Some notes how to visualize mathematical functions.

Domain and Range

• Input space: domain
• Output space: range or codomain

Graphs (single variable functions)

• Showing both ‘input space’ and ‘output space’
• limited in dimension
• used for:
  • single-variable functions
  • multivariable functions with a tow-dimensional input and a one-dimensional output (represent all points of the form $(x, y, f(x, y))$)

Contour Maps

• Show only the input space
• outputs values represent should be evenly spaced (easier to understand)
• useful for functions with a two-dimensional input and a one-dimensional output
• The Gradient is always perpendicular to the Contour lines

Parametric Curves/Surfaces

• Show only the output space
• used for functions whose output space has more dimensions than the input space
• input information is lost

Vector Fields

• Used for functions with same numbers of dimensions in the input space as in the output space
• Associates a vector with each point in space
• The vector lengths are usually not drawn to scale (but proportional to each other)
• Sometimes colours are used to indicate the length of the vectors

Transformations

• Watch (or imagine) how each input point moves to its corresponding output space
• have to be represented as animations or a schematic drawing
• useful for gaining conceptual understanding
• impractical for representing functions precisely
• Can be used for functions with any dimensions in the input- and output space

Plots

The plots were made with Octave with this script.