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Stoke's Theorem: 2. Evaluate the flux integral curl FdS by use of Stoke's Theorem: F(x, y, z) = (y², zy, zz), where S is the hemisphere r² + y²+z² = 1, z0, and is the "outward pointing" unit normal (Hint: First use Stoke's Theorem to convert the flux integral into a circulation integral). 3. Evaluate the flux integral ffs curl F.ñdS by use of Stoke's Theorem: F(x, y, z)=(y-z, yz,-12), where S is the five faces of the cube 0

Stoke's Theorem: 2. Evaluate the flux integral curl FdS by use of Stoke's Theorem: F(x, y, z) = (y², zy, zz), where S is the hemisphere r² + y²+z² = 1, z0, and is the "outward pointing" unit normal (Hint: First use Stoke's Theorem to convert the flux integral into a circulation integral). 3. Evaluate the flux integral ffs curl F.ñdS by use of Stoke's Theorem: F(x, y, z)=(y-z, yz,-12), where S is the five faces of the cube 0

Advanced Engineering Mathematics
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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Stoke's Theorem: 2. Evaluate the flux integral curl FdS by use of Stoke's Theorem: F(x, y, z) = (y², zy, zz), where S is the hemisphere r² + y²+z² = 1, z0, and is the "outward pointing" unit normal (Hint: First use Stoke's Theorem to convert the flux integral into a circulation integral). 3. Evaluate the flux integral ffs curl F.ñdS by use of Stoke's Theorem: F(x, y, z)=(y-z, yz,-12), where S is the five faces of the cube 0<y≤2, 0≤ y ≤2, 0 ≤ z <2 not lying in the zy-plane, and is the outward pointing unit normal (you may assume that Stoke's Theorem it true even for surfaces like S for which the parameterization is "piecewise C¹"). 1 4. Let C denote the curve of intersection of the sphere r²+y²+2² = a² and the plane+y+z=0. Consider the vector field F(x, y, z) = (y, z, z) Use Stoke's Theorem to evaluate the circulation integral fε F. dc. 5. Let C denote the curve of intersection of the cylinder ²+ y² = 4 and the plane + 1. Consider the vector field F(x, y, z) = (y-z, z-1,2-y) Use Stoke's Theorem to evaluate the circulation integral §. dc.
Transcribed Image Text:Stoke's Theorem: 2. Evaluate the flux integral curl FdS by use of Stoke's Theorem: F(x, y, z) = (y², zy, zz), where S is the hemisphere r² + y²+z² = 1, z0, and is the "outward pointing" unit normal (Hint: First use Stoke's Theorem to convert the flux integral into a circulation integral). 3. Evaluate the flux integral ffs curl F.ñdS by use of Stoke's Theorem: F(x, y, z)=(y-z, yz,-12), where S is the five faces of the cube 0<y≤2, 0≤ y ≤2, 0 ≤ z <2 not lying in the zy-plane, and is the outward pointing unit normal (you may assume that Stoke's Theorem it true even for surfaces like S for which the parameterization is "piecewise C¹"). 1 4. Let C denote the curve of intersection of the sphere r²+y²+2² = a² and the plane+y+z=0. Consider the vector field F(x, y, z) = (y, z, z) Use Stoke's Theorem to evaluate the circulation integral fε F. dc. 5. Let C denote the curve of intersection of the cylinder ²+ y² = 4 and the plane + 1. Consider the vector field F(x, y, z) = (y-z, z-1,2-y) Use Stoke's Theorem to evaluate the circulation integral §. dc.
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