Evaluate ff, (F n) dS in the following cases: (a) F = (2y+z, x+z, y − x) and S is the portion of the plane x + y + z = 1 lying in the first octant, and n points upward. (b) F = (x + y, y+z, z+x) and S is the surface of the cube bounded by the planes x = 0, y = 0, z = 0, x = 1, y = 1, z = 1. (Hint: combine the integrals for the two planes z = 0 and z = 1. Use symmetry for the others.) (c) F = (x, y, z) and S is the portion of the paraboloid z = 1- x² - y² lying in the first octant, and n points upward.
Evaluate ff, (F n) dS in the following cases: (a) F = (2y+z, x+z, y − x) and S is the portion of the plane x + y + z = 1 lying in the first octant, and n points upward. (b) F = (x + y, y+z, z+x) and S is the surface of the cube bounded by the planes x = 0, y = 0, z = 0, x = 1, y = 1, z = 1. (Hint: combine the integrals for the two planes z = 0 and z = 1. Use symmetry for the others.) (c) F = (x, y, z) and S is the portion of the paraboloid z = 1- x² - y² lying in the first octant, and n points upward.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Evaluate ff, (F n) dS in the following cases:
(a) F = (2y+z, x+z, y − x) and S is the portion of the plane x + y + z = 1 lying in
the first octant, and n points upward.
(b) F = (x + y, y+z, z+x) and S is the surface of the cube bounded by the planes
x = 0, y = 0, z = 0, x = 1, y = 1, z = 1. (Hint: combine the integrals for the two
planes z = 0 and z = 1. Use symmetry for the others.)
(c) F = (x, y, z) and S is the portion of the paraboloid z = 1- x² - y² lying in the first
octant, and n points upward.
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