Introduction to Multivariable Calculus

Real valued multivariable function

A function with more than 1 inputs or outputs.

denotes a n-dimensional space with each axis denoted by real numbers.

A function is a map between a ordered n-tuple to real numbers. The domain can also be . The same can be applied to the output of the function.

Vector-valued functions

Functions with more than 1 outputs are also called multivariable but they are commonly referred to as vector-valued functions.

Surfaces

Suppose . The graph of is the set of all points such that . The graph can be plotted on a 3D space.

Linear function

Functions of the below form are called linear functions.

Plane surfaces are resulted when these functions are plotted.

Common shapes of functions

Parabolic cylinder

A parabola shifted along a straight line

Ellipsoid

A surface in which all of its traces are ellipses.

When the surface becomes a sphere.

Cone

Horizontal traces are ellipses. Vertical traces in or are:

• Hyperbolas if
• Pair of lines if

Elliptic paraboloid

Horizontal traces are ellipses. Vertical traces are parabolas. The variable raised to first power ( in the example), indicates the axis of the paraboloid.

Hyperboloic Paraboloid

Horizontal traces are hyperbolas. Vertical traces are parabolas.

Hyperboloid of One Sheet

Horizontal traces are ellipses. Vertical traces are hyperbolas. Variable with the negative coefficient denotes the axis of symmetry.

Hyperboloid of Two Sheets

Horizontal traces are ellipses for where . Vertical traces are hyperbolas. Variable with the positive coefficient denotes the axis of symmetry.

Level curves

Suppose . The level curves of are the graphes where is a constant. Aka. level sets.

Suppose and .

epsilon-disk

An -disk around is the set of all points where the distance between the points is less than .

Interior point

is an interior point of iff there exists some -disk around that is contained in .

Isolated point

is an isolated point of iff there exists some -disk around that doesn’t contains no other points of .

Boundary point

is an boundary point of iff all -disks around contains points from and not from .

Open subset

is an open subset of iff all the points of are interior points of .

Closed subset

is an closed subset of iff contains all of its boundary points.

Closure

The set of boundary points of and the region . Denoted by .

Bounded subset

is a bounded subset of iff is contained in some -disk around some point.

Half-space

Either sides of a plane.

Neighbourhoods

Let be the domain of 2 variable function .

Circular neighbourhoods

The below set is called a -neighbourhood of the point and .

Square neighbourhoods

The below set is called a square-neighbourhood of the point and .