7. Let F= x cos(x² + y²) i + (y cos(x² + y²) + 3y) j a) Show that F is conservative by evaluating a potential function for F. b) Use the cross-derivative test to show that G = (cos(x2 + y²) + 3y)j is NOT conservative. c) Use Gauss-Green theorem to evaluate the integral of G over the boundary of the circle of center (0,0) and radius 1. cos(x² + y²) i + Hint: After you use polar coordinates, do the integral with respect to 0 first ...

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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7. Let F= x cos(x² + y²) i + (y cos(x² + y²) + 3y) j
a) Show that F is conservative by evaluating a potential function for F.
b) Use the cross-derivative test to show that G = cos(x² +y²) i+
(cos(x2 + y²) + 3y)j is NOT conservative.
c) Use Gauss-Green theorem to evaluate the integral of G over the boundary of the
circle of center (0,0) and radius 1.
Hint: After you use polar coordinates, do the integral with respect to 0 first ...
Transcribed Image Text:7. Let F= x cos(x² + y²) i + (y cos(x² + y²) + 3y) j a) Show that F is conservative by evaluating a potential function for F. b) Use the cross-derivative test to show that G = cos(x² +y²) i+ (cos(x2 + y²) + 3y)j is NOT conservative. c) Use Gauss-Green theorem to evaluate the integral of G over the boundary of the circle of center (0,0) and radius 1. Hint: After you use polar coordinates, do the integral with respect to 0 first ...
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