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Sahithyan's S2
Sahithyan's S2 — Methods of Mathematics

Surfaces

Graph of the set of all points such that (x,y,f(x,y))∈R3(x,y,f(x,y)) \in \mathbb{R}^3 where f:R2→Rf:\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+cf(x,y) = ax + by + c

Plot of a linear function is a plane surfaces.

Level curves

Aka. contours. Suppose f:R2→Rf: \mathbb{R}^{2} \rightarrow \mathbb{R}. The level curves of ff are the graphs f(x,y)=kf(x,y)=k where kk 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=x2z=x^2

Only 2 variables are related.

Ellipsoid

A surface in which all of its traces are ellipses.

x2a2+y2b2+z2c2=1\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)(a,0,0),(-a,0,0),(0,b,0) and so on. Becomes a sphere when a=b=ca=b=c.

Cone

z2c2=x2a2+y2b2\frac{z^2}{c^2} = \frac{x^2}{a^2} + \frac{y^2}{b^2}

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

  • Hyperbolas if k≠0k\neq 0
  • Pair of lines if k=0k=0

Elliptic paraboloid

zc=x2a2+y2b2\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 (zz in the example), indicates the axis of the paraboloid.

Hyperboloic Paraboloid

zc=x2a2−y2b2\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

x2a2+y2b2−z2c2=1\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

x2a2+y2b2−z2c2=−1\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = -1

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