Conservative Vector Field - Path Independence

Path Independence

A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that $S\subseteq\mathbb{R}^3$ is a region of three-dimensional space, and that is a rectifiable path in with start point and end point . If $\mathbf{v}=\nabla\varphi$ is a conservative vector field then the gradient theorem states that

This holds as a consequence of the Chain Rule and the Fundamental Theorem of Calculus.

An equivalent formulation of this is to say that

for every closed loop in S. The converse of this statement is also true: if the circulation of v around every closed loop in an open set S is zero, then v is a conservative vector field.

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