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

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

• – Mark S. Feb 22 at 19:32

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.

• 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 $t_i \in [a,b]$? – goblinb Feb 22 at 23:10
• Yes! Happy that I could help! ;) – Vinyl_cape_jawa Feb 22 at 23:11

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 $$x$$.