# New PDF release: Advanced Calculus of Several Variables By C. H. Edwards Jr.

ISBN-10: 0122325508

ISBN-13: 9780122325502

Modern conceptual remedy of multivariable calculus, emphasizing the interaction of geometry and research through linear algebra and the approximation of nonlinear mappings through linear ones. while, considerable awareness is paid to the classical functions and computational tools. thousands of examples, difficulties and figures. 1973 edition.

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Extra info for Advanced Calculus of Several Variables

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Let the matrices A, B, C be of dimensions k x /, / x m, and m x n respectively. Then let f:&l^&k,g: Mm -► M\ /? : &n -► Mm be the linear maps such that Mf = A,Mg = B, Mh = C. Then (/o(0oA))(x) =f(goh(x)) = f(9(h)x)))=(fog)(h(x)) = ((fog)oh)(x) n for all x e M , so f° (g ° h) = (f ° g) ° h. 2 therefore implies that A(BC) = MfMgûh = Mfo(goh) = (MfMg)Mh = (AB)C, thereby verifying associativity. To prove (b), let A be an / x m matrix, and B, C m x n matrices. Then let / : &m -> St1 and g, h : 0tn -+ \$m be the linear maps such that Mf = A, Mg = B, and Mh = C.

A„), (4) 36 I Euclidean Space and Linear Mappings Z)(a1? . , a4 + ra,·, . . , an) = Z)(al5 . . , a,, . . , a„), (5) and D ( a 1 ? . , a,·, . . , a i5 . . , a„) = - / ) ( a l 5 . . , a t , . . , a y , . . y. , An are as usual the column vectors of A. Then (4) above says that the determinant of A is multiplied by r if some column of A is multiplied by r, (5) that the determinant of A is unchanged if a multiple of one column is added to another column, while (6) says that the sign of det A is changed by an interchange of any two columns of A.

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. = { is a subsequence of the original sequence {(af, b , ) ^ } which converges to the point (a, b) e A x B. | We are now ready for the criterion that will serve as our recognition test for compact sets.