Vector-Valued and Multivariate Functions

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Introduction

• A (real) vector is an ordered $n$-tuple of real numbers
  • A vector is the coordinates of points in $n$-dimensional space
• The set of all vectors of arity $n$ is denoted ${\mathbb{R}}_n$

Preliminary Knowledge:
💫 Vectors Y11
💫 Vectors Y12
💫 Functions

The Vector Space

${\mathbb{R}}_2$ is the standard $xy$ plane, ${\mathbb{R}}_3$ is 3D space
But we can go beyond that into spaces we can’t visualise geometrically

• Vectors can only be added with vectors with the same arity
  • e.g. you cannot multiply a 2D and 3D vector

Vector-Valued Functions

A vector-valued function is a function that takes a real number and assigns a vector in ${\mathbb{R}}^n$

$n$ refers to the number of dimensions

Curve in 3D

We say that $C:x=f(t),y=g(t),z=h(t)$ for $t\in D$, are parametric equations of $C$ with parameter $t$, or equivalently, that $(f(t),g(t),h(t))$ is a parametrisation of $C$

• We can use parametrisation to describe curves that cannot be defined with a single function

Skills to Learn

You must be able to convert between parametrial equations into an equation with a relationship between $x$ and $y$ (and $z$) allowing you to sketch curves/shapes in 2D/3D space.

Functions of Two Variables

• A function of two variables is a rule that assigns to each $(x,y)\in D$ a real number $f(x,y)$
  • Converts a vector into a real number

Level Curves

• Gives visual information about a surface (read graphing functions of two variables on slides) in a 2D form

  

  • Level curves are traditionally found in maps in order to inform the reader on the shape of the terrain

• Imagine a horizontal plane $k$ cutting through the surface

  • $k$ is some real constant (in the range of $f$) that defines a curve in $D$ called a level curve of $f$
  • The level curve is created where the surface intersects the plane $k$

• When asked to sketch level curves, decide an integer constant to be the difference between each level of $k$

  • e.g. $k_0=0,k_1=1,...,k_n=n$

Level Surfaces

• Just as we can represent a 3D shape in 2D with level curves, we can represent a 4D shape, one that normally cannot be represented geometrically, in 3D, through level surfaces