What does this notation mean: $\vec{f} : [a,b] \times \mathbb{R}^n \rightarrow \mathbb{R}^n$?

Question

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

Answer 1

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.

Answer 2

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 $x$.