Does the Order of Differentiation Matter?

[quietrain]

is this the same? why are they the same? do the order of differential not matter?

d/dx_i (dy_j / dx_j) = d/dx_j (dy_j / dx_i)

where LHS : differentiate y_j w.r.t x_j first, then x_i

while RHS: differentiate y_j w.r.t x_i first, then x_j

thanks!

 

[HallsofIvy]

As long as f and its first and second derivatives are continuous, in some neighborhood of a point, the "mixed" derivatives
$$\frac{\partial f}{\partial x\partial y}$$
and
$$\frac{\partial f}{\partial y\partial x}$$
are equal at that point.

 

[quietrain]

is this a total differential function? or is it just a normal differential property?

i seem to be mixing everything up :(

 

[HallsofIvy]

I have no idea what you are asking. I don't know what you mean by "total differential function" (I do know what a total differential is) or 'normal differential property".

There is no "total differential" in this problem. It is entirely a property of partial derivatives.

 

[quietrain]

ah i see than kyou