Conservative Vector Fields
Test for conservative vector fields
Definition: Suppose a curve in (or ) is parametrized by for . Then is a simple curve if for any and with i.e. never intersects itself between its endpoints.
is closed if i.e. the initial and terminal points of are equal.
Definition: A open region in (or ) is connected if it is possible to connect any two points of by a continuous curve lying in .
is simply connected if every closed simple curve in can be deformed and contracted to a point in .
Recall that a vector field is said to be conservative on a region in (or if there exists a function such that . And is called a potential function of .
The following is the main theorem for determining whether a 2D vector field is conservative or not:
Theorem: Let be a 2D vector field defined on a connected and simply connected region , where and have continuous first partial derivatives on . Then is a conservative vector field on if and only if .
Proof: Omitted.
Example: Suppose on . Determine whether is conservative on .
Answer:
is obviously a connected and simply connected region.
and .
They are not equal. Hence is not conservative on .
Example: Suppose on . Show that is conservative on and find its potential function .
Answer:
is obviously a connected and simply connected region.
and .
They are equal. Hence is conservative on .
Let be its potential function. Then we have
Hence , which implies that
, where is a function of .
Since , using the above result, we have
, where is an arbitrary constant.
Hence .
Exercise: Determine whether the vector field is conservative on . Find its potential function if it is conservative.
Fundamental theorem for line integrals
The following theorem is a generalization of Fundamental theorem of calculus (FTC):
Theorem (Fundamental theorem for line integrals) Let be a continuous vector field on an open connected region in (or ). is conservative on i.e. there exists a potential function such that if and only if
for all points and in and all smooth oriented curves from to .
Proof: Suppose is conservative and is its potential function. Then . Therefore, we have
where for is a parametrization of the curve , , and
By chain rule, . Hence, we have
(Note: The second equality is due to FTC.)
Remark:
- This theorem is analogous to FTC - .
- The theorem implies that if is conservative, is path independent i.e. only the endpoints of matter.
- If is a simple closed smooth oriented curve i.e. and is conservative, then
The following theorem states the connections between conservative vector fields, line integrals over closed curves, and path independence:
Theorem: Let be a 2D vector field defined on a connected region . The following statements are equivalent:
- is conservative on .
- for any closed curve in .
- is path independent i.e. the line integral has the same value for any two points and and any curve from to
Applications of line integrals
Work done in a force field
Let be a continuous force field in a region in (or ). Suppose an object is moved by this force field along a curve in which is parametrized by . The work done by to move this object along the curve is
(Note: Recall that if a constant force is applied to an object so that it moves along a straight line with displacement , the work done by the force is . The above definition is a natural generalization to the case when the force is a variable force field and the object is moved along a curve instead of straight line.)
Example: How much work is required to move an object in the force field along the path for ?
Answer:
. Then we have
If the force field is a conservative force field i.e. there exists a potential function such that , the work done by on an object moving from point to point does not depend on the path taken by the object because of the above theorem. In fact, the work done is the difference between the potential function at point and point . The potential function is usually called the potential energy of the force.
Example: Gravitational force between point masses obeys inverse square laws - the force acts along the line joining the centers of two masses and they vary as , where is the distance between the centers. The gravitational force of attraction is given by the vector field
where is a physical constant. Find the work done in moving a mass from point to along the line segment joining them, where . Moreover, find the work done as .
Answer: We already knew that an inverse square field is a conservative field and its potential function (energy) is
Therefore, the work done along any path from to is
As , .
Flux and circulation of a 2D vector field
Let be a simple curve in and let be a 2D vector field, which models the velocity of a fluid flow across . We define the flux of the fluid across to be the rate at which the fluid is crossing toward the right as we traverse in the positive direction (the "outward" direction), which can be defined as follows:
where is the outward unit normal vector along the curve . Suppose the curve is parametrized by for . Its tangent vector is and hence is orthogonal to and pointing outward i.e. is an outward normal vector along . Hence, we have and the flux can be expressed as follows:
Example: Evaluate the flux of across a unit circle oriented counterclockwise.
Answer:
Let for be the parametrization of , the unit circle oriented counterclockwise. Then . Therefore, the flux is
(See the applet below for the visualization of the flux.)
The line integral of along an oriented closed curve ,
is called the circulation of along , which measures the tendency of the fluid to move in the direction of .
Example: Find the circulation on the unit circle oriented counterclockwise for the following vector fields:
(a) The radial flow
(b) The rotation flow
Answer:
Let for be the parametrization of , the unit circle oriented counterclockwise. Then .
(a)
(b)
(See the applet below for the visualization of circulation.)