# Exact Differential Equation and Green's theorem

• abcdefg10645
In summary, we can use Green's theorem to understand why the Exact Differential Equation satisfies the conditions it should have. This includes having "mixed partials" equal and looking at the vector equation with more than three variables. This can be further explored through \nabla\times f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k} and Stoke's theorem.
abcdefg10645
We can use Green's theorem to understand why the Exact Differential Equation satisfy the conditions it should have ...

How about a DE for more than two variables ?

Eg.dF=P(x,y,z,w)dx+Q(x,y,z,w)dy+R(x,y,z,w)dz+S(x,y,z,w)dw

IF the equation above is an Exact Differential Equation , what condition it would satisfy ?

An equation of the form f(x,y,z)dx+ g(x,y,z)dy+ h(x,y,z)dz= 0 is "exact" if and only if there exist some F(x,y,z) such that
$$dF= \frac{\partial F}{\p artial x}dx+ \frac{\partial F}{\partial y}dy+ \frac{\partial F}{\partial z}= f(x,y,z)dx+ g(x,y,z)dy+ h(x,y,z)dz$$

That means that we must have
$$\frac{\partial F}{\partial x}= f(x,y,z)$$
$$\frac{\partial F}{\partial y}= g(x,y,z)$$
and
$$\frac{\partial F}{\partial z}= h(x,y,z)$$

So, as long as those functions are continuous, we must have the "mixed partials" equal:
$$\frac{\partial^2 F}{\partial x\partial y}= \frac{\partial f}{\partial y}= \frac{\partial g}{\partial x}= \frac{\partial^2 F}{\partial y\partial x}$$
etc.

Since you mention "Green's theorem" (for two variables) you might want to look at $\nabla\times f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$ and Stoke's theorem.

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What u mentioned contains three variables ~and I've seen such a case in the book I read...

Suppose $$\vec{F}$$=P(x,y,z)$$\hat{i}$$+Q(x,y,z)$$\hat{j}$$+R(x,y,z)$$\hat{k}$$

dT=P(x,y,z)dx+Q(x,y,z)dy+R(x,y,z)dz is exact if only if $$\vec{F}$$$$\bullet$$($$\nabla$$$$\times$$$$\vec{F}$$)=0

I'm now requesting for the case which contains more than three variables~

HallsofIvy said:
An equation of the form f(x,y,z)dx+ g(x,y,z)dy+ h(x,y,z)dz= 0 is "exact" if and only if there exist some F(x,y,z) such that
$dF= \frac{\partial F}{\partial x}dx+ \frac{\partial F}{\partial y}dy+ \frac{\partial F}{\partial z}= f(x,y,z)dx+ g(x,y,z)dy+ h(x,y,z)dz$

That means that we must have
$$\frac{\partial F}{\partial x}= f(x,y,z)$$
$$\frac{\partial F}{\partial y}= g(x,y,z)$$
and
$$\frac{\partial F}{\partial z}= h(x,y,z)$$

So, as long as those functions are continuous, we must have the "mixed partials" equal:
$$\frac{\partial^2 F}{\partial x\partial y}= \frac{\partial f}{\partial y}= \frac{\partial g}{\partial x}= \frac{\partial^2 F}{\partial y\partial x}$$
etc.

Since you mention "Green's theorem" (for two variables) you might want to look at $\nabla\times f(x,y,z)\vec{i}+ g(x,y,z)\vec{j}+ h(x,y,z)\vec{k}$ and Stoke's theorem.

## 1. What is an exact differential equation?

An exact differential equation is a type of differential equation where the solution can be found by taking the partial derivatives of a function. In an exact differential equation, the total derivative of the function is equal to the sum of the partial derivatives with respect to each variable.

## 2. How do you determine if a differential equation is exact?

A differential equation is exact if the partial derivatives of the function involved satisfy the condition of equality called the Integrability condition. This means that the order in which the partial derivatives are taken does not matter.

## 3. What is Green's theorem?

Green's theorem is a mathematical tool used to calculate the line integral along a simple closed curve in a two-dimensional plane. It relates the line integral to a double integral over the region enclosed by the curve, and can be used to solve problems involving work, circulation, and flux.

## 4. How is Green's theorem related to exact differential equations?

Green's theorem can be used to solve exact differential equations by converting the line integral into a double integral, making it easier to solve and find the solution. This is because the double integral can be evaluated using standard techniques, such as integration by parts or substitution.

## 5. What are the applications of exact differential equations and Green's theorem?

Exact differential equations and Green's theorem have numerous applications in physics, engineering, and other fields of science. They are used to solve problems involving fluid flow, electric and magnetic fields, and heat transfer. They are also used in the study of curves and surfaces in mathematics.

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