By Sudhir R. Ghorpade, Balmohan V. Limaye
This self-contained textbook supplies 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 genuine research, released within the related sequence. The emphasis is on correlating common thoughts and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, while the overall definition of the amount of a pretty good is given utilizing triple integrals, the authors clarify why the shell and washing machine tools of one-variable calculus for computing the amount of a pretty good of revolution needs to supply a similar solution. additional, the ebook contains actual analogues of simple leads to one-variable calculus, resembling the suggest price theorem and the elemental theorem of calculus.
This ebook is amazing from others at the topic: it examines themes now not normally coated, akin to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature principles for approximate review of double integrals, and conditional in addition to unconditional convergence of double sequence and fallacious double integrals. furthermore, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle point.
Each bankruptcy comprises targeted proofs of proper effects, in addition to various examples and a large selection of routines of various levels of hassle, making the e-book important to undergraduate and graduate scholars alike. there's additionally an informative portion of "Notes and Comments’’ indicating a few novel beneficial properties of the remedy of issues in that bankruptcy in addition to references to appropriate literature. the single prerequisite for this article is a direction in one-variable calculus.
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Additional resources for A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics)
Xn ) is a finite sum of terms of the form cxi11 xi22 · · · xinn , where c ∈ R and i1 , . . , in are nonnegative integers; here c is called the coefficient of the term and in case c = 0, the sum i1 + · · · + in is called the total degree of the term. By a zero or a root of p(x1 , . . , xn ) in Rn we mean a point a = (a1 , . . , an ) ∈ Rn such that p(a1 , . . , an ) = 0, that is, by substituting ai in place of xi for each i = 1, . . , n in p(x1 , . . , xn ), we obtain the value 0. (i) Show that if n = 1 and p(x1 ) is a nonzero polynomial (that is, not all of its coefficients are zero), then it has at most finitely many zeros.
For more on quaternions, octonions, and the theorem of Frobenius, one can consult the book of Kantor and Solodovnikov  and the expository article of Baez . Yet another difference between R and Rn is the apparent absence of a natural order on Rn for n > 1. There are, of course, total orders on Rn , such as the lexicographic order on Rn (Exercise 1), that are compatible with the algebraic operations, but they fail to satisfy the archimedean property and the least upper bound property. In fact, H¨ older showed in 1901 that there cannot be an archimedean total order compatible with addition on Rn if n > 1.
Conversely, if ρ, ϕ, θ ∈ R are such that ρ > 0, ϕ ∈ (0, π), and θ ∈ (−π, π], then the real numbers x, y, z defined by x := ρ sin ϕ cos θ, y := ρ sin ϕ sin θ, z := ρ cos ϕ, are such that (x, y) = (0, 0), ρ = x2 + y 2 + z 2 , ϕ = cos−1 (z/ρ), and θ equals cos−1 (x/ρ sin ϕ) or − cos−1 (x/ρ sin ϕ) according as y ≥ 0 or y < 0. Proof. Suppose x, y, z ∈ R with (x, y) = (0, 0) are given. Define ρ, ϕ, and θ by the formulas displayed above. Since (x, y) = (0, 0), we see that ρ > 0 and |z/ρ| < 1. Consequently, ϕ := cos−1 (z/ρ) ∈ (0, π).
A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics) by Sudhir R. Ghorpade, Balmohan V. Limaye