Welcome to Math Insight

The Math Insight web site is a collection of pages and applets designed to shed light on concepts underlying a few topics in mathematics. The focus is on qualitative description rather than getting all technical details precise. Many of the pages were designed to be read even before students attend lecture on the topic, so they are intended to be somewhat readable introductions to the basic ideas.

You can browse the pages organized into threads, which are sequences through a subset of pages organized by particular topics. An index can help you find pages discussing a particular term. You can also search through the pages, applets, and image captions. A few pages allow you to change the notation system used to render the mathematics.

We hope Math Insight can help you understand key mathematical concepts. We welcome comments on how we can improve it.

Highlighted applets

Applet: Gradient and directional derivative on a mountain shown as level curves
The gradient and directional derivative are illustrated by a level curve plot of a mountain range. The applet interactively computes the gradient and the directional derivative at different points and directions.

Applet: Level curves of an elliptic paraboloid shown with graph
The level curves of an elliptic paraboloid are shown as the intersection of a horizontal plane with the graph.

Highlighted pages

• Subtleties about divergence
  Counterexamples illustrating how the divergence of a vector field may differ from the intuitive appearance of the expansion of a vector field.
• Parametrized curve and derivative as location and velocity
  Description of a parametrization of a curve as the position of a particle and the derivative as the particle's velocity. Illustrated with animated graphics.
• Parametrization of a line
  Introduction to how one can parametrize a line. Interactive graphics illustrate basic concepts.
• The idea behind Stokes' theorem
  Introduction to Stokes' theorem, based on the intuition of microscopic and macroscopic circulation of a vector field and illustrated by interactive graphics.
• Introduction to the multivariable chain rule
  Introduction to the multivariable chain rule. The basic concepts are illustrated through a simple example.

Recent news

• Interactive Gallery of Quadric Surfaces
  by Duane Q. Nykamp on March 14, 2012

  We've added the Interactive Gallery of Quadric Surfaces

• Calculus Refresher added
  by Duane Q. Nykamp on Jan. 19, 2012

  Material from Calculus Refresher by Paul Garrett has been added to Math Insight.

• More recent news

Recent pages

• How linear transformations map parallelograms and parallelepipeds
  Why linear transformations map parallelograms onto parallelograms and parallelepipeds onto parallelepipeds.
  Added April 4, 2012
• Volume calculation for changing variables in triple integrals
  A derivation of how a mapping that changes variables in triple integrals transforms volume.
  Added April 2, 2012
• Introduction to changing variables in triple integrals
  Introduction to the concepts behind a change of variables in triple integrals.
  Added April 2, 2012
• More new items