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. $$ u_i(x_1,x_2,x_3,t) $$