Explanation For finding the flux across the net, we have to find the surface integral by parameterizing and then we can evaluate it. Given: The force function and the points are given below. v = ( z y , x z , x y ) a n d y = 1 - x - z a n d x , y , z ≥ 0. Formula Used: ∬ s v . d s = ∬ R v ( G ( x , z ) . N ( x , z ) d x d z Calculation: Let x = x , z = z ,then y=1-x-z with 0 ≤ x ≤ 1 ,0 ≤ z ≤ 1-x Now G ( x , z ) = ( x , 1 − x − z , z ) T x ( x , z ) = ( 1 , − 1 , 0 ) T z ( x , z ) = ( 0 , − 1 , 1 ) Then, the normal vector is N ( x , z ) = T x ( x , z ) x T z ( x , z ) = | i j k 1 − 1 0 0 − 1 1 | = i ( − 1 − 0 ) − j ( 1 ) + k ( − 1 − 0 ) = ( − 1 , − 1 , − 1 ) Since, -1 < 0 so the normal pointing out toward negative direction, so we have to take normal as N ( x , z ) = ( 1 , 1 , 1 ) N o w v = ( y z , x z , x y ) = ( z − y z − z 2 , x z , x − x 2 − x z ) t h e n v ( G ( x , z ) ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

See similar textbooks

Videos

Question

Chapter 17.5, Problem 28E

To determine

To calculate: The flux across the net in the river.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 17.5, Problem 27E

Chapter 17.5, Problem 29E

Students have asked these similar questions

(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 17 Solutions

Calculus - Standalone book

Ch. 17.1 - Prob. 1PQ Ch. 17.1 - Prob. 2PQ Ch. 17.1 - Prob. 3PQ Ch. 17.1 - Prob. 4PQ Ch. 17.1 - Prob. 1E Ch. 17.1 - Prob. 2E Ch. 17.1 - Prob. 3E Ch. 17.1 - Prob. 4E Ch. 17.1 - Prob. 5E Ch. 17.1 - Prob. 6E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

The flux across the net in the river.

Solution Summary: The author calculates the flux across the net in the river by finding the surface integral by parameterizing and then evaluating it.

Videos

To calculate: The flux across the net in the river.

Want to see the full answer?

Chapter 17 Solutions