By Sudhir R. Ghorpade, Balmohan V. Limaye
This self-contained textbook offers an intensive exposition of multivariable calculus. it may be considered as a sequel to the one-variable calculus textual content, A direction in Calculus and genuine research, released within the related sequence. The emphasis is on correlating normal strategies and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the overall definition of the quantity 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 an effective of revolution needs to provide an analogous solution. extra, the ebook contains actual analogues of easy ends up in one-variable calculus, resembling the suggest worth theorem and the basic theorem of calculus.
This ebook is wonderful from others at the topic: it examines issues no longer ordinarily lined, corresponding 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 incorrect double integrals. additionally, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle point.
Each bankruptcy comprises distinct proofs of proper effects, in addition to quite a few examples and a large selection of routines of various levels of hassle, making the booklet necessary to undergraduate and graduate scholars alike. there's additionally an informative component to "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 single 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 (Undergraduate Texts in Mathematics)
Example text
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, . . , n, then the polynomial p(x1 , . . , xn ) is uniquely determined by the function f . 37. Let (x0 , y0 ) and (x1 , y1 ) be any two points in R2 and let Γ be a path joining them, that is, Γ is the path given by (x(t), y(t)), t ∈ [α, β], where x, y : [α, β] → R are continuous functions with (x(α), y(α)) = (x0 , y0 ) and (x(β), y(β)) = (x1 , y1 ).
Xn ) ∈ Rn , define the p-norm of x by 1/p x p := |x1 |p + · · · + |xn |p . Show that x p ≥ 0, and moreover, x for any x, y ∈ Rn and r ∈ R, we have x+y p ≤ x p + y p p = 0 ⇐⇒ x = 0. Also, show that and rx p = |r| x p . ) 34. Extend the p-norm defined in Exercise 33 to the case p = ∞ as follows. x ∞ for x = (x1 , . . , xn ) ∈ Rn . := max (|x1 |, . . , |xn |) Given any p, q with 1 ≤ p ≤ q ≤ ∞, prove that x q ≤ x p ≤ nλ x q for all x ∈ Rn , where λ := 1/p if p < q = ∞ and λ := (q − p)/pq if p ≤ q < ∞, while λ := 0 if p = q = ∞.
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). 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).