1.
Compute \(\int_{C_1}\vec{F}\cdot d\vec{r}\)
Answer.0
Section 3.1 Intro to Conservative Vector Fields
In this video we define the basic idea of a conservative vector field, and compute out an example modelling gravity that shows this is a conservative vector field.
Learning Objectives:
- Define a conservative vector field
- Demonstrate a field is conservative by computing line integrals along a generic path (note: this method is only possible in specially chosen cases - a more general approach will come later)
Post-Video Activities Post-Video Activities
Demonstrate that the spin field \(\vec{F}=-y\hat{i}+x\hat{j}\) is not conservative. We will do this by taking two curves from the point (1,0) to (-1,0). The first \(C_1\) is the straight line. The second \(C_2\)goes counterclockwise along the top half of the unit circle.
2.
Compute \(\int_{C_2}\vec{F}\cdot d\vec{r}\)
Answer.\(\pi\)
3.
Why does the above two computations demonstrate the field is not conservative?
4.
Suppose you computed the same result along two different curves. Would that be enough to deduce it was conservative?
