MathLiveGradient theorem
The line integral of the gradient of a scalar field along a curve equals the difference of the field's values at the curve's endpoints. It shows that such integrals depend only on the endpoints, not on the path taken.
\int_{\boldsymbol{c}} \nabla f \cdot \mathrm{d}\boldsymbol{s} = f(\boldsymbol{c}(b)) - f(\boldsymbol{c}(a))What the symbols mean
| scalar field (differentiable function of position) | |
| gradient of f | |
| smooth curve parametrized over the interval [a, b] | |
| infinitesimal displacement vector along the curve | |
| parameter value at the start of the curve | |
| parameter value at the end of the curve |
About this formula
- FieldMathematics
- TopicVector calculus
- LevelUniversity
- Also calledline integral of the gradient of a function, fundamental theorem of calculus for line integrals, fundamental theorem for line integrals, path independence of gradient fields, line integral of a conservative field, gradient theorem for line integrals
- SourceWikidata Q287347
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