Mathematical methods for physicists: a concise introduction by Tai L. Chow

By Tai L. Chow

This article is designed for an intermediate-level, two-semester undergraduate path in mathematical physics. It presents an available account of many of the present, very important mathematical instruments required in physics nowadays. it really is assumed that the reader has an enough education commonly physics and calculus. The ebook bridges the distance among an introductory physics direction and extra complex classes in classical mechanics, electrical energy and magnetism, quantum mechanics, and thermal and statistical physics. The textual content incorporates a huge variety of labored examples to demonstrate the mathematical suggestions constructed and to teach their relevance to physics. The booklet is designed essentially for undergraduate physics majors, yet may be utilized by scholars in different topics, equivalent to engineering, astronomy and arithmetic.

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Let the new coordinates u1 ; u2 ; u3 be de®ned by specifying the Cartesian coordinates (x1 ; x2 ; x3 ) as functions of (u1 ; u2 ; u3 †: x1 ˆ f …u1 ; u2 ; u3 †; x2 ˆ g…u1 ; u2 ; u3 †; x3 ˆ h…u1 ; u2 ; u3 †; …1:54† where f, g, h are assumed to be continuous, di€erentiable. A point P (Fig. 16) in space can then be de®ned not only by the rectangular coordinates (x1 ; x2 ; x3 ) but also by curvilinear coordinates (u1 ; u2 ; u3 ). If u2 and u3 are constant as u1 varies, P (or its position vector r) describes a curve which we call the u1 coordinate curve.

Alternatively dV can be written as     @…x1 ; x2 ; x3 †  @r @r @r   du du du ;  Á  du du du ˆ dV ˆ  @u1 @u2 @u3  1 2 3  @…u1 ; u2 ; u3 †  1 2 3 where   @x1   @u1   @…x1 ; x2 ; x3 †  @x2 ˆ Jˆ @…u1 ; u2 ; u3 †  @u1   @x3   @u1 @x1 @u2 @x2 @u2 @x3 @u2 …1:57†  @x1   @u3   @x2   @u3   @x3   @u3  is called the Jacobian of the transformation. 54) is one to one in the neighborhood of a point. We are now ready to express the gradient, divergence, and curl in terms of u1 ; u2 , and u3 .

Tensors of second rank From two contravariant vectors A and B we may form the N 2 quantities A B . This is known as the outer product of tensors. These N 2 quantities form the components of a contravariant tensor of the second rank: any aggregate of N 2 quantities T  which, under a coordinate change, transform like the product of 48 BASIC OPERATIONS WITH TE NS ORS two contravariant vectors T  ˆ @x @x H T ; @x H @x H …1:97† is a contravariant tensor of rank two. We may also form a covariant tensor of rank two from two covariant vectors, which transforms according to the formula T ˆ @x H @x H H T : @x @x …1:98† Similarly, we can form a mixed tensor T   of order two that transforms as follows: T  ˆ @x @x H H T : @x H @x …1:99† We may continue this process and multiply more than two vectors together, taking care that their indexes are all di€erent.

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