In one of my tutorial question about $1$-dim heat equation,a question about heat equation with pefectly insulated end at $x=0$ and $x=l$ with ${\rm u}\left(\, x,t\,\right)$ as temperature function,TAs used as perfectly insulated end implies $\frac{\partial u}{\partial x}|_{(0,t)}=0$ and $\frac{\partial u}{\partial x}|_{(l,t)}=0$,but if perfectly insulated end means temperature at that point remains constant,then correct expression should be like $\frac{\partial u}{\partial t}|_{(0,t)}=0 $and $\frac{\partial u}{\partial t}|_{(l,t)}=0$ right? (it means $u$ is independent of t at that point, am I right?)
How did the former equation represent perfectly insulated ends? What is the physical meaning of these each expression, i.e., $$\frac{\partial u}{\partial x}|_{(0,t)}=0 \ \hbox{ and } \frac{\partial u}{\partial t}|_{(0,t)}=0$$ also similar expression like $\frac{\partial u}{\partial x}|_{(x,0)}=0$ and $\frac{\partial u}{\partial t}|_{(x,0)}=0$ in context of heat transfer environment, what does each quantity represent?
What is wrong with my understanding about partial derivative? please help me... note:$u(x,t)$ is temperature at cordinate $x$ at time $t$
It is a consequence of the physics. The heat flow $Q$ is proportional to the gradient of the temperature $u$:
$$Q = -k \mathbf{\nabla}u$$
where $k$ is a heat conductivity. In a perfectly insulated material, there is no heat flow. Hence, the temperature gradient is zero. In 1D, this translates to the spatial derivative of $u$, as stated in the problem.