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d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V through F after choosing orientations of F and its boun- dary curve aF that fulfil the right-hand rule.

d) An open surface F is given by F = {(x,y, z) |z = 4 – x² – y² , z > 0} . Compute the flux of V through F after choosing orientations of F and its boun- dary curve aF that fulfil the right-hand rule.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.4: Linear Transformations
Problem 34EQ
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Consider in (x, y, z) space the vector field V(x, y,z) = (2x + 3y, 2y + 3x, –4z) and the function F(x,у, 2) — а - х* +b.y? +c:2?+d х-у, (х,у, 2) € R'. - с. = a · a) Find constants a, b,c and d so that V = VF. b) Compute the tangential line integral of V along the right half of the unit circle in the (y,z) plane centered at (0,1,1) with an orientation of your choice. c) Show that the vector field W(x, y, z) = (2yz + xz – x², –2xz – yz + y?, y² – x² + z²) is a vector potential for V. d) An open surface F is given by F = {(x, y, z) | z = 4 – x² – y? , z > 0} . Compute the flux of V through F after choosing orientations of F and its boun- dary curve ƏF that fulfil the right-hand rule.
Transcribed Image Text:Consider in (x, y, z) space the vector field V(x, y,z) = (2x + 3y, 2y + 3x, –4z) and the function F(x,у, 2) — а - х* +b.y? +c:2?+d х-у, (х,у, 2) € R'. - с. = a · a) Find constants a, b,c and d so that V = VF. b) Compute the tangential line integral of V along the right half of the unit circle in the (y,z) plane centered at (0,1,1) with an orientation of your choice. c) Show that the vector field W(x, y, z) = (2yz + xz – x², –2xz – yz + y?, y² – x² + z²) is a vector potential for V. d) An open surface F is given by F = {(x, y, z) | z = 4 – x² – y? , z > 0} . Compute the flux of V through F after choosing orientations of F and its boun- dary curve ƏF that fulfil the right-hand rule.
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