Mixed Partial Derivatives

The first partial derivatives of $u$, $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial y}$, can be differentiated with respect to the other variable as well:

$$\frac{\partial^2 u}{\partial y\partial x}=\frac{\partial}{\partia... ...t(\frac{\partial u}{\partial x}\right)=\frac{\partial u}{\partial y}(10x+3y)=3.$$ (9)

Respectively:

$$\frac{\partial^2 u}{\partial x\partial y}=\frac{\partial}{\partia... ...(\frac{\partial u}{\partial y}\right)=\frac{\partial u}{\partial x}(3x+4y^3)=3.$$ (10)

As the first partial derivatives of $u$ can be expressed as a vector, the second partial derivatives of $u$ can similarly be expressed using a so-called
Hessian-Matrix $H$:

$$\displaystyle{H=\left(\begin{array}{cc} \frac{\partial^2 u}{\part... ...{\partial y\partial x} & \frac{\partial^2 u}{\partial y^2} \end{array}\right)}.$$ (11)

For Example 2, the Hessian-Matrix is:

$$\displaystyle{H=\left(\begin{array}{cc} 10 & 3\\ 3 & 12y^2 \end{array}\right)}.$$ (12)