What does the notation $xf(x)$ and $f_{xxx}$ or $f_{xxy}$ mean?

Question

Hi I'm wondering about notations I encountered when studying differential equations and oscillation.

1. $V''' = \frac{3\pi}{4b}(f_{xxx} + f_{xyy} +g_{xxy}+g_{yyy})+\frac{3\pi}{4b^2}[f_{xy}(f_{xx}+f_{yy})+g_{xy}(g_{xx}+g_{yy})+f_{xx}g_{xx}-f_{yy}g_{yy}]$

This is a Calculation of Stability of the limit Cycle in Hopf bifurcation, and I don't know what $f_{xxx}$ etc., means.

2. A differential equation is given by $\frac{dx}{dt}=xf(x,y)$ What does the $xf$ stand for?

Thanks Josi

Answer 1

Note that

• $f_{xxx}=\frac{\partial^3 f}{\partial x^3}$ is the third partial derivative with respect to $x$

• $f_{xxy}=\frac{\partial^3 f}{\partial x^2y}$ is the third partial derivative with respect twice to $x$ and once to $y$

• $xf(x,y)$ is simply the product of $x$ times $f$

Answer 2

The notation $f_{\dots}$ means the order of which you have differentiated your expression. For example :

$$f_{xxx} = \frac{\partial^3 f}{\partial x^3}$$

$$f_{xyx} = \frac{\partial}{\partial x}\bigg(\frac{\partial}{\partial y}\bigg(\frac{\partial f}{\partial x}\bigg) \bigg)$$

As for the $x f(x,y)$, it simply stands for the product expression $x \cdot f(x,y)$.