Questions from General Calculus

Q: Evaluate the surface integral ∫∫S F ∙ dS for the

Evaluate the surface integral ∫∫S 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) orientat...

Q: Let r = x i + y j + z k and

Let r = x i + y j + z k and r = |r |. Verify each identity. (a). ∇r = r/r (b). ∇ × r = 0 (c). ∇ (1/r) = -r/r3 (d). = ln r = r/r2

Q: Evaluate the surface integral. ∫∫S (x + y

Evaluate the surface integral. ∫∫S (x + y + z) dS, S is the parallelogram with parametric equations x = u + v, y = u - v, z = 1 + 2u + v, 0 < u < 2, 0 < v < 1

Q: Use Stokes’ Theorem to evaluate ∫∫S curl F  dS.

Use Stokes’ Theorem to evaluate ∫∫S curl F  dS. F (x, y, z) = exy i + exz j + x2z k, S is the half of the ellipsoid 4x2 + y2 + 4z2 = 4 that lies to the right of the xz-plane, oriented in the directio...

Q: Use Stokes’ Theorem to evaluate ∫C F ∙ dr.

Use Stokes’ Theorem to evaluate ∫C F ∙ dr. In each case C is oriented counterclockwise as viewed from above. F (x, y, z) = (x + y2) i + (y + z2) j + (z + x2) k, C is the triangle with vertices (1, 0,...

Q: Use Stokes’ Theorem to evaluate ∫C F ∙ dr.

Use Stokes’ Theorem to evaluate ∫C F ∙ dr. In each case C is oriented counterclockwise as viewed from above. F (x, y, z) = i + (x + yz) j + (xy - √z) k, C is the boundary of the part of the plane 3x...

Q: Evaluate the surface integral. ∫∫S x2yz dS, S

Evaluate the surface integral. ∫∫S x2yz dS, S is the part of the plane z = 1 + 2x + 3y that lies above the rectangle [0, 3g] × [0, 2]

Q: Verify that the Divergence Theorem is true for the vector field F

Verify that the Divergence Theorem is true for the vector field F on the region E. F (x, y, z) = 3x i + xy j + 2xz k, E is the cube bounded by the planes x = 0, x = 1, y = 0, y = 1, z = 0, and z = 1

Q: Use the Divergence Theorem to calculate the surface integral ∫∫S F

Use the Divergence Theorem to calculate the surface integral ∫∫S F dS; that is, calculate the flux of F across S. S is the surface of the tetrahedron enclosed...

Q: (a). What is a conservative vector field? (

(a). What is a conservative vector field? (b). What is a potential function?