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Consider the following region R and the vector field F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. F = (4y, -4x); R is the triangle with vertices (0,0), (1,0), and (0,1).

Consider the following region R and the vector field F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. F = (4y, -4x); R is the triangle with vertices (0,0), (1,0), and (0,1).

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.CM: Cumulative Review
Problem 12CM
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Consider the following region R and the vector field F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. F = (4y, - 4x); R is the triangle with vertices (0,0), (1,0), and (0,1).
Transcribed Image Text:Consider the following region R and the vector field F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency. F = (4y, - 4x); R is the triangle with vertices (0,0), (1,0), and (0,1).
a. The two-dimensional curl is (Type an exact answer.) b. Set up the integral over the region R. S Į Į ( ) dy dx 0 0 (Type exact answers.) Set up the line integral for the line segment between (0,0) and (1,0). 1 jo 0 (Type an exact answer.) Write the line integral for the line segment between (1,0) and (0,1). 1 S dt (Type an exact answer.) Write the line integral for the line segment between (0,1) and (0,0). 1 Fill O 0 (Type an exact answer.) Evaluate these integrals and check for consistency. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type an exact answer.) A. The integrals are consistent because they both evaluate to B. The integrals are not consistent. The double integral evaluates to but evaluating the line integrals and adding the results yields
Transcribed Image Text:a. The two-dimensional curl is (Type an exact answer.) b. Set up the integral over the region R. S Į Į ( ) dy dx 0 0 (Type exact answers.) Set up the line integral for the line segment between (0,0) and (1,0). 1 jo 0 (Type an exact answer.) Write the line integral for the line segment between (1,0) and (0,1). 1 S dt (Type an exact answer.) Write the line integral for the line segment between (0,1) and (0,0). 1 Fill O 0 (Type an exact answer.) Evaluate these integrals and check for consistency. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type an exact answer.) A. The integrals are consistent because they both evaluate to B. The integrals are not consistent. The double integral evaluates to but evaluating the line integrals and adding the results yields
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