Conservative Vector Field

Topics: Vector Field

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(definition)

A conservative vector field is a vector field that has one (or several) potential functions.

All conservative fields are irrotational (i.e. their curl is $0$).

Line Integral §

Path Independence §

(theorem)

The line integral of a conservative vector field is path-independent.

That means that given a conservative vector field $F$, its line integral will only depend on the endpoints of the path, regardless of the actual path chosen. That is, if $P_1$ and $P_2$ are any two paths that have the same endpoints:

$${\int }_{P_1}F\cdot ds={\int }_{P_2}F\cdot ds$$

(theorem)

The above expression is equivalent to:

$${\int }_{P_c}F\cdot ds=0$$

…where $P_c$ is a closed path (i.e. its endpoints are one same point).

Fundamental Theorem of Calculus for Line Integrals §

(theorem)

Given $F$ a conservative vector field on a region $R\subseteq {\mathbb{R}}^3$, then the integral of $F$ on a curve in $R$ is equal to:

$${\int }_{c}F\cdot ds=f(B)-f(A)$$

…where $f$ is a potential function of $F$, and $A$ and $B$ are the extreme points of the curve.

Notice that this is basically the fundamental theorem of calculus for line integrals.