Questions from General Calculus

Q: If C is any piecewise-smooth simple closed plane curve and

If C is any piecewise-smooth simple closed plane curve and f and t are differentiable functions, show that ∫C f (x) dx + g (y) dy = 0

Q: If f and t are twice differentiable functions, show that ∇

If f and t are twice differentiable functions, show that ∇2(f g) = f ∇2g + g ∇2 f + 2 ∇f ∙ ∇g

Q: If f is a harmonic function, that is, ∇2

If f is a harmonic function, that is, ∇2 f = 0, show that the line integral ∫ fy dx - fx dy is independent of path in any simple region D.

Q: Find the work done by the force field F (x,

Find the work done by the force field F (x, y, z) = z i + x j + y k in moving a particle from the point (3, 0, 0) to the point (0, π/2, 3) along (a). a straight line (b). the helix x = 3 cos t, y = t,...

Q: Find the area of the part of the surface z = x2

Find the area of the part of the surface z = x2 + 2y that lies above the triangle with vertices (0, 0), (1, 0), and (1, 2).

Q: Suppose F is a vector field on R3. (a

Suppose F is a vector field on R3. (a). Define curl F. (b). Define div F. (c). If F is a velocity field in fluid flow, what are the physical interpretations of curl F and div F?

Q: Show that F is a conservative vector field. Then find a

Show that F is a conservative vector field. Then find a function f such that F = ∇f.

Q: Verify that Green’s Theorem is true for the line integral ∫C

Verify that Green’s Theorem is true for the line integral ∫C xy2 dx - x2 y dy, where C consists of the parabola y = x2 from (-1, 1) to (1, 1) and the line segment from (1, 1) to (-1, 1).

Q: The vector field F is shown in the xy-plane and

The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a). Is div F positive, negative, or...

Q: The vector field F is shown in the xy-plane and

The vector field F is shown in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (a). Is div F positive, negative, or...