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Gradient field in Euclidean spaces

Vector fields can be constructed out of scalar fields using the gradient operator (denoted by the del: ∇).

A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that V = ∇ f = (∂ f ∂ x 1, ∂ f ∂ x 2, ∂ f ∂ x 3, …, ∂ f ∂ x n). {\displaystyle V=\nabla f=\left({\frac {\partial f}{\partial x_{1}}},{\frac {\partial f}{\partial x_{2}}},{\frac {\partial f}{\partial x_{3}}},\dots,{\frac {\partial f}{\partial x_{n}}}\right).}

The associated flow is called the gradient flow, and is used in the method of gradient descent.

The path integral along any closed curve γ (γ (0) = γ (1)) in a conservative field is zero: ∮ γ V (x) ⋅ d x = ∮ γ ∇ f (x) ⋅ d x = f (γ (1)) − f (γ (0)). {\displaystyle \oint _{\gamma }V(\mathbf {x})\cdot \mathrm {d} \mathbf {x} =\oint _{\gamma }\nabla f(\mathbf {x})\cdot \mathrm {d} \mathbf {x} =f(\gamma (1))-f(\gamma (0)).}