Elliptic differential operator I am given the differential operator $D(f):=-(fg)'+hf$ and $D^* (f) = g \cdot f' + hf$ where $h,g$ are some smooth functions and want to find out under which conditions, these two operators are elliptic. Does anybody know how to do this, I am really puzzled by the wikipedia definition of elliptic differential operators. 

The heat equation is parabolic:
$$
       \frac{\partial f}{\partial t} = \frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}}
$$
Laplace's equation is elliptic:
$$
               \frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}} = 0
$$
The Wave equation is hyperbolic:
$$
    \frac{\partial^{2}f}{\partial t^{2}}=\frac{\partial^{2}f}{\partial^{2}x}+\frac{\partial^{2}f}{\partial y^{2}}+\frac{\partial^{2}f}{\partial z^{2}}
$$
These designations are related to $y-x^{2}=C$, $x^{2}+y^{2}=C$ and $y^{2}-x^{2}=C$, whose level surfaces are generally parabolic, elliptic and hyperbolic, respectively.
Your equation is never classified as elliptic because such designations are applied only to Partial Differential Equations.
