# Potential function for conservative vector field

• kasse
In summary, the conversation discusses finding a potential function for the conservative vector field F = <x + y, x - z, z - y>. A potential function is found by integrating the given equations, and further steps are taken to determine the function. The conversation also mentions a preferred method of integrating each partial differential equation separately before comparing them.

#### kasse

[SOLVED] Potential function for conservative vector field

## Homework Statement

Find a potential function for the conservative vector field F = <x + y, x - z, z - y>

2. The attempt at a solution

OK, we know that

(1) fx = x + y
(2) fy = x - z og
(3) fz = z - y

We can then integrate (1) with respect to x, and we get

(4) f = (1/2)x2 + xy + C(y,z)

We then differentiate (4) wrt y and get:

(5) fy = x + C'(y,z)

Comparing (2) and (5):

(6) C'(y,z) = -z

Integrate this wrt z:

(7) C(y,z) = -(1/2)z2 + C(y)

We then have the following expression for the potential function so far:

(8) f = (1/2)x2 + xy + -(1/2)z2 + C(y)

Then I'm stuck. Is my method correct so far?

In step (6) when you write C'(y,z)=(-z) the prime means derivative wrt y. So I wouldn't integrate wrt z. You're on the right track. Just pay more attention to what variable the derivatives are taken wrt.

kasse said:
OK, we know that

(1) fx = x + y
(2) fy = x - z og
(3) fz = z - y

We can then integrate (1) with respect to x, and we get

(4) f = (1/2)x2 + xy + C(y,z)

We then differentiate (4) wrt y and get:

(5) fy = x + C'(y,z)

Comparing (2) and (5):

(6) C'(y,z) = -z

I was with you up to here. Why did you compare (2) and (5) and then integrate with respect to z? It seems to me that if you'd integrated with respect to y first, you'd have gotten

C(y,z) = -yz + C(z) ,

from which point you should be all right...

I guess I prefer integrating each partial differential equation separately and then intercomparing all three. As long as the equations are too complicated, things usually fall into place pretty well.

## 1. What is a conservative vector field?

A conservative vector field is a type of vector field in which the path taken by a particle moving through the field does not depend on the initial position of the particle. This means that the work done by the field on the particle is independent of the path taken.

## 2. What is the significance of a potential function in a conservative vector field?

A potential function is a scalar function that can be used to describe the behavior of a conservative vector field. It helps to determine the potential energy of a particle at any given point in the field and allows for the calculation of the work done by the field on the particle.

## 3. How is a potential function related to the gradient of a vector field?

In a conservative vector field, the gradient of the potential function is equal to the vector field itself. This means that the potential function is the "antiderivative" of the vector field, and the direction of steepest increase of the potential function is the same as the direction of the vector field.

## 4. Can a vector field be both conservative and non-conservative?

No, a vector field can only be either conservative or non-conservative. If a vector field is conservative, it means that it has a potential function and follows the properties of a conservative vector field. If a vector field is non-conservative, it does not have a potential function and does not follow the properties of a conservative vector field.

## 5. What are some real-life applications of conservative vector fields?

Conservative vector fields have many applications in physics, engineering, and economics. Some examples include the study of conservative forces in mechanics, the flow of fluids in pipes or channels, and the analysis of economic systems such as supply and demand.