Index notation in vector functions

Suppose I have a vector $\vec u$ that depends on a vector $\vec x$ and a scalar t, so each component of $\vec u$ depend on all components of $\vec x$. How can I show this relationship with index notation?

$$\vec u(\vec x, t)$$

From my understanding $u_i(x_i, t)$ implies:

$$u_1(x_1, t)$$ $$u_2(x_2, t)$$ $$u_3(x_3, t)$$

wich is not correct, and $u_i(x_j, t)$ indicates: $$u_1(x_1, t)$$ $$u_1(x_2, t)$$ $$...$$

which doesn't seem right either. I think the solution can be to write all the components for $\vec x$ but it is not very compact. $$u_i(x_1,x_2,x_3,t)$$

• I'd write $\vec{u}(\vec{x},t) = (u_1(\vec{x},t),\cdots,u_n(\vec{x},t))$, but I'm not sure if that's what you want. May 26, 2016 at 19:13
• @IvoTerek I was looking for a form who doesn't require ellipsis, and doesn't mix vector notation ( $\vec x$ ) with index notation ($x_i$), but maybe is not possible. May 26, 2016 at 19:16

If the involved dimensions are not specified (and small!) you cannot do without ellipsis. We are talking here about a vector-valued function $${\bf u}(\cdot,\cdot):\qquad ({\bf x},t)\mapsto {\bf u}({\bf x},t)\in{\mathbb R}^m\ ,$$ depending on the vector variable ${\bf x}\in{\mathbb R}^n$ and time $t\in{\mathbb R}$, or in coordinates: $$(x_1,\ldots,x_n,t)\mapsto u_i(x_1,\ldots, x_n,t)\qquad(1\leq i\leq m)\ .$$