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In particular I don't understand how to interpret the domain.

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Usually the $\times $ sign means Cartesian product in the context of sets.

Let $$ A=\{x,y,z \}\qquad B=\{1,2 \} $$

then the set $C=A\times B$ is the set of all ordered pairs created from the elements of $A$ and $B$, in formal mathematical language

$$ C=\{(a,b)\mid a\in A, b\in B \}. $$

In the above example $C$ is the set $$ \{(x,1),(x,2),(y,1),(y,2),(z,1),(z,2) \}. $$

I hope I could help.

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  • $\begingroup$ Thank you. So in this case $[a,b] \times \mathbb{R}^n$ would be like all ordered pairs $(t_1,x_1,\dots,x_n), (t_2,x_1,\dots,x_n), \dots$ for all $t_i \in [a,b]$? $\endgroup$ – goblinb Feb 22 at 23:10
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    $\begingroup$ Yes! Happy that I could help! ;) $\endgroup$ – Vinyl_cape_jawa Feb 22 at 23:11
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The arrow means that we consider a time-dependent vector field. Hence $f(t,x)$ is a vector at the point $x$ depending on the time $t$. A good example is the wind : at each point of the earth $\bf R^2$ the wind is represented by a vector. This vector gives the direction and strength of the wind at the instant $t$ and at the point $x$.

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