In pde, it seems to me all kinds of equations about nature phenomena have the property that time and space derivatives are separate. For example, u_t = u_xx, heat equation. So I wonder, is that always the case in nature? I mean, do you guys ever see equation describing real nature mechanism but has mixed derivatives involved, ie, has a term like u_tx? If not, is there a reason for that? This might be a silly question, but thanks in advance.
It's not clear t me what you mean by "separate". Do you mean "no mixed derivatives", like u_xt?The heat equation in 3 space dimensions, [itex]u_t= u_{xx}+ u_{yy}+ u_{zz}[/itex] have the other space variables as "separate" as x and t. That depends entirely on your choice of coordinate system- not "nature".
Yes, I mean mixed derivatives both in time and space, like [itex] u_{tx} [/itex]. For example, an equation like [itex] u_{t} = u_{tx} + u_{xx}. [/itex]
The ultimate example is General Relativity. It mixes up all kind of derivatives, and in a non-linear way!!!
Then, as I said above, it depends entirely upon your choice of coordinates. It is always possible to find coordinate axes, in the "principal directions" that avoid mixed derivatives. It has nothing to do with "nature" or space and time.
Is it possible to give an example? Say, heat equation, can we change coordinate to come up a mixed derivative term? Or any example to illustrate your idea? thanks