(2) Let F zi + xj+yk and consider the integral vx Fi n dS where S is the surface of the paraboloid z = 1-x2-y2 corr...
(2) Let F-1 + rj + yk and consider the integral- , ▽ × F. т. dS where s is the surface of the paraboloid z = 1-12-y2 corresponding to z 0, and n is a unit normal vector to S in the positive z-direction (a) Apply Stokes' theorem to evaluate the integral. (b) Evaluate the integral directly over the surface S rectlv over the new surface
(2) Let F-1 + rj + yk and consider the integral- , ▽...
Please show full working. Only answer if you know how.
Regards
(2) Let F-~itrj yk and consider the integral JTs ▽ x F·ń dS where s is the surface of the paraboloid z = 1-2.2-y2 corresponding to z > 0, and n is a unit normal vector to S in the positive z-direction. (a) Apply Stokes' theorem to evaluate the integral. (b) Evaluate the integral directly over the surface S (c) Evaluate the integral directly over the new surface S...
Compute in two ways the flux integral ‹ S F~ · N dS ~ for F=
<2y, y, z2> and S the closed surface
formed by the paraboloid z = x2 + y2 and the
disk x2 + y2 ≤ 4 at z = 4. Use divergence
theorem to solve one way, and use SSs F * N ds to solve the other
way. (This is a Calculus 3 problem.)
* 36.3. Compute in two ways the fux integral ф...
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...
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...
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...
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
Verify Stokes, Theorem for the surface S that is the paraboloid given by z = 6-x2-y2 that lies above the plane z 2 (oriented upward) and the vector field F(x, y, z)2yzi+yj+3xk.
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...
2. Consider the conical surface
S={(x,y,z)∈R3 : x2 + y2 =
z2, 0 ≤ z ≤ 1},
and the vector field
(a) Carefully sketch S, and identify its boundary ∂S.
(b) By parametrising S appropriately, directly compute the flux
integral
S (∇ × f) · dS.
(c) By computing whatever other integral is necessary (and
please be careful about explaining any orien- tation/direction
choices you make), verify Stokes’ theorem for this case.
Evaluate the surface integral (x2 + y' +52 ) ds where S is the part of the cone z = 2- x2 + y2 above the z = 0 plane. The surface integral equals
Evaluate the surface integral (x2 + y' +52 ) ds where S is the part of the cone z = 2- x2 + y2 above the z = 0 plane. The surface integral equals