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.

Read more about this topic:  Conservative Vector Field

Famous quotes containing the words path and/or independence:

    “Every path but your own is the path of fate. Keep on your own track, then.”
    —Henry David Thoreau (1817–1862)

    “The [nineteenth-century] young men who were Puritans in politics were anti-Puritans in literature. They were willing to die for the independence of Poland or the Manchester Fenians; and they relaxed their tension by voluptuous reading in Swinburne.”
    —Rebecca West (1892–1983)