Sahithyan's S2 — Methods of Mathematics

Surfaces

Graph of the set of all points such that $(x,y,f(x,y)) \in \mathbb{R}^3$ where $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ . Can be plotted on a 3D space.

Half-space

One side of a plane.

Linear function

Function of the below form.

f(x,y) = ax + by + c

Plot of a linear function is a plane surfaces.

Level curves

Aka. contours. Suppose $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ . The level curves of $f$ are the graphs $f(x,y)=k$ where $k$ is a constant. Aka. level sets.

Common shapes of functions

I have created a visualization of all these common shapes in Geogebra. Use that as an aid when learning about these shapes.

Parabolic cylinder

A parabola shifted along a straight line. In the form:

z=x^2

Only 2 variables are related.

Ellipsoid

A surface in which all of its traces are ellipses.

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1

Goes through $(a,0,0),(-a,0,0),(0,b,0)$ and so on. Becomes a sphere when $a=b=c$ .

Cone

\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}

Horizontal traces are ellipses. Vertical traces in $x=k$ or $y=k$ are:

Hyperbolas if $k\neq 0$
Pair of lines if $k=0$

Elliptic paraboloid

\frac{z}{c} = \frac{x^2}{a^2} + \frac{y^2}{b^2}

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

Hyperboloic Paraboloid

\frac{z}{c} = \frac{x^2}{a^2} - \frac{y^2}{b^2}

Horizontal traces are hyperbolas. Vertical traces are parabolas.

Hyperboloid of One Sheet

\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1

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

Hyperboloid of Two Sheets

\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1

Horizontal traces are ellipses for $z=k$ where $k \not\in [-c,c]$ . Vertical traces are hyperbolas. Variable with the positive coefficient denotes the axis of symmetry.