Conservative Vector Fields

Vector fields, that are the gradient field of a differentiable function, are described with as conservative.

Line Integrals in Conservative Field

If $C$ is a smooth curve joining $A$ and $B$ parametrized by $r(t)$ and $F$ a gradient vector of the differentiable function $f$ on a domain $D$ containing $C$, then

$$\int \_CF\cdot dr=f(B)-f(A)$$

holds.

We denote conservative fields with $\int \_A^B$ instead of $\int \_C$

Properties of Conservative Fields

For a conservative field $F$ that is defined throughout the simply connected region $D$

• $F=\nabla f\text{on}D$
• $\nabla \times F=0$ throughout $D$
• The line integral $\int \_CF\cdot dr$ yields the same value over every path $C$
• Every integral $\oint \_CF\cdot dr=0$ over any closed path in D
• It may be the gradient of a Potential Function