# Exact Differentials: Proving Existence of u(x,y) in Connected Open Region

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In summary, we can prove the existence of a function u(x,y) defined on a connected open region R such that ∇u = F if F = (M,N) is a vector function on R that is path independent. This is known as the fundamental theorem of line integrals and makes use of the fact that R is a simply connected region.
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## Homework Statement

Let R be a connected open region ( in the plane ). Suppose that F = (M,N) is a vector function defined on R and is such that for any ( piecewise smooth ) curve C in R :

$\int_C Fdp$

depends on only the endpoints of C ( that is, any two curves from P1 to P2 in R give the same value for the integral).

Prove that there exists a function u(x,y) defined on R such that ∇u = F.

( i.e ux = M and uy = N )

## Homework Equations

Err I think this may have to do with simply connected regions?

## The Attempt at a Solution

I'm not quite sure where to start with this one? I'm having trouble seeing how the info provided leads to what I need.

I think it has to do with if R is a simply connected open region and Mdx + Ndy is such that My = Nx in R, then the differential is exact.

Any push in the right direction would be great.

You are correct in thinking that the information provided may have to do with simply connected regions. To prove the existence of a function u(x,y) such that ∇u = F, we can use the fact that R is a connected open region and F = (M,N) is a vector function defined on R. From this, we know that the integral of F along any curve C in R only depends on the endpoints of C. This property is known as path independence.

To prove the existence of u(x,y), we can use the fundamental theorem of line integrals, which states that if a vector function F = (M,N) is path independent, then there exists a function u(x,y) such that ∇u = F. This function u(x,y) is known as the potential function for F.

To show that u(x,y) exists, we can start by considering any two points P1 and P2 in R. Since R is a connected open region, we can choose a path C from P1 to P2 that lies entirely within R. Since the integral of F along any curve C in R only depends on the endpoints of C, we can say that the integral of F along this path C is equal to the difference in values of u at P2 and P1. This can be written as:

∫_C Fdp = u(P2) - u(P1)

Since this is true for any two points P1 and P2 in R, we can say that u(x,y) is a potential function for F, and therefore ∇u = F.

To summarize, we have shown that because R is a connected open region and F = (M,N) is path independent, there exists a function u(x,y) such that ∇u = F. This function u(x,y) is known as the potential function for F. We can also use the fact that R is a simply connected region to show that the differential is exact, which further supports our proof.

## 1. What is an exact differential?

An exact differential is a type of differential that satisfies the condition of being the total differential of a function. This means that the value of the differential at a given point is equal to the change in the function at that point.

## 2. How is the existence of u(x,y) proven in a connected open region?

In order to prove the existence of u(x,y) in a connected open region, we can use the condition that the partial derivatives of u with respect to x and y must be continuous in the region. This ensures that the total differential is well-defined and thus u(x,y) exists.

## 3. What is the significance of a connected open region in proving the existence of u(x,y)?

A connected open region is important because it allows us to define a path between any two points in the region. This path is necessary for proving that the partial derivatives of u with respect to x and y are continuous in the region, which is a requirement for the existence of u(x,y).

## 4. What are the applications of exact differentials?

Exact differentials have many applications in mathematics and physics. They are used to solve partial differential equations, determine the work done in thermodynamic processes, and study the stability of dynamical systems, among others.

## 5. Can an inexact differential be transformed into an exact differential?

Yes, an inexact differential can be transformed into an exact differential by finding a suitable integrating factor. This factor is a function that multiplies the inexact differential to make it exact. However, the process of finding this integrating factor can be complex and may not always be possible.

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