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Let the surface S be part of the sphere r +y² +z² = 4 (oriented away from the origin) that lies within the cylinder r² + y² = 1 and above the plane z = 0, and let n denote the unit normal vector in the direction of the orientation. Let C be the boundary curve of S. Consider the vector ficld F(x, y, z) = ri + yj+ xyzk. %3D (a) Evaluate the line integral F.T dS directly, without using Stokes' Theorem, where T is a unit tangent vector to the curve C. (b) Evaluate the surface integral | S (V × F) - n dS directly, without using Stokes' Theorem.

Let the surface S be part of the sphere r +y² +z² = 4 (oriented away from the origin) that lies within the cylinder r² + y² = 1 and above the plane z = 0, and let n denote the unit normal vector in the direction of the orientation. Let C be the boundary curve of S. Consider the vector ficld F(x, y, z) = ri + yj+ xyzk. %3D (a) Evaluate the line integral F.T dS directly, without using Stokes' Theorem, where T is a unit tangent vector to the curve C. (b) Evaluate the surface integral | S (V × F) - n dS directly, without using Stokes' Theorem.

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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Let the surface S be part of the sphere r? + y² +z² = 4 (oriented away from the origin) that lies within the cylinder x? + y? = 1 and above the plane z = the unit normal vector in the direction of the orientation. Let C be the boundary curve of S. Consider the vector field F(x, y, z) = xi+ yj+ xyzk. 0, and let n denote (a) Evaluate the line integral / F ·T dS directly, without using Stokes' Theorem, where T is a unit tangent vector to the curve C. (b) Evaluate the surface integral || (V × F) · n dS directly, without using Stokes' Theorem.
Transcribed Image Text:Let the surface S be part of the sphere r? + y² +z² = 4 (oriented away from the origin) that lies within the cylinder x? + y? = 1 and above the plane z = the unit normal vector in the direction of the orientation. Let C be the boundary curve of S. Consider the vector field F(x, y, z) = xi+ yj+ xyzk. 0, and let n denote (a) Evaluate the line integral / F ·T dS directly, without using Stokes' Theorem, where T is a unit tangent vector to the curve C. (b) Evaluate the surface integral || (V × F) · n dS directly, without using Stokes' Theorem.
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