1 Functions of several variables and partial derivatives

These functions were first studied in HELM booklet  18. As a reminder:

• a function of the two independent variables $x$ and $y$ may be written as $f\left(x,y\right)$
• the first and second order partial derivatives are $\frac{\partial f}{\partial x}$ ,  $\frac{\partial f}{\partial y}$ ,  $\frac{{\partial }^{2}f}{\partial x^2}$ ,  $\frac{{\partial }^{2}f}{\partial y^2}$ and $\frac{{\partial }^{2}f}{\partial x\partial y}$ .

Consider, for example, the function $f\left(x,y\right)=x^2+5xy+3y^4+1$ . The first and second partial derivatives are

The number of independent variables is not restricted to two. For example, if $u$ is a function of the three variables $x$ , $y$ and $z$ , say $u=x^2+y^2+z^2$ then:

$\frac{\partial u}{\partial x}=2x,\frac{\partial u}{\partial y}=2y,\frac{\partial u}{\partial z}=2z,\frac{{\partial }^{2}u}{\partial x^2}=2,\frac{{\partial }^{2}u}{\partial y^2}=2,\frac{{\partial }^{2}u}{\partial z^2}=2$

Similarly, if $u$ is a function of the four variables $x$ , $y$ , $z$ and $t$ say $u=x{y}^2{z}^3{e}^t$ then

$\frac{\partial u}{\partial x}=y^2{z}^3{\text{e}}^t,\frac{\partial u}{\partial t}=x{y}^2{z}^3{\text{e}}^t,\frac{{\partial }^{2}u}{\partial z^2}=6x{y}^{2}z{\text{e}}^t$ , etc.