What does $~ u(\cdot, t)$ mean when referring to a function?

Question

I sometimes stumble over professors defining a function $u$ using regular (but quite sloppy) notation like $u(x,t) = A\sin(x)e^{-kt}$. Later in their notes, they state something like

$u(\cdot, t)$ = ...

What does the $\cdot$ in this notation mean? Why not just write $u(x,t)$ as before?

In statistics, the dot indicates, summation. It seems like $\int_{x}u(x,t)dx$ to me. I might be off-though I feel it to be so. The context does matter. — , Mar 28 '12 at 15:32
The $\cdot$ is effectively a placeholder. Yes, it's lazy, but it's quite common... — J. M. ain't a mathematician, Mar 28 '12 at 15:32

score 9 · Accepted Answer · answered Mar 28 '12 at 15:34

This indicates that we are viewing $u$ as a function of the first variable only, with the second variable fixed at $t$.

Another way to say this is that if $u$ is a function from $\mathbb{R}^2$ to $\mathbb{R}$, and $t \in \mathbb{R}$, then $u(\cdot,t)$ is another name for the function from $\mathbb{R}$ to $\mathbb{R}$ which we might define by $f(x) = u(x,t)$.

What does $~ u(\cdot, t)$ mean when referring to a function?

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