Find the function phi(r,t) given its partial derivatives.

In summary, the conversation discusses the possibility of defining t^* = \phi(r, t) with the given equation dt^* = \left( 1-\frac{k}{r} \right) dt + 0dr, where k is a constant. The speaker is looking for hints on how to make dt^* an exact form in order to integrate it, but the necessary condition for an exact form is not met. Therefore, t^*(r,t) does not exist.
  • #1
stedwards
416
46
I would like to define [itex]t^*= \phi(r, t)[/itex] given [itex]dt^* = \left( 1-\frac{k}{r} \right) dt + 0dr[/itex] where k is a constant.

Perhaps it doesn't exist. It appears so simple, yet I've been running around in circles. Any hints?
 
Physics news on Phys.org
  • #2
You want ##dt^*## to be an exact form, so that you can integrate it. An exact form can be expressed as

$$ df = M dx + N dy,~~~M = \frac{\partial f}{\partial x},~~~N = \frac{\partial f}{\partial y},$$

therefore a necessary condition that ##df## be exact is that

$$ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}.$$

The corresponding relation for ##dt^*## fails, so we conclude that it is not exact and the corresponding ##t^*(r,t)## does not exist.
 
  • Like
Likes beyondthemaths, stedwards and HallsofIvy
  • #3
Nicely done. Thank you!
 

Related to Find the function phi(r,t) given its partial derivatives.

1. What is the purpose of finding the function phi(r,t) given its partial derivatives?

The function phi(r,t) represents a physical quantity or property in a system. By finding its values at different points (r) and times (t), we can understand how the quantity changes and behaves in the system.

2. What are partial derivatives and why are they important?

Partial derivatives represent how a function changes with respect to one of its variables while holding the other variables constant. They are important in understanding the behavior of multi-variable functions and in solving differential equations.

3. What information do we need to find the function phi(r,t) given its partial derivatives?

We need the values of the partial derivatives of phi with respect to r and t, as well as any initial or boundary conditions that may be given. These values can be obtained from experimental data or through mathematical analysis.

4. What are some methods for finding the function phi(r,t) given its partial derivatives?

There are several methods that can be used, such as the method of separation of variables, the method of characteristics, and using integral equations. The choice of method will depend on the specific problem and the available information.

5. Can we always find a unique function phi(r,t) given its partial derivatives?

In most cases, yes. However, there may be certain cases where multiple functions can satisfy the given partial derivatives, or where the solution is not unique. In these cases, additional information or assumptions may be needed to determine a unique solution.

Similar threads

  • Differential Equations
Replies
4
Views
2K
  • Differential Equations
Replies
1
Views
1K
Replies
2
Views
2K
  • Differential Equations
Replies
5
Views
2K
  • Calculus
Replies
2
Views
2K
  • Differential Equations
Replies
1
Views
2K
Replies
6
Views
2K
Replies
5
Views
1K
  • Differential Equations
Replies
1
Views
2K
Back
Top