Download Advanced Calculus of Several Variables by C. H. Edwards Jr. PDF

By C. H. Edwards Jr.

Modern conceptual remedy of multivariable calculus, emphasizing the interaction of geometry and research through linear algebra and the approximation of nonlinear mappings by means of linear ones. even as, plentiful consciousness is paid to the classical functions and computational tools. 1000s of examples, difficulties and figures. 1973 edition.

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In order to prove that lim x ^ (1 n / ( x , y) = 3, we first write \f(X9y)-3\ = \χ2 +Xy+y-3\ 2 ^ \x - l| + \y\f(x,y)-3\ 1[ + \xy; l| = | χ + ΐ| \χ-ι\ + ( j - i | + \xy-y Z\x+l\\x-\\ + 2 | ^ - l | + b | \x-\\. +y - i \ , (I) Given ε > 0, we want to find δ > 0 such that |(x, y) - (l, l)| = [(x - l) 2 + (y — l) 2 ] 1 / 2 < δ implies that the right-hand side of (I) is <ε. Clearly we need 7 Limits and Continuity 43 bounds for the coefficients |x + 11 and |>'| of \x - 11 in (1). So let us first agree to choose δ ^ 1, so | ( x , j O - ( l , 1)[ < ( 5 = > | x - 1| < 1 and =>0\x+ 1| < 3 \y - 1| < 1 0·) - 31 ^ 5U - 11 +2\y-l\ by (1).

Yn) e @m+n. If A c ^ m and 5 c @n9 then the Cartesian product A x B = {(a, b)e@m+n:*eA is a subset of @ and b e B} m+n . 5 If A is a compact subset of Mm and i? is a compact subset of 9tn9 then ,4 x £ is a compact subset of @m+n. PROOF Given a sequence {c,}f ={(a f , b,)}f of points of A x B, we want to show that it has a subsequence converging to a point of A x /?. ^ has a subsequence {a^JLi which converges to a point Sie A. }*= i has a subsequence {b(/;k)}^= { which converges to a point be B.

Ii') The intersection of any collection of closed sets is a closed set. (iii') The union of a finite number of closed sets is a closed set. Examples: (a) A closed interval is a closed subset of 01. 3). (c) A finite set F of points is a closed set. (d) The real line 01 is a closed subset of 0l2. ) The last example illustrates the following useful alternative characterization of closed sets. 1 The subset A of0ln is closed if and only if it contains all of its limit points. PROOF Suppose A is closed, and that a is a limit point of A.

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