Questions from General Calculus

Q: Let H be the hemisphere x2 + y2 + z2 = 50

Let H be the hemisphere x2 + y2 + z2 = 50, z > 0, and suppose f is a continuous function with f (3, 4, 5) = 7, f (3, -4, 5) = 8, f -3, 4, 5) = 9, and f (-3, -4, 5) = 12. By dividing H into four patche...

Q: Show that F is conservative and use this fact to evaluate ∫

Show that F is conservative and use this fact to evaluate ∫C F ∙ dr along the given curve.

Q: Identify the surface with the given vector equation. r (

Identify the surface with the given vector equation. r (u, v) = u2 i + u cos v j + u sin v k

Q: Evaluate ∫C (y + sin x) dx + (

Evaluate ∫C (y + sin x) dx + (z2 + cos y) dy + x3 dz where C is the curve r (t) =〈sin t, cos t, sin 2t〉, 0 < t < 2π. [Hint: Observe that C lies on the surface z = 2xy.]

Q: Evaluate the surface integral. ∫∫S (x2 + y2

Evaluate the surface integral. ∫∫S (x2 + y2 + z2) dS, S is the part of the cylinder x2 + y2 = 9 between the planes z = 0 and z = 2, together with its top and bottom disks

Q: (a). If C is the line segment connecting the point

(a). If C is the line segment connecting the point (x1, y1) to the point (x2, y2), show that (b). If the vertices of a polygon, in counterclockwise order, are (x1, y1), (x2, y2), . . . , (xn , yn),...

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: 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) = -2yz i + y j + 3x k, S is the part of the paraboloid z = 5 - x2 - y2 that lies above the plane z = 1, orie...

Q: Match the equations with the graphs labeled I–VI and give

Match the equations with the graphs labeled I–VI and give reasons for your answers. Determine which families of grid curves have u constant and which have v constant.

Q: Let C be a simple closed smooth curve that lies in the

Let C be a simple closed smooth curve that lies in the plane x 1 y + z = 1. Show that the line integral ∫C z dx - 2x dy + 3ydz depends only on the area of the region enclosed by C and not on the shape...