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7.101. If A is the vector of Problem 7.100, prove that the divergence of A, i.e., r A, is an invariant (often called a scalar invariant), i.e., prove that
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The results of this and the preceding problem express an obvious requirement that physical quantities must not depend on coordinate systems in which they are observed. Such ideas when generalized lead to an important subject called tensor analysis, which is basic to the theory of relativity. 7.102. Prove that (a) A B; b A B; c r A are invariant under the transformation of Problem 7.100.
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7.103. If u1 ; u2 ; u3 are orthogonal curvilinear coordinates, prove that   @ u1 ; u2 ; u3 @r @r @r a ru1 ru2 ru3 b ru1 ru2 ru3 1 @ x; y; z @u1 @u2 @u3 and give the signi cance of these in terms of Jacobians. 7.104. Use the axiomatic approach to vectors to prove relation (8) on Page 155. 7.105. A set of n vectors A1 ; A2 ; ; An is called linearly dependent if there exists a set of scalars c1 ; c2 ; . . . ; cn not all zero such that c1 A1 c2 A2 cn An 0 identically; otherwise, the set is called linearly independent. (a) Prove that the vectors A1 2i 3j 5k, A2 i j 2k; A3 3i 7j 12k are linearly dependent. (b) Prove that any four three-dimensional vectors are linearly dependent. (c) Prove that a necessary and su cient condition that the vectors A1 a1 i b1 j c1 k, A2 a2 i b2 j c2 k; A3 a3 i b3 j c3 k be linearly independent is that A1 A2 A3 6 0. Give a geometrical interpretation of this. 7.106. A complex number can be de ned as an ordered pair a; b of real numbers a and b subject to certain rules of operation for addition and multiplication. (a) What are these rules (b) How can the rules in (a) be used to de ne subtraction and division (c) Explain why complex numbers can be considered as two-dimensional vectors. (d) Describe similarities and di erences between various operations involving complex numbers and the vectors considered in this chapter.
Applications of Partial Derivatives
APPLICATIONS TO GEOMETRY The theoretical study of curves and surfaces began more than two thousand years ago when Greek philosopher-mathematicians explored the properties of conic sections, helixes, spirals, and surfaces of revolution generated from them. While applications were not on their minds, many practical consequences evolved. These included representation of the elliptical paths of planets about the sun, employment of the focal properties of paraboloids, and use of the special properties of helixes to construct the double helical model of DNA. The analytic tool for studying functions of more than one variable is the partial derivative. Surfaces are a geometric starting point, since they are represented by functions of two independent variables. Vector forms of many of these these concepts were introduced in the previous chapter. In this one, corresponding coordinate equations are exhibited.
Fig. 8-1
1. Tangent Plane to a Surface. Let F x; y; z 0 be the equation of a surface S such as shown in Fig. 8-1. We shall assume that F, and all other functions in this chapter, is continuously di erentiable unless otherwise indicated. Suppose we wish to nd the equation of a tangent plane to S at the point P x0 ; y0 ; z0 . A vector normal to S at this point is N0 rFjP , the subscript P indicating that the gradient is to be evaluated at the point P x0 ; y0 ; z0 . If r0 and r are the vectors drawn respectively from O to P x0 ; y0 ; z0 and Q x; y; z on the plane, the equation of the plane is r r0 N0 r r0 rFjP 0 since r r0 is perpendicular to N0 . 183
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