Questions from General Calculus

Q: Match the functions f with the plots of their gradient vector fields

Match the functions f with the plots of their gradient vector fields labeled I–IV. Give reasons for your choices. f (x, y) = sin √ (x^2 + y^2)

Q: Sketch the vector field F by drawing a diagram like Figure 5

Sketch the vector field F by drawing a diagram like Figure 5 or Figure 9. Figure 9: F (x, y) = 0.3 i - 0.4 j

Q: A thin wire has the shape of the first-quadrant part

A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a. If the density function is ρ (x, y) = kxy, find the mass and center of mass of the wire.

Q: (a). Write the formulas similar to Equations 4 for the

(a). Write the formulas similar to Equations 4 for the center of mass (z ̅, y ̅, z ̅) of a thin wire in the shape of a space curve C if the wire has density function ρ (x, y, z). (b). Find the center...

Q: Find the mass and center of mass of a wire in the

Find the mass and center of mass of a wire in the shape of the helix x = t, y = cos t, z = sin t, 0 < t < 2π, if the density at any point is equal to the square of the distance from the origin.

Q: If a wire with linear density ρ (x, y)

If a wire with linear density ρ (x, y) lies along a plane curve C, its moments of inertia about the x- and y-axes are defined as Find the moments of inertia for the wire in Example 3.

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

Find the work done by the force field F (x, y) = x i + (y + 2) j in moving an object along an arch of the cycloid r(t) = (t - sin t) i + (1 - cos t) j 0 < t < 2π

Q: Evaluate the line integral. ∫C yz cos x ds

Evaluate the line integral. ∫C yz cos x ds, C: x = t, y = 3 cos t, z = 3 sin t, 0 < t < π

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 x2y2 dx 1 xy dy, C consists of the arc of the parabola y = x2 from (0, 0) to (1, 1) and the line segments fro...

Q: Use Green’s Theorem to evaluate the line integral along the given positively

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫C yex dx + 2ex dy, C is the rectangle with vertices (0, 0), (3, 0), (3, 4), and (0, 4)