Page 191 - Schaum's Outline of Theory and Problems of Advanced Calculus

182 VECTORS [CHAP. 7

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; ðbÞ A B; ðcÞr A are invariant under the transformation of Problem 7.100.
7.103. If u 1 ; u 2 ; u 3 are orthogonal curvilinear coordinates, prove that
@r @r @r

@ðu 1 ; u 2 ; u 3 Þ
¼ru 1 ru 2 ru 3 ðru 1 ru 2 ru 3 Þ¼ 1
ðaÞ ðbÞ
@u 1 @u 2 @u 3
@ðx; y; zÞ
and give the significance 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 A 1 ; A 2 ; ; A n is called linearly dependent if there exists a set of scalars c 1 ; c 2 ; .. . ; c n not all
zero such that c 1 A 1 þ c 2 A 2 þ þ c n A n ¼ 0 identically; otherwise, the set is called linearly independent.
(a)Prove that the vectors A 1 ¼ 2i 3j þ 5k, A 2 ¼ i þ j 2k; A 3 ¼ 3i 7j þ 12k are linearly dependent.
(b)Prove that any four three-dimensional vectors are linearly dependent. (c)Prove that a necessary
and sufficient condition that the vectors A 1 ¼ a 1 i þ b 1 j þ c 1 k, A 2 ¼ a 2 i þ b 2 j þ c 2 k; A 3 ¼ a 3 i þ b 3 j þ c 3 k be
linearly independent is that A 1 A 2 A 3 6¼ 0. Give a geometrical interpretation of this.

7.106. Acomplex number can be defined as an ordered pair ða; bÞ 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)beused
to define subtraction and division? (c) Explain why complex numbers can be considered as two-dimen-
sional vectors. (d)Describe similarities and differences between various operations involving complex
numbers and the vectors considered in this chapter.