By Sudhir R. Ghorpade, Balmohan V. Limaye
This self-contained textbook supplies an intensive exposition of multivariable calculus. it may be seen as a sequel to the one-variable calculus textual content, A direction in Calculus and actual research, released within the similar sequence. The emphasis is on correlating basic options and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the final definition of the amount of an effective is given utilizing triple integrals, the authors clarify why the shell and washing machine equipment of one-variable calculus for computing the amount of a pretty good of revolution needs to supply a similar solution. extra, the e-book contains actual analogues of simple leads to one-variable calculus, reminiscent of the suggest worth theorem and the elemental theorem of calculus.
This e-book is individual from others at the topic: it examines subject matters now not regularly lined, comparable to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature principles for approximate evaluate of double integrals, and conditional in addition to unconditional convergence of double sequence and flawed double integrals. additionally, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle point.
Each bankruptcy includes specified proofs of correct effects, besides various examples and a large number of routines of various levels of hassle, making the e-book worthwhile to undergraduate and graduate scholars alike. there's additionally an informative component of "Notes and Comments’’ indicating a few novel good points of the remedy of subject matters in that bankruptcy in addition to references to proper literature. the one prerequisite for this article is a direction in one-variable calculus.
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Extra info for A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics)
N and we have p(a1 , . . , an ) = 0 for all (a1 , . . , an ) ∈ E1 × · · · × En , then show that p(x1 , . . , xn ) must be the zero polynomial, that is, all its coefficients are zero. Now suppose f is a polynomial function on a subset D of Rn , that is, suppose there is a polynomial p(x1 , . . , xn ) in n variables with coefficients in R such that f (a1 , . . , an ) = p(a1 , . . , an ) for all (a1 , . . , an ) ∈ D. Show that if D = E1 × · · · × En , where Ej is an interval containing more than one point in R for each j = 1, .
Let (x1 , y1 ), (x2 , y2 ) ∈ [a, b] × [c, d] with (x1 , y1 ) ≤ (x2 , y2 ). 11 to the restriction f |[a,x2 ]×[c,y2 ] , we see that vf (x1 , y1 ) + |f (x2 , y2 ) − f (x1 , y1 )| ≤ vf (x2 , y2 ), and hence g(x2 , y2 ) − g(x1 , y1 ) = 1 [vf (x2 , y2 ) − vf (x1 , y1 ) + f (x2 , y2 ) − f (x1 , y1 )] ≥ 0 2 as well as h(x2 , y2 ) − h(x1 , y1 ) = 1 [vf (x2 , y2 ) − vf (x1 , y1 ) − f (x2 , y2 ) + f (x1 , y1 )] ≥ 0. 2 Thus g and h are monotonically increasing. The uniqueness of g and h is obvious from the conditions f = g − h and vf = g + h.
Xn , yn ) ∈ R2 satisfying (a, c) = (x0 , y0 ) ≤ (x1 , y1 ) ≤ · · · ≤ (xn , yn ) = (x, y), we have n i=1 |f (xi , yi ) − f (xi−1 , yi−1 )| + |f (b, d) − f (x, y)| ≤ V (f ). Hence f |[a,x]×[c,y] is of bounded variation and its total variation is at most V (f ) − |f (b, d) − f (x, y)|. ⊔ ⊓ If f : [a, b] × [c, d] → R is of bounded variation, then we define the corresponding total variation function vf : [a, b] × [c, d] → R by vf (x, y) := V f |[a,x]×[c,y] . The following result gives the so-called Jordan decomposition of a function of bounded variation.
A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics) by Sudhir R. Ghorpade, Balmohan V. Limaye