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Let F(x, y, z) = (yza, x, xy +z) and answer the following questions. Show all...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?,...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top
hemisphere of the unit sphere oriented upward, and C the unit
circle in the xy-plane with positive orientation.
(a) Compute div(F) and curl(F).
(b) Is F conservative? Briefly explain.
(c) Use Stokes’ Theorem to compute ? F · dr by converting it to
a surface integral. (The integral is easy if C
you set it up correctly)
4. (23 pts)...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Begin with the paraboloid z = x2 + y², for 0 < < 4, and slice...
Begin with the paraboloid z = x2 + y², for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 < 0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2x + y, 2x + 2,2y + x>. ZA С 4. w S z...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sk...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr
10. Stokes' Theorem and Surface Integrals...
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice...
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2z+y, 2x + z, 2y + 2 > ZA C 4 S 2 = x2...
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of cu...
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y, Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)ーーーーーーーーーーーーー Compute N(u,v) e. Use Stokes' Theorem to compute Jc F dr
10. Stokes Theorem and Surface...
(c) Let F be the vector field on R given by F(x, y, z) = (2x...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
ie Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = x2 sin(z)i +...
ie Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = x2 sin(z)i + y2 + xyk, S is the part of the paraboloid z = upward. - x2 - y2 that lies above the xy-plane, oriented
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisph...
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
10. Stokes' Theorem and Surfac e Integrals of Vector Fields a. Stokes' Theorem: F-dr= b. Let S be th ky-plane. Draw a sketch of curve C in the xy-plane. et be the surface of the paraboloi...
10. Stokes' Theorem and Surfac e Integrals of Vector Fields a. Stokes' Theorem: F-dr= b. Let S be th ky-plane. Draw a sketch of curve C in the xy-plane. et be the surface of the paraboloid z 4-x-y and Cis the trace of S in the c Let Fox.y.z) <2z, x, y>, Compute the curl (F) d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) F-dr Use Stokes' Theorem to compute , e.
10. Stokes' Theorem and Surfac...
(23 pts) Let F(x, y, z) = ?x + y, x + y, x2 + y2?, S be the top
hemisphere of the unit sphere oriented upward, and C the unit
circle in the xy-plane with positive orientation.
(a) Compute div(F) and curl(F).
(b) Is F conservative? Briefly explain.
(c) Use Stokes’ Theorem to compute ? F · dr by converting it to
a surface integral. (The integral is easy if C
you set it up correctly)
4. (23 pts)...
Questions. Please show all work. 1. Consider the vector field F(x, y, z) (-y, x-z, 3x + z)and the surface S, which is the part of the sphere x2 + y2 + z2 = 25 above the plane z = 3. Let C be the boundary of S with counterclockwise orientation when looking down from the z-axis. Verify Stokes' Theorem as follows. (a) (i) Sketch the surface S and the curve C. Indicate the orientation of C (ii) Use the...
Begin with the paraboloid z = x2 + y², for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 < 0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2x + y, 2x + 2,2y + x>. ZA С 4. w S z...
10. Stokes' Theorem and Surface Integrals of Vector Fields a. Stokes' Theorem: F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y?». Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) e. Use Stokes' Theorem to computec F dr
10. Stokes' Theorem and Surface Integrals...
Begin with the paraboloid z = 22 + y2 for 0 < < 4, and slice it with the plane y = 0. Let S be the surface that remains for y> 0 (excluding the planar surface in xz-plane) oriented downward (i.e. n3 <0). Let C be the Semicircle and the line segment in the plane z = 4 with counterclockwise orientation and F =< 2z+y, 2x + z, 2y + 2 > ZA C 4 S 2 = x2...
10. Stokes Theorem and Surface Integrals of Vector Fields a Stokes Theorem:J F dr- b. Let S be the surface of the paraboloid z 4-x2-y2 and C is the trace of S in the xy-plane. Draw a sketch of curve C in the xy-plane. Let F(x,y,z) = <2z, x, y, Compute the curl (F) c. d. Find a parametrization of the surface S: G(u,v)ーーーーーーーーーーーーー Compute N(u,v) e. Use Stokes' Theorem to compute Jc F dr
10. Stokes Theorem and Surface...
(c) Let F be the vector field on R given by F(x, y, z) = (2x +3y, z, 3y + z). (i) Calculate the divergence of F and the curl of F (ii) Let V be the region in IR enclosed by the plane I +2y +z S denote the closed surface that is the boundary of this region V. Sketch a picture of V and S. Then, using the Divergence Theorem, or otherwise, calculate 3 and the XY, YZ...
ie Use Stokes' Theorem to evaluate curl F. ds. F(x, y, z) = x2 sin(z)i + y2 + xyk, S is the part of the paraboloid z = upward. - x2 - y2 that lies above the xy-plane, oriented
Consider the vector field F(x, y, z) -(z,2x, 3y) and the surface z- /9 - x2 -y2 (an upper hemisphere of radius 3). (a) Compute the flux of the curl of F across the surface (with upward pointing unit normal vector N). That is, compute actually do the surface integral here. V x F dS. Note: I want you to b) Use Stokes' theorem to compute the integral from part (a) as a circulation integral (c) Use Green's theorem (ie...
10. Stokes' Theorem and Surfac e Integrals of Vector Fields a. Stokes' Theorem: F-dr= b. Let S be th ky-plane. Draw a sketch of curve C in the xy-plane. et be the surface of the paraboloid z 4-x-y and Cis the trace of S in the c Let Fox.y.z) <2z, x, y>, Compute the curl (F) d. Find a parametrization of the surface S: G(u,v)- Compute N(u,v) F-dr Use Stokes' Theorem to compute , e.
10. Stokes' Theorem and Surfac...
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