ASSIGNMENT 10 1. [3 marks] Show that the integral Z (1,p/2) (0,0) e x sin y dx + e x cos y dy is independent of the path, and find its value by any method. 2. [3 marks] Use Green’s Theorem to calculate Z C (e x + y 2 )dx + (e y + x 2 ) dy, where C is the boundary of the region between y = x 2 and y = x oriented counterclockwise. 3. [4 marks] Evaluate Z Z s F · n dS where F(x, y, z) = (x+y)i+ (y +z)j+ (z +x)k and s is the portion of the plane x+y +z = 1 in the first octant, oriented by unit normals with positive components. 4. [4 marks] Use the Divergence Theorem to evaluate RR s F · n dS, where n is the outer unit normal to s. (a) F(x, y, z) = 2xi + 2yj + 2zk, s is the sphere x 2 + y 2 + z 2 = 9. (b) F(x, y, z) = z 3 i - x 3 j + y 3k, s is the sphere x 2 + y 2 + z 2 = a 2 . 5. [3 marks] Use Stokes’ Theorem to evaluate R C F · dr, where F(x, y, z) = -3y 2 i + 4zj + 6xk, C is the triangle in the plane z = 1 2 y with vertices (2, 0, 0), (0, 2, 1) and (0, 0, 0), with a counterclockwise orientation looking down the positive z-axis. 6. [3 marks] Find the net volume of the fluid that crosses the surface s in the direction of the orientation in one unit of time where the flow field is given by F(x, y, z) = x 2 i+yxj+ zxk, s is the portion of the plane 6x+3y+2z=6 in the first octant oriented by upward unit normals.
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ASSIGNMENT 10 Z (1;=2) x x 1. [3 marks] Show that the integral e sinydx +e cosydy is independent of the (0;0) path, and nd its value by any method. Z x 2 y 2 2. [3 marks] Use Green's Theorem to calculate (e +y )dx + (e +x )dy; where C is C 2 the boundary of the region between y =x and y =x oriented counterclockwise. ZZ 3. [4 marks] Evaluate FndS where F(x;y;z) = (x+y)i+(y +z)j+(z +x)k and is the portion of the planex+y +z = 1 in the rst octant, oriented by unit normals with positive components. RR 4. [4 marks] Use the Divergence Theorem to evaluate FndS, where n is the outer unit normal to . 2 2 2 (a) F(x;y;z) = 2xi + 2yj + 2zk, is the sphere x +y +z = 9. 3 3 3 2 2 2 2 (b) F(x;y;z) =z ix j +y k, is the sphere x +y +z =a . R 5. [3 marks] Use Stokes' Theorem to evaluate Fdr, where C 2 1 F(x;y;z) = 3y i + 4zj + 6xk, C is the triangle in the plane z = y with vertices 2 (2; 0; 0), (0; 2; 1) and (0; 0; 0), with a counterclockwise orientation looking down the posi- tive z-axis. 6. [3 marks] Find the net volume of the uid that crosses the surface in the direction of 2 the orientation in one unit of time where the ow eld is given byF(x;y;z) =x i +yxj + zxk, is the portion of the plane 6x+3y+2z=6 in the rst octant oriented by upward unit normals.