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12P SHOW ALL WORK. Support papers must be uploaded for full credit. Consider the 2-dimensional vector...
Consider the following region R and vector field a. Evaluate both integrals in Green's Theorem - Circulation Form and check for consistency. b. Is the vector field conservative? 7) (16 points) F...
Consider the following region R and vector field a. Evaluate both integrals in Green's Theorem - Circulation Form and check for consistency. b. Is the vector field conservative? 7) (16 points) F = 〈x4, xy〉, R is the triangular region with vertices (0,0), (1,0) and (0,1).
Consider the following region R and vector field a. Evaluate both integrals in Green's Theorem - Circulation Form and check for consistency. b. Is the vector field conservative? 7) (16 points) F = 〈x4,...
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 the circulation form of Green's Theorem and check for...
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 the circulation form of Green's Theorem and check for consistency. c. State whether the vector field is conservative. F-3y,3x); R is the triangle with vertices (0, 0), (1, 0), and (0, 1) a. The two-dimensional curl is D (Type an exact answer, using π as needed.) b. Set up the integral over the region R. dy...
Consider the vector field F2(x, y)-(-y,z) and the closed curve C which is the square with corners...
Consider the vector field F2(x, y)-(-y,z) and the closed curve C which is the square with corners (-1,-1), (1,-1), (1,1), and (-1,1) and is traversed counter-clockwise starting at (-1,-1) (a) Compute the outward flux across the curve C by calculating a line integral. (b) Use an appropriate version of Green's Theorem to compute the above flux as a (c) Compute the circulation of the vector field around the curve by computing a line (d) Use an appropriate version of Green's...
5. Let F (y”, 2xy + €35, 3yes-). Find the curl V F. Is the vector...
5. Let F (y”, 2xy + €35, 3yes-). Find the curl V F. Is the vector field F conservative? If so, find a potential function, and use the Fundamental Theorem of Line Integrals (FTLI) to evaluate the vector line integral ScF. dr along any path from (0,0,0) to (1,1,1). 6. Compute the Curl x F = Q. - P, of the vector field F = (x4, xy), and use Green's theorem to evaluate the circulation (flow, work) $ex* dx +...
Consider the following region R and the vector field F. a. Compute the two-dimensional curl of...
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 = (3y, - 3x); R is the triangle with vertices (0,0), (1,0), and (0,2). . a. The two-dimensional curl is (Type an exact answer.) b. Set up the integral over the region R. JO dy dx 0 0 (Type exact answers.) Set up the line integral for the line...
Consider the following region and the vector field F. a. Compute the two-dimensional divergence of the...
Consider the following region and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency F= (2x-2y); R=(x,y): x2 + y²59 a. The two-dimensional divergence is (Type an exact answer.) b. Set up the integral over the region. Write the integral using polar coordinates with r as the radius and O as the angle SO rdr d0 (Type exact answers.) 0 o Set up the line...
Consider the following region R and the vector field F. a. Compute the two-dimensional divergence of...
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,9x2 - 4y?); R is the region bounded by y = x(6 - x) and y=0. .- a. The two-dimensional divergence is 0 b. Set up the integral over the region. dy dx 0 Set...
2. EXTRA CREDIT OPTION: GREEN'S THEOREM Fix θ > 0, Consider the region A bounded by the straight ...
2. EXTRA CREDIT OPTION: GREEN'S THEOREM Fix θ > 0, Consider the region A bounded by the straight line segment from (0,0) to (1,0), the portion of the hyperbola parametrized by r(t) (cosh(t), sinh(t)) for 0 t 0, and the straight line segment from P-(cosh(9), sinh(9) back to the origin. Using the vector field F-1/2(-y,z) and Green's Theorem, find the area of A in terms of θ. Show all work in an organized, well-written manner for full credit.
2. EXTRA...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vec...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vector field F(x.y)-(r-yi+r+y)j and the circle C: r y-9. Verify Green's Theorem by calculating the outward flux of F across C (12 points) Find the absolute extreme values of the function .-2-4--3 on the closed triangular region in the xy-plane bounded by the lines x...
