Let c be the perimeter of the Twith vertices Coo ), (3, o) and (3, 2)...
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5. Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove dx dy. riangle A. [Hint: the lect when A is a rectangle. So, the idea is is to give a similar proof where we have this triangle A in place of a rectangle.] 3 marks
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-8y). Use Green's Theorem to calculate the circulation of Faround the perimeter of the triangle C oriented counter-clockwise with (1 point) Suppose F(x, y) = (x2 + 5y, 4 vertices (10,0). (0,5), and (-10,0). JË. dr =
9.) (12 pts.) Let loop C be the triangle with vertices (0,0), (2,0), and (2,6). Evaluate the line integral $ ay dx + (x - y) dy using one of Green's Theorems.
Let A be the inside and boundary of the triangle in R2 whose vertices are (0,0), (1,0) and (0,1). Let C be the curve obtained by proceeding around the boundary of A in an anti- clockwise direction. Prove İ}!").lx (ly İ)(2 dr dy. Pdr+Qdy That is, prove Green's Theorem for the triangle A. [Hint: the lecture notes have a proof for when A is a rectangle. So, the idea is is to give a similar proof where we have this...
Use Stokes' Theorem to evaluate fe(x+y)dx + (2x – 3)dy +(y +z)dz over the boundary of the triangle with vertices (2,0,0), (0,3,0), (0,0,6) traversed in the counter clockwise direction.
2. Let S be the interior of the triangle with vertices (0,0,0), (1,0,0) and (0,1,0). a) Given F(x, y, z)=(x+1)i +(y+1)] +(2+1)k, calculate the flux of through S without using an integral b) F(x, y, z) = (z+1)7 +(y+1) 7+(x+1)k , set up an iterated integral in dx dy or dy dx to calculate the flux of F through S. You do not need to evaluate your integral
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 +...
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The path C is defined as the counter clockwise along the sides of the triangle with vertices at (0,0), (1,0) and (1,2). Use Green's Theorem to evaluate $c (2x+y) dx + (3x-2y) dy).
Find the perimeter of the triangle whose vertices are the following specified points in the plane. (0,-3), (-2, 1) and (-4, 6) Answer
Problem. Use Green's Theorem, to evaluate the line integral, 5. Pdr + Qdy = 1] (e. - SP) da, 1. (=x+ + e* In y)dx + (x + y + ) dy, where C is the triangle with the vertices (1,1), (2.1), and (2, 2), and the positive (counter- clockwise) orientation. (10 points)