Chapter 16.3 Review

Question 1

When is a vector field F on domain D conservative?

Answer 1

A vector field F on domain D is conservative when there exists a function V such that ∇V=F on D. The function V is called a potential function of F.

Question 2

When is a vector field F on domain D path independent?

Answer 2

A vector field F on domain D is path independent if for any two points P, Qϵ D, we have

∫_c1F*ds= ∫_c2F*ds

for any two paths c₁ and c₂ in domain D from P to Q

Question 3

The fundamental theorem for conservative Vector Fields

Answer 3

If F=∇V, then ∫cF*ds=V(Q)-V(P) for any path c from P to Q in the domain of F. This shows that conservative fector fields are path-independent.

Question 4

what happens if c is a closed path?

Answer 4

In particular, if c is a closed path (P=Q), then ∫_c1F*ds=0

Question 5

What happens on an open connected domain?

Answer 5

On an open connected domain, a path-independent vector field is conservative.

Question 6

Show the cross-partial condition for conservative vector fields please?

Answer 6