Section 16.6
Conservative Vector Fields
Determining if a vector field is a gradient field.
1
Introduction
A vector field is called Conservative if for some scalar function .
If is conservative, line integrals are path-independent, and integrals around closed loops are zero.
2
The Component Test
Clairaut's Theorem Check
If , then must equal . Thus:
In 3D: .
3
Visualizing "Irrotational"
Interactive: Conservative Flow
4
Worked Examples
Example 1: Finding Potential
Determine if is conservative. If so, find .
1. Check Partials:
.
.
Equal! Conservative.
2. Integrate P with respect to x:
.
3. Differentiate wrt y and match Q:
.
Match with .
So .
Result: .
Example 2: Failing the Test
Test .
1. Check Partials:
.
.
2. Conclusion:
. Not conservative.
Calculus: Work depends on path. Physics: Field has "curl".
Example 3: 3D Potential
Find for .
1. Integrate x:
.
2. Match y:
.
.
So far: .
3. Match z:
.
Match with R: . So .
Result: .
5
Practice Quiz
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