Calculus Volume 3 · free preview
Conservative Vector Fields
A field is conservative when its line integrals are path-independent, equivalently the mixed partials' condition holds, and the potential is recovered by integrating along a path, the closed loops' integrals all zero.
1. The test for conservatism
The test for conservatism
2. Recovering the potential
Recovering the potential
3. Path independence and closed loops
Path independence and closed loops
4. The curl's emergence
The curl's emergence
5. Conservatism test
For F = (P, Q) on a simply connected region, the field is conservative when which condition holds?
6. Simply connected necessity
Why is the simply connected condition, no holes, needed for the conservatism test?
7. Potential recovery
To recover the potential of a conservative field, one integrates P with respect to x and then determines the unknown function of y by which step?
8. Path independence content
For a conservative field, the line integral between two points is path-independent, equal to what?
9. Energy conservation
The physics' energy conservation is the conservatism's expression through which quantity?
10. Mixed partials equality
The conservatism test compares P_y with Q_x. As a single word, what must they be? Answer in lowercase.
11. Potential freedom
The potential is determined up to which additive constant? Answer with two words.
12. Closed loop integral
A conservative field's integral around every closed loop is what number? Answer with a number.
13. Three dimensional conservatism
In three dimensions, the field F = (P, Q, R) is conservative when which conditions hold?
14. Rotation field curl
The two-dimensional curl of the rotation field (-y, x) is the constant which number? Answer with a number.
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