Questions from General Calculus

Q: Find (a) the curl and (b) the divergence

Find (a) the curl and (b) the divergence of the vector field. F (x, y, z) = xy2z2 i + x2yz2 j + x2y2z k

Q: Evaluate the surface integral. ∫∫S xz dS, S

Evaluate the surface integral. ∫∫S xz dS, S is the part of the plane 2x + 2y + z = 4 that lies in the first octant

Q: Evaluate the surface integral. ∫∫S x dS, S

Evaluate the surface integral. ∫∫S x dS, S is the triangular region with vertices (1, 0, 0), (0, -2, 0), and (0, 0, 4)

Q: Evaluate the surface integral. ∫∫S y dS, S

Evaluate the surface integral. ∫∫S y dS, S is the surface z = 2/3 (x3/2 + y3/2), 0 < x < 1, 0 < y < 1

Q: Evaluate the surface integral. ∫∫S z2 dS, S

Evaluate the surface integral. ∫∫S z2 dS, S is the part of the paraboloid x = y2 + z2 given by 0 < x < 1

Q: Evaluate the surface integral. ∫∫S y2z2 dS, S

Evaluate the surface integral. ∫∫S y2z2 dS, S is the part of the cone y = √x2 + z2 given by 0 < y < 5

Q: Verify that Stokes’ Theorem is true for the given vector field F

Verify that Stokes’ Theorem is true for the given vector field F and surface S. F (x, y, z) = y i + z j + x k, S is the hemisphere x2 + y2 + z2 = 1, y > 0, oriented in the direction of the positive y-...

Q: State the Divergence Theorem.

State the Divergence Theorem.

Q: Evaluate the surface integral. ∫∫S y2 dS, S

Evaluate the surface integral. ∫∫S y2 dS, S is the part of the sphere x2 + y2 + z2 = 1 that lies above the cone z = √x2 + y2

Q: Evaluate the surface integral. ∫∫S (x2z + y2z

Evaluate the surface integral. ∫∫S (x2z + y2z) dS, S is the hemisphere x2 + y2 + z2 = 4, z > 0