Can Cross Derivatives be Equal in Partial Derivatives?

[chicago77]

So this is a theoretical question I am not sure about with partial derivatives.

Say we have function x(i,j) with a=dx/di and b=dx/dj

Now is this logical when the cross derivatives are required to be equal?

db/da --> substitute b
= d^2x/(dadj) substitute da
= d^2x/((d^2x/di)dj)
=di/dj

Is this allowed?

 

[Mark44]

So this is a theoretical question I am not sure about with partial derivatives.

Say we have function x(i,j) with a=dx/di and b=dx/dj

Now is this logical when the cross derivatives are required to be equal?

db/da --> substitute b

Right off the bat I'm not following this. This says you are taking the (partial) derivative with respect to dx/dj, which makes not sense at all.

A small part of my difficulty is with your notation - dx/di - which is supposed to be the partial derivative of x with respect to i.

= d^2x/(dadj) substitute da
= d^2x/((d^2x/di)dj)
=di/dj

Is this allowed?

Presumably i and j (which are really bad choices for the names of variables) are independent variables, so you wouldn't ordinarily take the derivative (partial or otherwise) of one with respect to the other.

 

[chicago77]

nvm I have solved this with a lagrangian.