Extra (ancillary) information in math from the physics

To illustrate the way we use extra information from physics above and beyond the information represented in the mathematics, consider the following example.

Given the following function, where K is a known constant,

A(x, y) = K(x2 + y2)

what is

A(r, θ) = ?

When this problem is given to physicists, they almost always give the answer:

A(r, θ) = Kr2.

In choosing this, they are typically using the interpretation of x and y as the coordinates of a point in a plane, and of r and θ as the polar coordinates of that point. Thus, the bring in the extra (ancillary) information defining polar coordinates:

r2 = x2 + y2
(the Pythagorean theorem) and
tan θ= y/x.

A mathematician would typically not use the ancillary information and say: "The function A(.,.) is K times the sum of the squares of the first variable and the second variable. Therefore the answer should be:

A(r, θ) = K(r2 + θ2).

The physicist would respond,"That's awful! You can't add r2 and θ2; they have different units."

The mathematician would reply, "If you mean only r2, you have to give it a different name, since the functional dependence is different. Something like:"

A(x, y) = B(r, θ) = Kr2.

The physicist says, "I don't want to use a different symbol. The A reminds me that I am calculating a vector potential. If I used B, I might forget and interpret it later as a magnetic field instead. In any case, I know what I mean even it it isn't mathematically precise about the functional dependence."

The moral here is that in physics, you can't just look at what the math equation is saying. We often load physical meaning on a mathematical symbol in a way that changes the way we interpret the math.

This problem was shown to me by Corinne Manogue of Oregon State University.

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This page prepared by

Edward F. Redish
Department of Physics
University of Maryland
College Park, MD 20742
Phone: (301) 405-6120

Last revision 5 September, 2005.