Is there any geometric reason for why these equations are also called "elliptic", "parabolic", and "hyperbolic"? Like what does the wave equation have to do with a hyperbola?

niyiqiul

Is there a system of PDEs?

YutianW

Is there any specific applications for elliptic equation, just as heat equation for heat and wave equation for wave simulation?

Starboy

Isn't the transport equation also one kind of important PDE?

dab

How can both the first derivative of time and second derivative of time both be the Laplacian?

fzeng

What are other examples of fundamental model equations?

jefftan

Is there any geometric reason for why these equations are also called "elliptic", "parabolic", and "hyperbolic"? Like what does the wave equation have to do with a hyperbola?

Is there any geometric reason for why these equations are also called "elliptic", "parabolic", and "hyperbolic"? Like what does the wave equation have to do with a hyperbola?

Is there a system of PDEs?

Is there any specific applications for elliptic equation, just as heat equation for heat and wave equation for wave simulation?

Isn't the transport equation also one kind of important PDE?

How can both the first derivative of time and second derivative of time both be the Laplacian?

What are other examples of fundamental model equations?

Is there any geometric reason for why these equations are also called "elliptic", "parabolic", and "hyperbolic"? Like what does the wave equation have to do with a hyperbola?