00059___9c73f1b8cb2b89656d30b1b890021412

00059___9c73f1b8cb2b89656d30b1b890021412 - whose components...

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38 TENSOR ANALYSIS Chap. 2 2.5. Show that the length of the line element (d@,O,O) is fiIdO'1; that of the line element (0, do2, 0) is /do2 1. 2.6. Let the angle between the line elements (dO',O,O) and (0,d02,0) be denoted by a12. Show that Hint: Find the components of the line elements with respect to a system of rectangular Cartesian coordinates in which we know how to compute the angle between two vectors. 2.4. SCALARS, CONTRAVARIANT VECTORS, COVARIANT VECTORS~ In nonrelativistic physics there are quantities like mass and length which are independent of reference coordinates, and there are quantities like displacement and velocity whose components do dcpend on reference coor- dinates. The former are the scalars; the latter, vectors. Mathematically, we define them according to the way their components change under ad- missible transformations. In the following discussions we consider a system
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Unformatted text preview: whose components are defined in the general set of variables Bi and are functions of B1, B2, B3. If the variables Bi can be changed to ei by an ad- missible and proper transformation, then we can define new components of the system in the new variables @. The systcm will be given various names according to the way in which the new and the old components are related. A system is called a scalar if it has only a single component q5 in the variables Bi and a single component 6 in the variables @, and if q5 and 6 are numerically equal at the corresponding points, A system is called a contravariant vector field or a contravariant tensor field of rank one, if it has three components ti in the variables Bi and three components p in the variables ei, and if the components are related by the law tone may pass over the rest of this chapter and proceed directly to Chapter 3 on first reading....
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