Introduction to Multivariable Calculus

Real valued multivariable function

A function with more than 1 inputs or outputs.

$\mathbb{R}^n$ denotes a n-dimensional space with each axis denoted by real numbers. $$

A function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ is a map between a ordered n-tuple to real numbers. The domain can also be $D \subset \mathbb{R}^{n}$. The same can be applied to the output of the function.

Vector-valued functions

Functions with more than 1 outputs. Referred to as vector-valued functions, even though they are also multivariable functions.

Definitions

Suppose $P \subset \mathbb{R}^2$ and $Q \equiv (x_0,y_0)\in\mathbb{R}^2$.

epsilon-disk

An $\epsilon$-disk around $Q$ is the set of all points $(x,y)\in\mathbb{R}^2$ where the distance between the points is less than $\epsilon$.

Interior point

$Q$ is an interior point of $P$ iff there exists some $\epsilon$-disk around $Q$ that is contained in $P$.

Boundary point

$Q$ is an boundary point of $P$ iff all $\epsilon$-disks around $Q$ contains points from $P$ and not from $P$.

Isolated point

$Q$ is an isolated point of $P$ iff there exists some $\epsilon$-disk around $Q$ that doesn’t contains no other points of $P$. Subset of boundary points.

Open subset

$P$ is an open subset of $\mathbb{R}^2$ iff all the points of $P$ are interior points of $P$.

Closed subset

$P$ is an closed subset of $\mathbb{R}^2$ iff $P$ contains all of its boundary points.

Note

A subset can neither be open nor closed. A subset in $\mathbb{R}^2$ is open iff its complement is closed. $$

Closure

The set of boundary points of $P$ and the region $P$. Denoted by $\overline{P}$.

Bounded subset

$P$ is a bounded subset of $\mathbb{R}^2$ iff $P$ is contained in some $\epsilon$-disk around some point.

Circular neighbourhood

The below set is called a $\delta$-neighbourhood of the point $(a,b) \in D$ and $\delta > 0$.

$$\Big\{(x,y)\,\text{ s.t. }\,\sqrt{(x-a)^2 + (y-b)^2} \lt \delta\Big\}$$

Square neighbourhood

The below set is called a square-neighbourhood of the point $(a,b) \in D$ and $\delta > 0$.

$$\Big\{(x,y)\,\text{ s.t. }\,\lvert x-a \rvert \lt \delta \land |y-b| \lt \delta\Big\}$$