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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.
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...
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...
7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-< x2 sin(z), y2, xy >, and s is the part of the paraboloid z = 1-2-1/2 that lies above the xy-plane. 7. (a) State Stoke's Theorem. (b) Use Stoke's theorem to evaluate curl(F)d where F(x, y, z)-, and s is the part of the paraboloid z = 1-2-1/2 that lies above the xy-plane.
Use Stokes' Theorem to evaluate. 8. Use Stokes, Theorem to evaluate J, ▽ x ที่ do, where F(x, y, z)-(z2yz,yz2,23ezy and s is part of the sphere x2 + y2 + z-5 that lies above the plane z-1. Also, s is oriented upward. 8. Use Stokes, Theorem to evaluate J, ▽ x ที่ do, where F(x, y, z)-(z2yz,yz2,23ezy and s is part of the sphere x2 + y2 + z-5 that lies above the plane z-1. Also, s is oriented...
Let F = { – 4y, 16z, xạy?). Use Stokes Theorem to evaluate S[ curlſ - aš, where S is the part of the paraboloid z = = 36 x2 - y2 that lies above the xy-plane answer = 7T
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...
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
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...
Let S be the surface of the 'liptic paraboloid z = 4 - 22 - y2 above the plane z = 0, and with upward orientation. Let F =< yetan(z), -xcos > be a vector field in R3. 9 + Use Stoke's Theorem to compute: Sf curlĒ. ds. S