If M=∂F/∂x and N=(∂F/∂y) then ∂M/∂y=(∂^2 F/∂y)∂x and (∂^2 F/∂x)∂y = ∂N/∂x

[Integrate]

I understand that (∂F/∂x)dx+(∂F/∂y)dy=0 is the derivative of a function set to a constant. My calc 3 is rusty so I forget how this general form is created but not the topic of this post.

I understand that if M=∂F/∂x and then ∂M/∂y=(∂^2 F/∂y)∂x and (∂^2 F/∂x)∂y = ∂N/∂x


I also understand that fₓᵧ = fᵧₓ which is how this statement is true.

I just don't know how we get from

if M=∂F/∂x and then ∂M/∂y=(∂^2 F/∂y)∂x and (∂^2 F/∂x)∂y = ∂N/∂x

What are the steps inbetween or context that makes this true?


I want to understand not memorize.

 

[Steven G]

It's good that you want to understand vs memorize.
This should be in all differential equations text books which I assume that you have one. What steps are you not understanding?

 

[Integrate]

It's good that you want to understand vs memorize.
This should be in all differential equations text books which I assume that you have one. What steps are you not understanding?

After working through a problem I think I cleared quite a bit up myself.



I guess I just want to know why ∂M/∂y = ∂N/∂x proves that a differential equation is exact?

I understand that if ∂M/∂y /= ∂N/∂x and we tried to work backwards to find the original parent function it wouldn't work because well there is no parent function.


Am I on the right track?

 

[topsquark]

After working through a problem I think I cleared quite a bit up myself. View attachment 36298



I guess I just want to know why ∂M/∂y = ∂N/∂x proves that a differential equation is exact?

I understand that if ∂M/∂y /= ∂N/∂x and we tried to work backwards to find the original parent function it wouldn't work because well there is no parent function.


Am I on the right track?

An exact differential equation is one such that
[imath]M(x,y) \, dx + N(x,y) \, dy = 0[/imath]

Such that
[imath]M(x,y) = \dfrac{\partial F}{\partial x}[/imath]

and
[imath]N(x,y) = \dfrac{\partial F}{\partial y}[/imath]

for some function F(x,y). It's called this due to the "exact differential" (sometimes total differential)
[imath]dF = \dfrac{\partial F}{\partial x} \, dx + \dfrac{\partial F}{\partial y} \, dy[/imath]

Now, if we want an expression to relate M and N, note that if we take partial y and partial x (respectively) of M and N, the equality of the operators
[imath]\dfrac{\partial^2}{\partial x \partial y} = \dfrac{\partial^2}{\partial y \partial x}[/imath]

says that, if we have an exact differential for F(x,y), then
[imath]\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}[/imath]

Note: Given an arbitrary M(x,y) and N(x,y) we are not guaranteed that this might be the F(x,y) we are looking for. F(x,y) needs to be required to be exact for M and N to have this property. And, as you noted, given an arbitrary M and N, it may not be possible to construct an exact form for F.

-Dan

 

[Steven G]

After working through a problem I think I cleared quite a bit up myself. View attachment 36298



I guess I just want to know why ∂M/∂y = ∂N/∂x proves that a differential equation is exact?

I understand that if ∂M/∂y /= ∂N/∂x and we tried to work backwards to find the original parent function it wouldn't work because well there is no parent function.


Am I on the right track?

Please use equal signs.

 

[Integrate]

An exact differential equation is one such that
[imath]M(x,y) \, dx + N(x,y) \, dy = 0[/imath]

Such that
[imath]M(x,y) = \dfrac{\partial F}{\partial x}[/imath]

and
[imath]N(x,y) = \dfrac{\partial F}{\partial y}[/imath]

for some function F(x,y). It's called this due to the "exact differential" (sometimes total differential)
[imath]dF = \dfrac{\partial F}{\partial x} \, dx + \dfrac{\partial F}{\partial y} \, dy[/imath]

Now, if we want an expression to relate M and N, note that if we take partial y and partial x (respectively) of M and N, the equality of the operators
[imath]\dfrac{\partial^2}{\partial x \partial y} = \dfrac{\partial^2}{\partial y \partial x}[/imath]

says that, if we have an exact differential for F(x,y), then
[imath]\dfrac{\partial M}{\partial y} = \dfrac{\partial N}{\partial x}[/imath]

Note: Given an arbitrary M(x,y) and N(x,y) we are not guaranteed that this might be the F(x,y) we are looking for. F(x,y) needs to be required to be exact for M and N to have this property. And, as you noted, given an arbitrary M and N, it may not be possible to construct an exact form for F.

-Dan

Okay I think I understand it.


We are given a differential equation.


We wonder of it is an exact or "total" differential equation, so we manipulate it into the general form of an exact differential equation Mdx+Ndy=0.



Because an exact differential is taking the rate of change in each direction we take a partial derivative in each direction and then add them together.

Rise plus run.


Because mixed partial derivatives will end up with the same result regardless of the order the variables are derived.



If we are to take the partial derivative again for the next variable they equal the same.


If they dont then they are not an exact differential equation.


Oh and the parent function, the exact equation, creates a plane that intersects a surface.


The line the intersection creates is called a level curve.