Explanation Given: The mapping G ( u , v ) = ( 2 u + v , 5 u + 3 v ) Calculation: Since ( x , y ) = G ( u , v ) = ( 2 u + v , 5 u + 3 v ) . Therefore, x = 2 u + v v = x − 2 u Substitute v = x − 2 u in y = 5 u + 3 v we get y = 5 u + 3 ( x − 2 u ) y = 5 u + 3 x − 6 u y = 3 x − u u = 3 x − y Therefore,

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.6, Problem 7E

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The image of the line $v = 4 u$ under the mapping $G (u, v) = (2 u + v, 5 u + 3 v)$ .in slope intercept form.

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Chapter 16.6, Problem 6E

Chapter 16.6, Problem 8E

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(12 points) Let E={(x, y, z)|x²+ y² + z² ≤ 4, x, y, z > 0}. (a) (4 points) Describe the region E using spherical coordinates, that is, find p, 0, and such that (x, y, z) (psin cos 0, psin sin 0, p cos) € E. (b) (8 points) Calculate the integral E xyz dV using spherical coordinates.

(10 points) Let f(x, y, z) = ze²²+y². Let E = {(x, y, z) | x² + y² ≤ 4,2 ≤ z < 3}. Calculate the integral y, f(x, y, z) dV.

(14 points) Let f: R3 R and T: R3. →R³ be defined by f(x, y, z) = ln(x²+ y²+2²), T(p, 0,4)=(psin cos 0, psin sin, pcos). (a) (4 points) Write out the composition g(p, 0, 4) = (foT)(p,, ) explicitly. Then calculate the gradient Vg directly, i.e. without using the chain rule. (b) (4 points) Calculate the gradient Vf(x, y, z) where (x, y, z) = T(p, 0,4). (c) (6 points) Calculate the derivative matrix DT(p, 0, p). Then use the Chain Rule to calculate Vg(r,0,4).

Chapter 16 Solutions

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The image of the line v = 4 u under the mapping G ( u , v ) = ( 2 u + v , 5 u + 3 v ) .in slope intercept form.

Solution Summary: The author explains that the image of the line v = 4 u in slope intercept form is y = 17 6 x.

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The image of the line $v = 4 u$ under the mapping $G (u, v) = (2 u + v, 5 u + 3 v)$ .in slope intercept form.

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Chapter 16 Solutions

The image of the line v = 4 u under the mapping G ( u , v ) = ( 2 u + v , 5 u + 3 v ) .in slope intercept form.

Solution Summary: The author explains that the image of the line v = 4 u in slope intercept form is y = 17 6 x.

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To describe: The image of the line v=4uunder the mapping G(u,v)=(2u+v,5u+3v).in slope intercept form.

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Chapter 16 Solutions

To describe:

The image of the line $v = 4 u$ under the mapping $G (u, v) = (2 u + v, 5 u + 3 v)$ .in slope intercept form.