Undergrad Understanding the Notation: Deciphering the Equation in the Image

[lucdj3]

Can you help me figure out what the equation in the attached image means?
What I do know is: h(p) is a Monge parameterization height function depending on radius (p), this is a boundary condition for a differential equation, and p0 is the outer boundary, with the inner boundary creating a boundary condition of h(pi)=0.
Thanks!

 

[lucdj3]

I wasn't sure where this thread should be posted, so if there would have been a better place, please let me know and I will re-post there

 

[Mr-R]

To me this read: The partial derivative with respect to $\rho$ evaluated at $\rho=\rho_0$.
As in take the derivative first and then set $\rho=\rho_0$.
I could be wrong though.

 

[lucdj3]

That seems reasonable. Short term it is yielding something that doesn't look wrong, once I get through everything else I'll be able to tell if it's right. Thanks!
Any idea why it wouldn't be d/dp as the operator out front?

 

[Mr-R]

Maybe the author is being sloppy because it looks like it only depends on $\rho$ and therefore it could be derived with $\frac{d}{d\rho}$.

 

[lucdj3]

Well it worked it, thanks!