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(1) Evaluate the integral of the normal component of F = 2xyi - yj around the...
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly,...
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
xi+ yj + zk 3. Given the vector field in space F(x, y, z) = or...
xi+ yj + zk 3. Given the vector field in space F(x, y, z) = or more conveniently, (.x2 + y2 + 22)3/2 1 Fr) where r = xi + yj + zk and r= ||1|| = x2 + y2 + x2 (instead of p) 73 r (a) [10 pts) Find the divergence of F, that is, V.F. (b) (10 pts] Directly evaluate the surface integral [/F F.Nds where S is the unit sphere 22 + y2 + z2 1...
zi+yj + zk 3. Given the vector field in space F(x, y, z) or more conveniently,...
zi+yj + zk 3. Given the vector field in space F(x, y, z) or more conveniently, (x2 + y2 + 22)3/2 f where r = ci + yj + zk and r= |||| = V2 + y2 + z2 (instead of p) 1 F(r) = r2 (a) [10 pts] Find the divergence of F, that is, V.F. (b) (10 pts] Directly evaluate the surface integral lle F.NDS where S is the unit sphere x2 + y2 + z2 = 1...
Evaluate the surface integral of IjF.dS where F-xi+yj-zk and σ is the (a) 2, oriented by a downward portion of z-Vx2+y which lies between z1 and normal. Evaluate the surface integral of IjF.dS w...
Evaluate the surface integral of IjF.dS where F-xi+yj-zk and σ is the (a) 2, oriented by a downward portion of z-Vx2+y which lies between z1 and normal.
Evaluate the surface integral of IjF.dS where F-xi+yj-zk and σ is the (a) 2, oriented by a downward portion of z-Vx2+y which lies between z1 and normal.
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk,...
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk, S is the part of the paraboloid z - x2 + y2, 251 (b) F(x, y, z) = (z, x-z, y), S is the triangle with vertices (1,0,0), (0, 1,0), and (0,0,1), oriented downward (c) F-(y. -x,z), S is the upward helicoid parametrized by r(u, v) = (UCOS v, usin v,V), osus 2, OSVS (Hint: Tu x Ty = (sin v, -cos v, u).)...
13. Use the divergence theorem to evaluate Sis Fonds where F(x, y, z) - Xi+yj+zk and...
13. Use the divergence theorem to evaluate Sis Fonds where F(x, y, z) - Xi+yj+zk and S is the unit cube in the first octant bounded by the planes x-0, x= 1, y = 0, y - 1,2-0, z - 1. The vector n is the unit outward normal to the cube.
Use Stokes' Theorem to evaluate the line integral $cF. dr, where F(x, y, z) = xyzi+yj...
Use Stokes' Theorem to evaluate the line integral $cF. dr, where F(x, y, z) = xyzi+yj + zk. S is the surface 3x + 4y + 2z = 12 in the first octant, and is the boundary of S with counterclockwise orientation (from above).
(1 point) Use Stokes' Theorem to evaluate / (2xyi + zj+ 3yk) dr where C is the intersection of th...
(1 point) Use Stokes' Theorem to evaluate / (2xyi + zj+ 3yk) dr where C is the intersection of the plane x z 8 and the cylinder x2 y9oriented counterclockwise as viewed from above. Since the ellipse is oriented counterclockwise as viewed from above the surface we attach is oriented upwards curl(2xyi+zj +3yk)- 2,0,-2x The easiest surface to attach to this curve is the interior of the cylinder that lies on the plane x + z-8. Using this surface in...
15. Use the Divergence Theorem to evaluate the surface integral F dS triple iterated integral where...
15. Use the Divergence Theorem to evaluate the surface integral F dS triple iterated integral where as a F= (-2rz 2yz, -ry,-xy 2rz - yz) and E is boundary of the rectangular box given by -1< x< 3, -1<y< 3 and z2 1
15. Use the Divergence Theorem to evaluate the surface integral F dS triple iterated integral where as a F= (-2rz 2yz, -ry,-xy 2rz - yz) and E is boundary of the rectangular box given by -1
Use the surface integral in Stokes' Theorem to calculate the circulation of the field F=x^2i+5xj+z^2k around...
