13 Conservative Vector Fields, Greens Theorem, Line Integrals III

Question 1

What is a conservative vector field?

Answer 1

• if a vector field F can be written as a gradient field F = ∇f
• The function f is called the potential function/ potential of vector field F
• Is written b (int)C F * dr = (int) C ∇f * dr

Question 2

What is the fundamental theorem of conservative vector fields?

Answer 2

The integral can be computed as…
(int)C ∇f * dr = f(Q) - f(P)
where q is the initial point on curve C, and Q is final point on curve
path independent. if this equals 0, the circle is a closed loop

Question 3

A conservative vector is notated by…

Answer 3

F(x,y,z) = ∇f(x,y,z) = ⟨∂f/∂x,∂f/∂y,∂f/∂z⟩
The mixed partial derivatives are equal

Question 4

What’s another way to check if a vector field is conservative?

Answer 4

Check if the curl is zero. curl(F) = ∇ X F

Question 5

How do I find a potential function?

Answer 5

Make sure its conservative
Need to find a single function f(x,y,z) where

∂f/∂x = F1
∂f/∂y = F2
∂f/∂z = F3

• find anti derivatives of F1, 2, 3

Question 6

What does Green’s Theorem do?

Answer 6

It relates a vector lie integral around a closed loop to a double integral of the region inside

(int w circle) C F * dr = (int)(int) D (∂F2/∂x - ∂F1/∂y) dA
Circle on integral side means C is a single look with positive orientation (loop moved one in counter clockwise direction)
means that (int w circle) C (F1dx + F2dy) helpful for turning vector line integrals into double integrals

Question 7

What are the conditions in order to use Green’s Theorem?

Answer 7

• Are you computing a vector line integral over simple closed loop?
• Does loop enclose a region you can integrate?
• Do the partial derivatives of the vector field exist over the enclosed region?

Question 8

How do I use Green’s Theorem to write the vector line integral as a double integral?

Answer 8

1. Compute z component Curl of vector field Curlz(F) = ∂F2/∂x - ∂F1/∂y
2. Find bounds of enclosed domain D
3. Write double integral over region D of z component of curl
4. Compute new double integral (gives value of vector line integral, or work done by this field around simple closed loop)
   * If F was conservative, curl would be 0, energy conserved