Let F(x, y,z) = < x + y2,y + z2,z + x2 >, let S be a surface with boundary C. C is the triangle with vertices...
Given the following vectors F=[y2, x2,x-z] and surface S: the triangle surface with vertices (0,0,1), (1,0,1), (1,1,1) in first octave. A. Evaluate the surface integral F(F) . dA B. Evaluate the surface integral VxF(F) dA C. Evaluate the line integral F() di where C is the curve enclosing the triangle. (Don't apply Green's theorem and integrate directly) Given the following vectors F=[y2, x2,x-z] and surface S: the triangle surface with vertices (0,0,1), (1,0,1), (1,1,1) in first octave. A. Evaluate the...
Let Ě = < 2x + y², 8y + z2, 6z + x2 >. Use Stokes' Theorem to evaluate so F. dri where C is the triangle with vertices (1,0,0), (0,1,0), and (0,0,1), oriented counterclockwise as viewed from above.
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...
9. Let S be the triangle below with vertices (0,1,0), (1,0,0) and (0,0,1). (For all three of these points x+y+z=1) Let F =(x+y+z)i + (x + y)] +xk. Set up an integral in dy and dx that will calculate the flux of F through S.
1 Use Stokes' theorem to evaluate the integrals: F(x, y, z) dr a) where F(r, y,z)(3yz,e, 22) and C is the boundary of the triangle i the plane y2 with vertices b) where F(x, y,z (-2,2,5xz) and C is in the plane 12- y and is the boundary of the region that lies above the square with vertices (3,5, 0), (3,7,0),(4,5,0), (4,7,0) c) where F(x, y,z(7ry, -z, 3ryz) and C is in the plane y d) where intersected with z...
4. Let F(x, y, z)=(y,x,z2). Let S be the surface of the tetrahedron with the vertices (0,0,0), (2,0,0), (0,2,0), and (0,0,2). Use the divergence theorem to evaluate SS F.dS. (13 points)
Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2 Evaluatef(x, y, z) dS. f(x, y, z) = x2 + y2 +z2
please help with both a and b 16 an Let F(x, y) = (x2 - y2) it (x²+y²); let C be the path that starts at (-1,0), travels a long the xaxis to (1,0) then along the circle ² + y²= 1 counter cockwise back to (-1,0) compute the work down along the path b) Let F(x, y, z) = (x+y)i + (y-2)j + (x2-52)k Lets be the solid tetrahedron in the first octant with vertices (0,0,0), (1,0,0), (0, 1,0)...
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.
(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)...