Electricity and Magnetism for Mathematicians: A Guided Path by Thomas A. Garrity

By Thomas A. Garrity

This article is an creation to a couple of the mathematical wonders of Maxwell's equations. those equations ended in the prediction of radio waves, the conclusion that mild is one of those electromagnetic wave, and the invention of the specified thought of relativity. in reality, just about all present descriptions of the elemental legislation of the universe might be considered as deep generalizations of Maxwell's equations. much more incredible is that those equations and their generalizations have resulted in one of the most very important mathematical discoveries of the prior thirty years. it appears the math in the back of Maxwell's equations is unending. The target of this publication is to provide an explanation for to mathematicians the underlying physics at the back of electrical energy and magnetism and to teach their connections to arithmetic. beginning with Maxwell's equations, the reader is resulted in such issues because the certain conception of relativity, differential kinds, quantum mechanics, manifolds, tangent bundles, connections, and curvature.

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2 ∂t ∂x (Since we know that ∂y ∂t = −v ∂∂ xy , we know that ∂2y ∂2y = −v 2 . ∂ x∂t ∂x 4 Special Relativity Summary: We develop the basics of special relativity. Key is determining the allowable coordinate changes. This will let us show in the next chapter not just how but why magnetism and electricity must be linked. 1. Special Theory of Relativity The physics and mathematics of Maxwell’s equations in the last chapter were worked out during the 1800s. In these equations, there is no real description for why electricity and magnetism should be related.

7. ) There are a number of measurable quantities that tell us about the world. Velocity (distance over time) and acceleration (velocity over time) are two such quantities, whose importance is readily apparent. The importance of other key quantities, such as momentum and the various types of energy, is more difficult to recognize immediately. In fact, Jennifer Coopersmith’s recent Energy, the Subtle Concept: The Discovery of Feynman’s Blocks from Leibniz to Einstein [11] gives a wonderful history of the slow, if not tortuous, history of the development of the concept of energy and, to a lesser extent, momentum.

Fix a value of y(x, t) = y0 . Then we have some x 0 such that y0 = y(x 0 , 0). This means that for all x and t such that x 0 = x − vt, then y0 = y(x, t). The speed of the wave is how fast this point with y-coordinate y0 moves along the x-axis with respect to t. Then dy dt ∂ y dx ∂ y dt + = ∂ x dt ∂t dt ∂ y dx ∂ y + = ∂ x dt ∂t ∂y ∂y + , =v ∂x ∂t giving us that ∂ y/∂t = −v∂ y/∂ x. Person A is looking at the same wave but measures the wave as having speed v + α. We want to see explicitly that under the appropriate change of coordinates this indeed happens.

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