Questions from General Calculus

Q: Determine whether or not F is a conservative vector field. If

Determine whether or not F is a conservative vector field. If it is, find a function f such that F = ∇f. F (x, y) = (y2 cos x + cos y) i + (2y sin x - x sin y) j

Q: Evaluate the line integral by two methods: (a) directly

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C y2 dx + x2y dy, C is the rectangle with vertices (0, 0), (5, 0), (5, 4), and (0, 4)

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

Use Green’s Theorem to evaluate ∫C F ∙ dr. (Check the orientation of the curve before applying the theorem.) F (x, y) =〈√ (x^2 + 1), tan^ (-1) x〉, C is the triangle from (0, 0) to (1, 1) to (0, 1) to...

Q: (a) Find a function f such that F = ∇

(a) Find a function f such that F = ∇f and (b) use part (a) to evaluate ∫C F ∙ dr along the given curve C. 12. F (x, y, z) = yz i + xz j + (xy + 2z) k, C is the line segment from (1, 0, -2) to (4, 6...

Q: Show that the line integral is independent of path and evaluate the

Show that the line integral is independent of path and evaluate the integral. ∫C 2xe-y dx + (2y - x2e-y) dy, C is any path from (1, 0) to (2, 1)

Q: Show that there is no vector field G such that curl G

Show that there is no vector field G such that curl G = 2x i + 3yz j - xz2 k

Q: Evaluate the line integral by two methods: (a) directly

Evaluate the line integral by two methods: (a) directly and (b) using Green’s Theorem. ∮C y dx - x dy, C is the circle with center the origin and radius 4

Q: If a circle C with radius 1 rolls along the outside of

If a circle C with radius 1 rolls along the outside of the circle x2 + y2 = 16, a fixed point P on C traces out a curve called an epicycloid, with parametric equations x = 5 cos t - cos 5t, y = 5 sin...

Q: Suppose you’re asked to determine the curve that requires the least work

Suppose you’re asked to determine the curve that requires the least work for a force field F to move a particle from one point to another point. You decide to check first whether F is conservative, an...

Q: Let D be a region bounded by a simple closed path C

Let D be a region bounded by a simple closed path C in the xy-plane. Use Green’s Theorem to prove that the coordinates of the centroid (x, y) of D are where A is the area of D.