n d h_{1} (y) 鈮� x 鈮� h_{2} (y)}$ ?

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} {鈭�}_{- 3}^{3} f (x, y, z) d z d y d x$

In Exercises, let $S = \{(x, y) 鈭� x^{2} + y^{2} 鈮� 1 and x 鈮� 0\} .$

If the density at each point in S is proportional to the point鈥檚 distance from the origin, find the moments of inertia about the x-axis, the y-axis, and the origin. Use these answers to find the radii of gyration of S about the x-axis, the y-axis, and the origin.

Explain how to construct a midpoint Riemann sum for a function of two variables over a rectangular region for which each $(x_{j}^{*}, y_{k}^{*})$ is the midpoint of the subrectangle

R_{j k} = {(x, y) 鈭� x_{j - 1} 鈮� x_{j}^{*} 鈮� x_{j} and y_{k - 1} 鈮� y_{k}^{*} 鈮� y_{k}} .

Refer to your answer to Exercise 10 or to Definition 13.3.

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$f (x, y) = \sin x \cos y W h e r e R = {(x, y) | 0 鈮� x 鈮� 蟺, 0 鈮� y 鈮� 蟺}$

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