Mathematics · University
Conservative-field line integral · formula reference
This applies when the vector field is the gradient of a single-valued potential on the path.
Formula
This applies when the vector field is the gradient of a single-valued potential on the path.
∫C ∇φ · dr = φ(B) − φ(A)
Worked example
Let F = ∇φ with φ(x,y) = 1x² + 6y². Find the line integral of F from (0,0) to (3,9). Round the final result to 4 decimal places if needed.
The fundamental theorem for line integrals gives φ(3,9) − φ(0,0).
495
Prerequisites and checking
Algebra; calculus or linear algebra as appropriate to the topic. Keep units and rounding consistent. State the model conditions before applying the formula.
Further reading and review references
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