Exercise7.1.1
For each of the functions below, state what \(\vec x\) and \(\vec y\) should be so that the function can be written in the form \(\vec y = \vec f (\vec x)\text{.}\) In addition, identify what type of function each is from the list in Section 6.6.
The point to this exercise is to help you learn to recognize the dimensions of the domain and codomain of the function. If we write \(\vec f:\R^n\to \R^m\text{,}\) then \(\vec x\) is a vector in \(\R^n\) with \(n\) components, and \(\vec y\) is a vector in \(\R^m\) with \(m\) components.
(a)
\(f(x,y,z)=w\)
(b)
\(\vec r(t)=(x,y,z)\)
(c)
\(\vec r(u,v)=(x,y,z)\)
(d)
\(\vec F(x,y)=(M,N)\)
(e)
\(\vec F(\rho,\phi,\theta)=(x,y,z)\)