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 direction in Calculus and genuine Analysis, released within the similar sequence. The emphasis is on correlating normal recommendations and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, while the overall definition of the amount of a fantastic 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 high-quality of revolution needs to supply an identical solution. extra, the e-book contains actual analogues of easy leads to one-variable calculus, equivalent to the suggest worth theorem and the elemental theorem of calculus.
This booklet is unique from others at the topic: it examines themes now not usually lined, corresponding to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature principles for approximate overview of double integrals, and conditional in addition to unconditional convergence of double sequence and incorrect double integrals. in addition, the emphasis is on a geometrical method of such easy notions as neighborhood extremum and saddle aspect.
Each bankruptcy includes distinctive proofs of appropriate effects, in addition to a number of examples and a large number of routines of various levels of trouble, making the ebook beneficial to undergraduate and graduate scholars alike. there's additionally an informative element of "Notes and Comments’’ indicating a few novel good points of the therapy of issues in that bankruptcy in addition to references to correct literature. the one prerequisite for this article is a path in one-variable calculus.
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Extra resources for A Course in Multivariable Calculus and Analysis
Clearly f and g are bimonotonic on [0, 1]× [0, 1]; indeed, each is a function of the form φ(x) + ψ(y). In particular, f and g are of bounded bivariation. However, f g is not of bounded bivariation. Indeed, if f g were of bounded bivariation, then by considering grids of the form (x0 , y0 ), . . , (xn , ym ), where m = 1, y0 = 0, and y1 = 1, we see that f would be of bounded variation. 10, f would have to be bounded, which is not the case. 15. y (0, 1) (1, 1) f (x, y) = 1 f (x, y) = 0 (0, 0) (1, 0) x Fig.
Since (x, y) = (0, 0), we see that ρ > 0 and |z/ρ| < 1. Consequently, ϕ := cos−1 (z/ρ) ∈ (0, π). 26, we see that (ρ sin ϕ, θ) are the polar coordinates of (x, y). Hence θ ∈ (−π, π] and moreover, x = ρ sin ϕ cos θ and y = ρ sin ϕ sin θ. Notes and Comments 33 Conversely, suppose ρ, ϕ, θ ∈ R are such that ρ > 0, ϕ ∈ (0, π), and θ ∈ (−π, π]. Define x := ρ sin ϕ cos θ, y := ρ sin ϕ sin θ, and z := ρ cos ϕ. 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).
It follows that f is not of bounded bivariation on [0, 1] × [0, 1]. 20. The concepts of bimonotonicity and bounded bivariation introduced in this chapter for functions of two variables can be extended to n-fold monotonicity and bounded n-fold variation for functions of n variables. To this end, it is useful to consider the difference operator △ defined as follows. Given any a = (a1 , . . , an ) and b = (b1 , . . , bn ) in Rn with a ≤ b, that is, ai ≤ bi for i = 1, . . , n, and any f : [a, b] → R, define △b a f := k(c)f (c), c where the summation is over all c = (c1 , .