Select statements that are correct. Green's Theorem calculate the circulation in R^2 which convert the line...
Select statements that are correct. Green's Theorem calculate the circulation in R^2 which convert the line integral into a double integral over the region Din R^2 formed by the simple and closed curve C To compute the work done by a vector field in moving a particle around a simple and closed curve Cin R^2, we apply the Green's Theorem U line integral of a vector field computes the work done to move a particle along a space curve C...
Consider the following region R and vector field a. Evaluate both integrals in Green's Theorem - Circulation Form and check for consistency. b. Is the vector field conservative? 7) (16 points) F = 〈x4, xy〉, R is the triangular region with vertices (0,0), (1,0) and (0,1).
Consider the following region R and vector field a. Evaluate both integrals in Green's Theorem - Circulation Form and check for consistency. b. Is the vector field conservative? 7) (16 points) F = 〈x4,...
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 the circulation form of Green's Theorem and check for consistency. c. State whether the vector field is conservative. F-3y,3x); R is the triangle with vertices (0, 0), (1, 0), and (0, 1) a. The two-dimensional curl is D (Type an exact answer, using π as needed.) b. Set up the integral over the region R. dy...
Consider the vector field F2(x, y)-(-y,z) and the closed curve C which is the square with corners (-1,-1), (1,-1), (1,1), and (-1,1) and is traversed counter-clockwise starting at (-1,-1) (a) Compute the outward flux across the curve C by calculating a line integral. (b) Use an appropriate version of Green's Theorem to compute the above flux as a (c) Compute the circulation of the vector field around the curve by computing a line (d) Use an appropriate version of Green's...
5. Let F (y”, 2xy + €35, 3yes-). Find the curl V F. Is the vector field F conservative? If so, find a potential function, and use the Fundamental Theorem of Line Integrals (FTLI) to evaluate the vector line integral ScF. dr along any path from (0,0,0) to (1,1,1). 6. Compute the Curl x F = Q. - P, of the vector field F = (x4, xy), and use Green's theorem to evaluate the circulation (flow, work) $ex* dx +...
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 = (3y, - 3x); R is the triangle with vertices (0,0), (1,0), and (0,2). . a. The two-dimensional curl is (Type an exact answer.) b. Set up the integral over the region R. JO dy dx 0 0 (Type exact answers.) Set up the line integral for the line...
Consider the following region and the vector field F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Green's Theorem and check for consistency F= (2x-2y); R=(x,y): x2 + y²59 a. The two-dimensional divergence is (Type an exact answer.) b. Set up the integral over the region. Write the integral using polar coordinates with r as the radius and O as the angle SO rdr d0 (Type exact answers.) 0 o Set up the line...
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,9x2 - 4y?); R is the region bounded by y = x(6 - x) and y=0. .- a. The two-dimensional divergence is 0 b. Set up the integral over the region. dy dx 0 Set...
2. EXTRA CREDIT OPTION: GREEN'S THEOREM Fix θ > 0, Consider the region A bounded by the straight line segment from (0,0) to (1,0), the portion of the hyperbola parametrized by r(t) (cosh(t), sinh(t)) for 0 t 0, and the straight line segment from P-(cosh(9), sinh(9) back to the origin. Using the vector field F-1/2(-y,z) and Green's Theorem, find the area of A in terms of θ. Show all work in an organized, well-written manner for full credit.
2. EXTRA...
3) (11 points) Consider the vector field Use the Fundamental Theorem of lLine Integrals to find the work done by F along any curve from 41. 1Le) to B(2. el) 4) (10 points) Consider the vector field F(x.y)-(r-yi+r+y)j and the circle C: r y-9. Verify Green's Theorem by calculating the outward flux of F across C (12 points) Find the absolute extreme values of the function .-2-4--3 on the closed triangular region in the xy-plane bounded by the lines x...
Select statements that are correct. Green's Theorem calculate the circulation in R^2 which convert the line integral into a double integral over the region Din R^2 formed by the simple and closed curve C To compute the work done by a vector field in moving a particle around a simple and closed curve Cin R^2, we apply the Green's Theorem U line integral of a vector field computes the work done to move a particle along a space curve C...
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