Evaluate c Fdr where given vector field F yi-xj + (x2 +y2)k and the curve C is parameterized by
Evaluate the surface integral F.ds for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation F(x, y, z) = yi - xj + Szk, S is the hemisphere x2 + y2 + z2 = 4, z 20, oriented downward -8751 x
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized by (u,v)-(ucos v, u sin v, hu) x2+y2 a at height h above the xy-plane Z = a V 0<vsa, OSvs 2n, and as is the curve parametrized by ē(f) =(acost,asint, h), 0sis27 as x2+ a
3. Verify Stokes' Theorem for the vector field F(x, y, z)= (x2)ĩ+(y2)]+(-xy)k where S is the surface of the cone +y parametrized...
(a) Find the flux of the vector field F=yi-xjtk across the surface σ which is 4. x2 +y2 and below z the portion of z 4 and is oriented by the outward normal. _t7г (b) Use Stokes' Theorem to evaluate the line integral of J F.dr of F--уз ì_x3 j+(x+z)k where C is the clockwise path along the triangle with vertices (0,0,0). (1.0,0)and (1.i.o) aong the thiangle with(i) t)
(a) Find the flux of the vector field F=yi-xjtk across the...
(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 corresponding to 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 which is given by the disk
(2) Let F zi + xj+yk...
6-Let F(x, y,z) = yi - xj+zx°y?k. Evaluate (V x F) dS where S is the surfacex2232 = 1, z < 0 oriented by the upward- pointing unit normal.
6-Let F(x, y,z) = yi - xj+zx°y?k. Evaluate (V x F) dS where S is the surfacex2232 = 1, z
Help Given the following vector field and orientated curve C, evaluate • SF.Tds. c F= (x,y) on the curve r(t) = (12,612), for 1sts3 (x² + y2) 312 The value of the line integral of F over C is (Type an exact answer, using radicals as needed.)
Show that vector field F(x,y) = 2x cos yi + (1 - zsiny) is a gradient field and then find the function f(x,y) such that F = VS. Use it to evaluate line integral SF. dr where the curve C is the arc of the circle 12 + y2 = 4 from (2,0) to (0,2)
1/ Given the vector field F and curve C with vector valued function rt() . How to evaluate the line integral C F dr ? a) Conservative field F and curve C from t = a to t = b, where a b . b) Non conservative field F and curve C from t = a to t = b, where a b . c) Non conservative field F and closed curve C. d) Conservative field F and closed curve...
Consider the given vector field. F(x, y, z) = (9 / sqrt(x2 + y2 + z2)) (x i + y j + z k) Find the curl of the vector field. Then find Divergence
(a3, y3,4z3). Let Si be the disk in the 12. Consider the vector field in space given by F(x, y, z) xy-plan described by x2 + y2 < 4, z = 0; and let S2 be the upper half of the paraboloid given by z 4 y2, z 2 0. Both Si and S2 are oriented upwards. Let E be the solid region enclosed by S1 and S2 (a) Evaluate the flux integral FdS (b) Calculate div F div F...