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

Surfaces

Graph of the set of all points such that where . Can be plotted on a 3D space.

Half-space

One side of a plane.

Linear function

Function of the below form.

Plot of a linear function is a plane surfaces.

Level curves

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

Common shapes of functions

Parabolic cylinder

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

Only 2 variables are related.

Ellipsoid

A surface in which all of its traces are ellipses.

Goes through and so on. Becomes a sphere when .

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.