What does $\Bbb{R}^2\to\Bbb{R}^2$ mean?

Question

Find $\dfrac{\partial}{\partial s}(f\circ T)(1, 0)$, where $f(u, v) = \cos u \sin v$ and $T:\Bbb{R}^2\to\Bbb{R}^2$ is defined by $T(s, t) = (\cos (t^2s), \log\sqrt{1+s^2})$

I apologize if this is a very basic concept, but what does the $\Bbb{R}^2\to\Bbb{R}^2$ mean?

Answer 1

It means from pairs of reals to pairs of reals.

Answer 2

$T\colon\mathbb{R}^2\rightarrow\mathbb{R}^2$ means that $T$ is a map from $\mathbb{R}^2$ to $\mathbb{R}^2$, $T$ associates a point of $\mathbb{R}^2$ to a unique point of $\mathbb{R}^2$.

This is surprizing that you never encountered this notation before but you still now partial derivatives. There is no shame though.

Answer 3

It means your function $T$ takes an element from $\mathbb R^2$ and maps it to an element in $\mathbb R^2$. The first set is called the domain and the second set is called the codomain.