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Download A Course in Multivariable Calculus and Analysis by Sudhir R. Ghorpade, Balmohan V. Limaye PDF

By Sudhir R. Ghorpade, Balmohan V. Limaye

This self-contained textbook offers a radical exposition of multivariable calculus. it may be seen as a sequel to the one-variable calculus textual content, A path in Calculus and actual research, released within the similar sequence. The emphasis is on correlating common innovations and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the final definition of the amount of an exceptional is given utilizing triple integrals, the authors clarify why the shell and washing machine tools of one-variable calculus for computing the quantity of a superb of revolution needs to provide an analogous resolution. additional, the e-book contains actual analogues of simple ends up in one-variable calculus, resembling the suggest worth theorem and the basic theorem of calculus.

This publication is uncommon from others at the topic: it examines themes now not regularly coated, comparable to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature principles for approximate assessment of double integrals, and conditional in addition to unconditional convergence of double sequence and mistaken double integrals. in addition, the emphasis is on a geometrical method of such easy notions as neighborhood extremum and saddle point.

Each bankruptcy includes specified proofs of proper effects, besides quite a few examples and a large number of routines of various levels of hassle, making the e-book necessary to undergraduate and graduate scholars alike. there's additionally an informative element of "Notes and Comments’’ indicating a few novel good points of the remedy of themes in that bankruptcy in addition to references to appropriate literature. the single prerequisite for this article is a path in one-variable calculus.

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Extra info for A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics)

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Moreover, a new and interesting phenomenon emerges, namely, that of a saddle point. To define the latter, we first introduce some terminology concerning paths in R2 . Let Γ be a path in R2 given by (x(t), y(t)), t ∈ [α, β]. The path Γ is said to pass through a point (x0 , y0 ) ∈ R2 if there is t0 ∈ (α, β) such that (x(t0 ), y(t0 )) = (x0 , y0 ). 1 of ACICARA), we say that the tangent to Γ at a point (x(t0 ), y(t0 )), where t0 ∈ (α, β), is defined if x, y are differentiable at t0 and (x′ (t0 ), y ′ (t0 )) = (0, 0).

Then x2 + y 2 = ρ2 sin2 ϕ > 0, and hence (x, y) = (0, 0). Also, it is clear that ρ= x2 + y 2 + z 2 and ϕ = cos−1 (z/ρ). 26 with r := ρ sin ϕ, we readily see that θ equals cos−1 (x/ρ sin ϕ) or − cos−1 (x/ρ sin ϕ) according as y ≥ 0 or y < 0. 27, we obtain a one-toone correspondence between the sets {(x, y, z) ∈ R3 : (x, y) = (0, 0)} and {(ρ, ϕ, θ) ∈ R3 : ρ > 0, ϕ ∈ (0, π), θ ∈ (π, π]}. 28. 27 also give the following relation between the cylindrical and the spherical coordinates of a point. Let P be a point of R3 that is not on the z-axis.

Let f : [a, b] × [c, d] → R be a function on the rectangle [a, b] × [c, d]. For a fixed y ∈ [c, d], let φy : [a, b] → R denote the function defined by φy (x) := f (x, y). Also, for a fixed x ∈ [a, b], let ψx : [c, d] → R denote the function defined by ψx (y) := f (x, y). Assume that f is bimonotonically increasing and prove the following. (i) If φc is increasing on [a, b], then so is φy for every y ∈ [c, d]. (ii) If ψa is increasing on [c, d], then so is ψx for every x ∈ [a, b]. (iii) If φc is increasing on [a, b] and ψa is increasing on [c, d], then f is monotonically increasing on [a, b] × [c, d].

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