Gradient theorem

Evaluates a line integral through a gradient field using the original scalar field

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 (generally n-dimensional) rather than just the real line. If φ : U ⊆ Rn → R is a differentiable function and γ a differentiable curve in U which starts at a point p and ends at a point q, then ∫ γ ∇ φ ( r) ⋅ d r = φ ( q) − φ ( p) {\displaystyle \int _{\gamma }\nabla \varphi (\mathbf {r})\cdot \mathrm {d} \mathbf {r} =\varphi \left(\mathbf {q} \right)-\varphi \left(\mathbf {p} \right)} where ∇φ denotes the gradient vector field of φ.

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