6.3 Conservative vector fields  (Page 4/16)

Recall that the reason a conservative vector field F is called “conservative” is because such vector fields model forces in which energy is conserved. We have shown gravity to be an example of such a force. If we think of vector field F in integral ${\displaystyle {\oint }_C\text{F}·d\text{r}}$ as a gravitational field, then the equation ${\displaystyle {\oint }_C\text{F}·d\text{r}}=0$ follows. If a particle travels along a path that starts and ends at the same place, then the work done by gravity on the particle is zero.

The second important consequence of the Fundamental Theorem for Line Integrals is that line integrals of conservative vector fields are independent of path—meaning, they depend only on the endpoints of the given curve, and do not depend on the path between the endpoints.

The second consequence is stated formally in the following theorem.

Proof

Let D denote the domain of F and let $C_1$ and $C_2$ be two paths in D with the same initial and terminal points ( [link] ). Call the initial point $P_1$ and the terminal point $P_2.$ Since F is conservative, there is a potential function $f$ for F . By the Fundamental Theorem for Line Integrals,

Therefore, ${\displaystyle {\int }_{C_1}\text{F}·d\text{r}}={\displaystyle {\int }_{C_2}\text{F}·d\text{r}}$ and F is independent of path.

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To visualize what independence of path means, imagine three hikers climbing from base camp to the top of a mountain. Hiker 1 takes a steep route directly from camp to the top. Hiker 2 takes a winding route that is not steep from camp to the top. Hiker 3 starts by taking the steep route but halfway to the top decides it is too difficult for him. Therefore he returns to camp and takes the non-steep path to the top. All three hikers are traveling along paths in a gravitational field. Since gravity is a force in which energy is conserved, the gravitational field is conservative. By independence of path, the total amount of work done by gravity on each of the hikers is the same because they all started in the same place and ended in the same place. The work done by the hikers includes other factors such as friction and muscle movement, so the total amount of energy each one expended is not the same, but the net energy expended against gravity is the same for all three hikers.

We have shown that if F is conservative, then F is independent of path. It turns out that if the domain of F is open and connected, then the converse is also true. That is, if F is independent of path and the domain of F is open and connected, then F is conservative. Therefore, the set of conservative vector fields on open and connected domains is precisely the set of vector fields independent of path.