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Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F= (8xy,9x? - 4y?); R is the region bounded by y = x(5 – x) and y = 0. a. The two-dimensional divergence is b. Set up the integral over the region. Set up the line integral for the y = x(5 - x) boundary. jo. dt Set up the line integral for the y = 0 boundary. 5 dt Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice. O A. The integrals are not consistent because the integral over R evaluates to while the line integrals evaluate to O B. The integrals are consistent because they all evaluate to c. Is the vector field source-free? O A. No, because the two-dimensional divergence is not zero everywhere. O B. Yes, because the two-dimensional divergence is zero everywhere. OC. Yes, because the flux is zero for the given region. O D. No, because the flux is zero only for the given region.

Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F= (8xy,9x? - 4y?); R is the region bounded by y = x(5 – x) and y = 0. a. The two-dimensional divergence is b. Set up the integral over the region. Set up the line integral for the y = x(5 - x) boundary. jo. dt Set up the line integral for the y = 0 boundary. 5 dt Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice. O A. The integrals are not consistent because the integral over R evaluates to while the line integrals evaluate to O B. The integrals are consistent because they all evaluate to c. Is the vector field source-free? O A. No, because the two-dimensional divergence is not zero everywhere. O B. Yes, because the two-dimensional divergence is zero everywhere. OC. Yes, because the flux is zero for the given region. O D. No, because the flux is zero only for the given region.

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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Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F= (8xy,9x? - 4y?); R is the region bounded by y = x(5 - x) and y = 0. a. The two-dimensional divergence is b. Set up the integral over the region. dy dx Set up the line integral for the y = x(5 - x) boundary. 5 dt Set up the line integral for the y = 0 boundary. 5 dt Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice. O A. The integrals are not consistent because the integral over R evaluates to| , while the line integrals evaluate to B. The integrals are consistent because they all evaluate to c. Is the vector field source-free? O A. No, because the two-dimensional divergence is not zero everywhere. O B. Yes, because the two-dimensional divergence is zero everywhere. OC. Yes, because the flux is zero for the given region. O D. No, because the flux is zero only for the given region.
Transcribed Image Text:Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in the flux form of Green's Theorem and check for consistency. c. State whether the vector field is source free. F= (8xy,9x? - 4y?); R is the region bounded by y = x(5 - x) and y = 0. a. The two-dimensional divergence is b. Set up the integral over the region. dy dx Set up the line integral for the y = x(5 - x) boundary. 5 dt Set up the line integral for the y = 0 boundary. 5 dt Evaluate these integrals and check for consistency. Select the correct choice below and fill in any answer boxes to complete your choice. O A. The integrals are not consistent because the integral over R evaluates to| , while the line integrals evaluate to B. The integrals are consistent because they all evaluate to c. Is the vector field source-free? O A. No, because the two-dimensional divergence is not zero everywhere. O B. Yes, because the two-dimensional divergence is zero everywhere. OC. Yes, because the flux is zero for the given region. O D. No, because the flux is zero only for the given region.
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