Advanced Calculus fi..

Chapter 2 Differential Calculus of Functions of Several Variables 1679. Let f(x, y) and g(x, y, u) be such thatwhen u = f (x, y). Then show thatf(x,y) and g[x,v, f(x. y)lare functionally dependent.10. Let u(x, y ) and v(x, y) be harmonic in a domain D and have no critical points in D. Showthat if u and v are functionally dependent, then they are "linearly" dependent: u = av + bfor suitable constants a and b. [Hint: Assume a relation of form u = f (v) and take theLaplacian of both sides.]11. a) Show that if linear functions u = ax + by, v = cx + d y are functionally dependentthen they are linearly dependent: hu(x, y) + kv(x, y) _= 0 for suitable constants hand k. [Hint: Differentiate the identity F(ax + by, cx + dy) = 0 with respect to xand y .]b) Generalize the result of part (a) to n functions of m variables.12. With reference to Example 2 in Section 2.22, verify that A has rank 1 in E3, that the F,can be chosen as stated, that the range of u = f(x) is an ellipse and that (aF,/auk) hasrank 2 in a domain D, containing the ellipse. (Here u = f(x) is a submersion of E3 ontothe ellipse.)*2.23 REAL VARIABLE THEORY THEOREM ON MAXIMUMAND MINIMUMIn this section we prove some basic theorems on sets and on functions of realvariables. As will be seen, the concept of sequence is important for the development.(By "sequence" we here mean "infinite sequence.")We take the real number system itself as known and think of real numbersas those representable as decimal expressions such as 2.3175, -1.3333. . . , and3.14159.. . . These include the integers 0, k1, f 2, . . . , and every real number xsatisfies k 5 x < k + 1 for a unique integer k. They also include the rationalnumbers: numbers representable as plq, where p and q are integers with q # 0.If we represent the rational numbers as points on a line, the number axis, thenthese points do not fill the line: there are infinitely many points left out, such as thoserepresenting the real numbers 2/2, rr and e and, in general, all the irrational numbers(those which are not rational). For each irrational number x and each positive integerq, we can find a unique integer k such that k/q < x < (k + l)/q. We need onlychoose k as above so that k < qx < k + 1 (equality is excluded here); thus x isapproximated by rational numbers above and below within distance r = 1 /q.The sequence s, is said to converge to the real numbers (written s, + s) or havelimit s if for each r > 0 there is an integer N for which Is, - s I < r for n > N. Thelimit, when it exists, is unique, since if also s, + sf, thenfor n sufficiently large; this holds for every r > 0, so that s' = s.A sequence s, is said to be rnonotone increasing if s, _< s,+l for all n andmonotone decreasing if s, 2 s,+, for all n; if the inequality is strict (< or >), then