Vector Calculus - Independence of Path and Conservative Vector Fields

14. Vector Calculus

14.1 Vector Fields

14.2 Line Integrals

14.3 Independence of Path and Conservative Vector Fields

Definition 3.1

A region $D \subset R^n$ (for $n \ge 2$) is called connected if every pair of points in $D$ can be connected by a piecewise-smooth curve lying entirely in $D$.

Theorem 14.3 Independence of Path and Conservative Vector Fields 3.1

Suppose that the vector field $F(x, y) = \langle M(x, y), N(x, y) \rangle$ is continuous on the open, connected region $D \subset R^2$. Then, the line integral is independent of path in $D$ if and only if $F$ is conservative on $D$. Recall that a vector field $F$ is conservative whenever $F = \nabla f$, for some scalar function $f$ (called a potential function for $F$).

Theorem 14.3 Independence of Path and Conservative Vector Fields 3.2

Suppose that $F(x, y) = \langle M(x, y), N(x, y) \rangle$ is continuous in the open, connected region $D \subset R^2$ and that $C$ is any piecewise-smooth curve lying in $D$, with initial point $(x<em>1, y</em>1)$ and terminal point $(x<em>2, y</em>2)$. Then, if $F$ is conservative on $D$, with $F(x, y) = \nabla f(x, y)$, we have

Example 14.3 Independence of Path and Conservative Vector Fields 3.1 A Line Integral That Is Independent of Path

Show that for $F(x, y) = \langle 2xy - 3, x^2 + 4y^3 + 5 \rangle$, the line integral is independent of path. Then, evaluate the line integral for any curve $C$ with initial point at $(-1, 2)$ and terminal point at $(2, 3)$.

14.3 Independence of Path and Conservative Vector Fields Closed Curves

We consider a curve $C$ to be closed if its two endpoints are the same. That is, for a plane curve $C$ defined parametrically by $C = {(x, y) | x = g(t), y = h(t), a \le t \le b }$, $C$ is closed if $(g(a), h(a)) = (g(b), h(b))$.

Theorem 14.3 Independence of Path and Conservative Vector Fields 3.3

Suppose that $F(x, y)$ is continuous in the open, connected region $D \subset R^2$. Then $F$ is conservative on $D$ if and only if for every piecewise-smooth closed curve $C$ lying in $D$.

14.3 Independence of Path and Conservative Vector Fields Simply-Connected Regions

A region $D$ is simply-connected if every closed curve in $D$ encloses only points in $D$.

Theorem 14.3 Independence of Path and Conservative Vector Fields 3.4

Suppose that $M(x, y)$ and $N(x, y)$ have continuous first partial derivatives on a simply-connected region $D$. Then, is independent of path in $D$ if and only if $\frac{\partial M}{\partial y}(x, y) = \frac{\partial N}{\partial x}(x, y)$
for all $(x, y)$ in $D$.

Example 14.3 Independence of Path and Conservative Vector Fields 3.2 Testing a Line Integral for Independence of Path

Determine whether or not the line integral is independent of path.

14.3 Conservative Vector Fields

Let $F(x, y) = \langle M(x, y), N(x, y) \rangle$, where we assume that $M(x, y)$ and $N(x, y)$ have continuous first partial derivatives on an open, simply-connected region $D \subset R^2$. The following five statements are equivalent, meaning that for a given vector field, either all five statements are true or all five statements are false.

1. $F(x, y)$ is conservative on $D$.

2. $F(x, y)$ is a gradient field in $D$ (i.e., $F(x, y) = \nabla f(x, y)$, for some potential function $f$, for all $(x, y) \in D$).

3. is independent of path in $D$.

4. for every piecewise-smooth closed curve $C$ lying in $D$.

5. $M<em>y(x, y) = N</em>x(x, y)$, for all $(x, y) \in D$.

14.3 Conservative Force Fields in Space

For a three-dimensional vector field $F(x, y, z)$, we say that $F$ is conservative on a region $D$ whenever there is a scalar function $f(x, y, z)$ for which $F(x, y, z) = \nabla f(x, y, z)$, for all $(x, y, z) \in D$. As in two dimensions, $f$ is called a potential function for the vector field $F$.

Example 14.3 Independence of Path and Conservative Vector Fields 3.3 Showing That a Three-Dimensional Vector Field Is Conservative

Show that the vector field $F(x, y, z) = \langle 4xe^z, \cos y, 2x^2e^z \rangle$ is conservative on $R^3$, by finding a potential function $f$.

Theorem 14.3 Independence of Path and Conservative Vector Fields 3.5

Suppose that the vector field $F(x, y, z)$ is continuous on the open, connected region $D \subset R^3$. Then, the line integral is independent of path in $D$ if and only if the vector field $F$ is conservative on $D$; that is, $F(x, y, z) = \nabla f(x, y, z)$, for all $(x, y, z)$ in $D$, for some scalar function $f$ (a potential function for $F$).

Theorem 14.3 Independence of Path and Conservative Vector Fields 3.5

Further, for any piecewise-smooth curve $C$ lying in $D$, with initial point $(x<em>1, y</em>1, z<em>1)$ and terminal point $(x</em>2, y<em>2, z</em>2)$, we have

14.4 Green’s Theorem

14.5 Curl and Divergence

14.6 Surface Integrals

14.7 The Divergence Theorem

14.8 Stokes' Theorem

14.9 Applications of Vector Calculus