Maths - Multidimensional Algebra and Fields

The geometry of fields is discussed on this page here we discuss how to represent fields using multidimensional algebra types.

Vector Field

A vector field has a vector value for every point in a space:

vector gravity field

how this plot was produced.

For instance, if we have a very large mass at the origin, then we can plot the gravitational field, as in the diagram above. Its difficult to find a suitable way to show this, especially in 3 dimensions, an alternative might be to plot lines of constant field force.

Scalar Field

The above situation can be shown as a scalar field. That on the following diagram the height (y-dimension) shows the scalar value.

We can always get a scalar field from a vector field by using the divergence as shown on this page.

plot inverse square 3d

how this plot was produced.

Prerequisites

The geometry of fields is discussed on this page.

You may also want to review:

Vectors

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Symmetry

Symmetry is an important topic for maths and physics.

Symmetry is important for many branches of mathematics including geometry (see this page) and group theory (see this page). Its importance can become apparent in unexpected places, for example, solving quintic equations.

We say that an object is symmetric, with respect to a given mathematical operation, if this operation does not change the object.

Nothers Theorem (discussed further on this page) says that, for every symmetry exhibited by a physical law, there is a corresponding observable quantity that is conserved. Virtually every theory, including relativity and quantum physics is based on symmetry principles.

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see also:

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

Mathematics for 3D game Programming - Includes introduction to Vectors, Matrices, Transforms and Trigonometry. (But no euler angles or quaternions). Also includes ray tracing and some linear & rotational physics also collision detection (but not collision response).

Other Math Books

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Terminology and Notation

Specific to this page here:

program

I am working on a project which uses these principles, if you would like to help me with this you are welcome to join in, here:

http://sourceforge.net/projects/mjbworld/

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