Questions from General Calculus

Q: Suppose S and E satisfy the conditions of the Divergence Theorem and

Suppose S and E satisfy the conditions of the Divergence Theorem and f is a scalar function with continuous partial derivatives. Prove that These surface and triple integrals of vector functions are...

Q: The surface with parametric equations x = 2 cos θ +

The surface with parametric equations x = 2 cos θ + r cos (θ /2) y = 2 sin θ + r cos (θ /2) z = r sins (θ /2) where -1/2 < r < 12 and 0

Q: Evaluate ∫∫S (x2 + y2 + z2) dS correct

Evaluate ∫∫S (x2 + y2 + z2) dS correct to four decimal places, where S is the surface z = xey, 0 < x < 1, 0 < y < 1.

Q: Find the exact value of ∫∫S xyz dS, where S

Find the exact value of ∫∫S xyz dS, where S is the surface z = x2y2, 0 < x < 1, 0 < y < 2.

Q: Find the value of ∫∫S x2y2z2 dS correct to four decimal

Find the value of ∫∫S x2y2z2 dS correct to four decimal places, where S is the part of the paraboloid z = 3 - 2x2 - y2 that lies above the xy-plane.

Q: Compute the outward flux of F (x, y, z

Compute the outward flux of F (x, y, z) = x i + y j + z k/ (x2 + y2 + z2)3/2 through the ellipsoid 4x2 + 9y2 + 6z2 = 36.

Q: Find the flux of F (x, y, z)

Find the flux of F (x, y, z) = sin (xyz) i + x2y j + z2ex/5 k across the part of the cylinder 4y2 + z2 = 4 that lies above the xy-plane and between the planes x = -2 and x = 2 with upward orientation....

Q: Find an equation of the tangent plane to the given parametric surface

Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane.

Q: Find an equation of the tangent plane to the given parametric surface

Find an equation of the tangent plane to the given parametric surface at the specified point. Graph the surface and the tangent plane.

Q: Find the center of mass of the hemisphere x2 + y2 +

Find the center of mass of the hemisphere x2 + y2 + z2 = a2, z > 0, if it has constant density.