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Section7.1Introduction

In this section you will...

  • connect Calculus I/II to Calculus III: Multivariable via vectors

We'll find that throughout this course, the key difference between first-semester calculus and multivariate calculus is that we replace the input \(x\) and output \(y\) of functions with the vectors \(\vec x\) and \(\vec y\text{.}\) For the function \(f(x,y)=z\text{,}\) we can write \(f\) in the vector notation \(\vec y=\vec f(\vec x)\) if we let \(\vec x=(x,y)\) and \(\vec y=(z)\text{.}\) Notice that \(\vec x\) is a vector of inputs, and \(\vec y\) is a vector of outputs.

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)

\(\vec F(\rho,\phi,\theta)=(x,y,z)\)