Notation for separable partial differential equations

[frozenguy]

Hey.. I ran across some problems and the notation used is a little different from what I've seen before.

considering U(x,y)=X(x)Y(y)
Sometimes I'll see U_xx for $$\frac{d^{2}u}{dt^{2}}$$ which equals X''Y
Or U_x for $$\frac{du}{dt}$$ which equals X'Y

But what about U'_x

Is that a redundant way of saying the partial derivative of U with respect to x?

Or is it saying the derivative of the partial derivative of U with respect to x?

As I originally read it, I considered it X'Y, but now I'm wondering if maybe its X"Y.

Thanks!

 

[AlephZero]

Hey.. I ran across some problems and the notation used is a little different from what I've seen before.

considering U(x,y)=X(x)Y(y)
Sometimes I'll see U_xx for $$\frac{d^{2}u}{dt^{2}}$$ which equals X''Y
Or U_x for $$\frac{du}{dt}$$ which equals X'Y

I think you meant
U_xx for $$\frac{\partial^{2}u}{\partial x^{2}}$$
or
U_x for $$\frac{\partial u}{\partial x}$$

But what about U'_x

Out of context, I would take that to mean $$\frac{d}{dt} \left(\frac{\partial u}{\partial x}\right)$$, but the context might indicate it meant something different. (The difference between $$d$$ and $$\partial$$ is not a typo.)

 

[frozenguy]

I think you meant
U_xx for $$\frac{\partial^{2}u}{\partial x^{2}}$$
or
U_x for $$\frac{\partial u}{\partial x}$$



Out of context, I would take that to mean $$\frac{d}{dt} \left(\frac{\partial u}{\partial x}\right)$$, but the context might indicate it meant something different. (The difference between $$d$$ and $$\partial$$ is not a typo.)

Yes that was a bad typo I'm sorry. Glad you knew what I was talking about.

The context would be partial differential equations using separation of variables.

For example: $$U^{'}_{x}=U^{'}_{y}+U$$