3. This problem uses a common alternate notation for the surface in- tegral of a vector field: Consider the vector field S FdS = S F.ndo. F(x, y, z) = (e²+z – 2y) i + (xe³+* + y) j+eª+yk. J (a) Find curl F. (b) Let S₁ be the part of the plane z = 1/e that is inside of the cylinder x² + y² = 1, oriented with upward-pointing unit normal vectors. Evaluate curl F. ds. e−(x²+y²) that is inside of the cylinder (c) Let S₂ be the part of the surface z = e¯ x² + y² = 1, oriented with upward-pointing unit normal vectors. Evaluate J S2 Hint: Use Stokes' Theorem and part (b). curl F. ds.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3.
This problem uses a common alternate notation for the surface in-
tegral of a vector field:
Consider the vector field
S
FdS =
S
F.ndo.
F(x, y, z) = (e²+z – 2y) i + (xe³+* + y) j+eª+yk.
J
(a) Find curl F.
(b) Let S₁ be the part of the plane z
=
1/e that is inside of the cylinder
x² + y² = 1, oriented with upward-pointing unit normal vectors. Evaluate
curl F. ds.
e−(x²+y²) that is inside of the cylinder
(c) Let S₂ be the part of the surface z = e¯
x² + y² = 1, oriented with upward-pointing unit normal vectors. Evaluate
J
S2
Hint: Use Stokes' Theorem and part (b).
curl F. ds.
Transcribed Image Text:3. This problem uses a common alternate notation for the surface in- tegral of a vector field: Consider the vector field S FdS = S F.ndo. F(x, y, z) = (e²+z – 2y) i + (xe³+* + y) j+eª+yk. J (a) Find curl F. (b) Let S₁ be the part of the plane z = 1/e that is inside of the cylinder x² + y² = 1, oriented with upward-pointing unit normal vectors. Evaluate curl F. ds. e−(x²+y²) that is inside of the cylinder (c) Let S₂ be the part of the surface z = e¯ x² + y² = 1, oriented with upward-pointing unit normal vectors. Evaluate J S2 Hint: Use Stokes' Theorem and part (b). curl F. ds.
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