Gradient theorem

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. If F is a path-independent vector field, then F is the gradient of some scalar-valued function.Suppose U is an open, path-connected subset of ℝn, and F : U → ℝn is a continuous and path-independent vector field. Fix some element a of U, and define f : U → ℝ by 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. Let φ : U ⊆ ℝn → ℝ and γ is any curve from p to q. Then It is a generalization of the fundamental theorem of calculus to any curve in a plane or space (generally n-dimensional) rather than just the real line. The gradient theorem implies that line integrals through gradient fields are path independent. In physics this theorem is one of the ways of defining a conservative force. By placing φ as potential, ∇φ is a conservative field. Work done by conservative forces does not depend on the path followed by the object, but only the end points, as the above equation shows. The gradient theorem also has an interesting converse: any path-independent vector field can be expressed as the gradient of a scalar field. Just like the gradient theorem itself, this converse has many striking consequences and applications in both pure and applied mathematics. If φ is a differentiable function from some open subset U (of ℝn) to ℝ, and if r is a differentiable function from some closed interval to U, then by the multivariate chain rule, the composite function φ ∘ r is differentiable on (a, b) and for all t in (a, b). Here the ⋅ denotes the usual inner product. Now suppose the domain U of φ contains the differentiable curve γ with endpoints p and q, (oriented in the direction from p to q). If r parametrizes γ for t in , then the above shows that where the definition of the line integral is used in the first equality, and the fundamental theorem of calculus is used in the third equality.