Questions from General Calculus

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 = 3 (b). = ∙ (r r) = 4r (c). ∇2r3 = 12r

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

Let r = x i + y j + z k and r = |r |. If F = r/rp, find div F. Is there a value of p for which div F = 0?

Q: Use Green’s first identity (Exercise 33) to prove Green’s second

Use Green’s first identity (Exercise 33) to prove Green’s second identity: first identity: where D and C satisfy the hypotheses of Green’s The...

Q: Recall from Section 14.3 that a function t is called

Recall from Section 14.3 that a function t is called harmonic on D if it satisfies Laplace’s equation, that is, ∇2g = 0 on D. Use Green’s first id...

Q: Use Green’s first identity to show that if f is harmonic on

Use Green’s first identity to show that if f is harmonic on D, and if f (x, y) = 0 on the boundary curve C, then ∫∫D |∇f |2 dA...

Q: This exercise demonstrates a connection between the curl vector and rotations.

This exercise demonstrates a connection between the curl vector and rotations. Let B be a rigid body rotating about the z-axis. The rotation can be described by the vector w − k, whe...

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

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

Q: We have seen that all vector fields of the form F =

We have seen that all vector fields of the form F = ∇g satisfy the equation curl F = 0 and that all vector fields of the form F = curl G satisfy the equation div F = 0 (assuming continuity of the appr...

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

Find the work done by the force field F (x, y) = x2 i + yex j on a particle that moves along the parabola x = y2 + 1 from (1, 0) to (2, 1).

Q: Evaluate the line integral. ∫C y3 dx + x2

Evaluate the line integral. ∫C y3 dx + x2 dy, C is the arc of the parabola x = 1 - y2 from (0, -1) to (0, 1)