Evaluate the line integral in Stokes Theorem to evaluate the surface integral J J(VxF)-n ds. Assu...
Evaluate the line integral - dr by evaluating the surface integral in Stokes Theorem with an appropriate choice of metal Close counterdeckwite orientation when viewed from above С F-6²-82-x².7² -2²) C is the boundary of the square Ix 13. lys 13 in the plane 2 #0 Rewrite the given in integral as a | as a surface integral fro-SSO ds C 5 Evaluate the integral $r..-Type an exact answer
Evaluate the surface integral f(x,y,z) dS using a parametric description of the surface. f(xyz) = 4x, where S is the cylinder X + z2-25, 0 ys2 The value of the surface integral is (Type exact answers, using T as needed.) Find the area of the following surface using the given explicit description of the surface. The cone z2 = (x2 +y2) , for Oszs8 Set up the surface integral for the given function over the given surface S as a...
Verify that the line integral and the surface integral of Stokes Theorem are equal far the following vector field, surface S, and closed curve C. Assume that C has counterlockwise orientation and S has a consistentorientation F = 〈y,-x, 11), s is the upper half of the sphere x2 + y2 +22-1 and C is the circle x2 + y2-1 in the xy-plane Construct the line integral of Stokes' Theorem using the parameterization r(t)= 〈cost, sint, O. for 0 sts2r...
(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...
Evaluate the surface integral f(x,y,z) dS using a parametric description of the surface. 2 f(x,y,z) x 2 where S is the hemisphere x + y +z2 = 25, for z 2 0 The value of the surface integral is (Type an exact answers, using t as needed.) Evaluate the surface integral f(x,y,z) dS using a parametric description of the surface. 2 f(x,y,z) x 2 where S is the hemisphere x + y +z2 = 25, for z 2 0 The...
Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F across the surface in the direction away from the origin. F-3y + (5 - 5x)j + (z? - 2K S: 7,0) = (v10 sin 6 cos 0) (V10 sin sine))+ ( 10 cos •)*, 05058/2,050 2x The flux of the curl of the field F across the surface S in the direction of the outward unit normal nis I (Type an exact...
Verify Stokes' Theorem by evaluating the line integral and the double surface integral. Assume that the surface has an upward orientation. (a) F(x, y, z)= x’i + y²j+z?k; o is the portion of the cone below the plane z=l. (b) 7 (x, y, z)=(z - y){ +(z+x) ș- (x + y)k; o is the portion of the paraboloid z=9-r? - y2 above the xy-plane. [0, 187]
Use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field F = x2 yi + 2y3zj + 5zk across the surface S: r(1,0) =r cos ei +r sin 0j +rk, Osrs3,0 s0s 2n in the direction with a positive k-component for n. The flux of the curl of the field F is (Type an exact answer, using a as needed.)
Use Stokes' Theorem to evaluate the line integral $cF. dr, where F(x, y, z) = (-y+z)i + (x – z)j + (x – y)k. S is the surface z = V1 – 22 – y2, and C is the boundary of S with counterclockwise orientation (from above).
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...