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The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space rather than just the real line.
For φ : U ⊆ R → R as a differentiable function and γ as any continuous curve in U which starts at a point p and ends at a point q, then
where ∇φ denotes the gradient vector field of φ.
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