Use the surface integral in Stokes' Theorem to calculate the circulation of the field F=x^2i+5xj+z^2k around the curve C: the ellipse 16x^2 + y^2 = 1 in the xy-plane, counterclockwise when viewed from above.
Q1. Evaluate the line integral f (x2 + y2)dx + 2xydy by two methods a) directly, b) using Green's Theorem, where C consists of the arc of the parabola y = x2 from (0,0) to (2,4) and the line segments from (2,4) to (0,4) and from (0,4) to (0,0). [Answer: 0] Q2. Use Green's Theorem to evaluate the line integral $. F. dr or the work done by the force field F(x, y) = (3y - 4x)i +(4x - y)j...
xi+ yj + zk 3. Given the vector field in space F(x, y, z) = or more conveniently, (.x2 + y2 + 22)3/2 1 Fr) where r = xi + yj + zk and r= ||1|| = x2 + y2 + x2 (instead of p) 73 r (a) [10 pts) Find the divergence of F, that is, V.F. (b) (10 pts] Directly evaluate the surface integral [/F F.Nds where S is the unit sphere 22 + y2 + z2 1...
zi+yj + zk 3. Given the vector field in space F(x, y, z) or more conveniently, (x2 + y2 + 22)3/2 f where r = ci + yj + zk and r= |||| = V2 + y2 + z2 (instead of p) 1 F(r) = r2 (a) [10 pts] Find the divergence of F, that is, V.F. (b) (10 pts] Directly evaluate the surface integral lle F.NDS where S is the unit sphere x2 + y2 + z2 = 1...
Evaluate the surface integral of IjF.dS where F-xi+yj-zk and σ is the (a) 2, oriented by a downward portion of z-Vx2+y which lies between z1 and normal.
Evaluate the surface integral of IjF.dS where F-xi+yj-zk and σ is the (a) 2, oriented by a downward portion of z-Vx2+y which lies between z1 and normal.
2. Evaluate the surface integral [[Fids. (a) F(x, y, z) - xi + yj + 2zk, S is the part of the paraboloid z - x2 + y2, 251 (b) F(x, y, z) = (z, x-z, y), S is the triangle with vertices (1,0,0), (0, 1,0), and (0,0,1), oriented downward (c) F-(y. -x,z), S is the upward helicoid parametrized by r(u, v) = (UCOS v, usin v,V), osus 2, OSVS (Hint: Tu x Ty = (sin v, -cos v, u).)...
13. Use the divergence theorem to evaluate Sis Fonds where F(x, y, z) - Xi+yj+zk and S is the unit cube in the first octant bounded by the planes x-0, x= 1, y = 0, y - 1,2-0, z - 1. The vector n is the unit outward normal to the cube.
Use Stokes' Theorem to evaluate the line integral $cF. dr, where F(x, y, z) = xyzi+yj + zk. S is the surface 3x + 4y + 2z = 12 in the first octant, and is the boundary of S with counterclockwise orientation (from above).
(1 point) Use Stokes' Theorem to evaluate / (2xyi + zj+ 3yk) dr where C is the intersection of the plane x z 8 and the cylinder x2 y9oriented counterclockwise as viewed from above. Since the ellipse is oriented counterclockwise as viewed from above the surface we attach is oriented upwards curl(2xyi+zj +3yk)- 2,0,-2x The easiest surface to attach to this curve is the interior of the cylinder that lies on the plane x + z-8. Using this surface in...
15. Use the Divergence Theorem to evaluate the surface integral F dS triple iterated integral where as a F= (-2rz 2yz, -ry,-xy 2rz - yz) and E is boundary of the rectangular box given by -1< x< 3, -1<y< 3 and z2 1
15. Use the Divergence Theorem to evaluate the surface integral F dS triple iterated integral where as a F= (-2rz 2yz, -ry,-xy 2rz - yz) and E is boundary of the rectangular box given by -1
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