-2

just want to be clear with partial derivative notation, and as to what respect we are taking the partial deriv of , for example; $$f_x, f_y, f_{xx}, f_{yy}, f_{xy}$$

so with what respect is each partial deriv being taken of? are they all with respect to x?

coa
  • 1

1 Answers1

3

Let $f \colon \mathbb{R}^2 \to \mathbb{R}$, $f \in C^{k}(\mathbb{R}^2)$. Then $$f_x(x_0,y_0) = \frac{\partial f}{\partial x}(x_0, y_0)$$ $$f_y(x_0,y_0) = \frac{\partial f}{\partial y}(x_0, y_0)$$ $$(f_x)_x(x_0,y_0) = f_{xx}(x_0,y_0) =\frac{\partial \Big(\frac{\partial f}{\partial x}\Big)}{\partial x}(x_0,y_0) = \frac{\partial f_x}{\partial x}(x_0,y_0) = \frac{\partial^2 f}{\left.\partial x\right.^2}(x_0, y_0)$$ $$(f_x)_y(x_0,y_0) = f_{xy}(x_0,y_0) =\frac{\partial \Big(\frac{\partial f}{\partial x}\Big)}{\partial y}(x_0,y_0) = \frac{\partial f_x}{\partial y}(x_0,y_0) = \frac{\partial^2 f}{\partial y \partial x}(x_0, y_0)$$

Is it clear now? I hope so!

LightninBolt74
  • 166
  • 2
    Errr... not quite so clear. The Legendre notation does work as you've written it, in which successive derivatives are added on to the subscripts from left to right. Unfortunatately, the other notation descends, I think, from Liebniz: the first two terms in your equation for the "mixed" partial derivative are correct, but successive derivatives thus get added on from the left. Thus $\frac{\partial (\frac{\partial f}{\partial x})}{\partial y} = \frac{\partial^2 f}{\partial y \partial x} $ . (I used to get confused by this, too...) – colormegone May 14 '13 at 17:32
  • 1
    I wasn't sure if this was another abandoned thread, since the OP evidently is inactive. But the last equation did need to be remarked upon. Thanks for fixing it! – colormegone May 14 '13 at 18:07