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

By Sudhir R. Ghorpade, Balmohan V. Limaye

This self-contained textbook provides 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 techniques and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the final definition of the quantity of an excellent 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 high-quality of revolution needs to supply an analogous resolution. extra, the booklet comprises real analogues of uncomplicated ends up in one-variable calculus, reminiscent of the suggest price theorem and the elemental theorem of calculus.

This booklet is wonderful from others at the topic: it examines issues now not as a rule lined, resembling 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 wrong double integrals. in addition, the emphasis is on a geometrical method of such easy notions as neighborhood extremum and saddle point.

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

Example text

Show that is a total order on Rn . Further, show that is compatible with the algebraic operations in the sense that for any x, y ∈ Rn with x y, we have x + z y + z for all z ∈ Rn , and also, cx cy or cx cy according as c ≥ 0 or c ≤ 0. 2. (Parallelogram Law) Show that |x + y|2 + |x − y|2 = 2|x|2 + 2|y|2 for all x, y ∈ Rn . 3. Let c = (c1 , . . , cn ) ∈ Rn and r ∈ R with r > 0. (i) Show that Br (c) ⊆ Sr (c) ⊆ Br√n (c). (ii) If we let Mr (c) := {x ∈ Rn : |x1 − c1 | + · · · + |xn − cn | < r}, then show that Mr (c) ⊆ Br (c) ⊆ Mrn (c).

A function φ : [a, b] → R of one variable is said to n be of bounded variation on [a, b] if the set S := { i=1 |φ(xi )−φ(xi−1 )| : n ∈ N and 1 = x0 ≤ x1 ≤ · · · ≤ xn = b} is bounded above in R. In this case, we denote the supremum of S by V (φ) and call this the total variation of φ on [a, b]. Prove that φ : [a, b] → R is of bounded variation if and only if φ is a difference of two monotonically increasing functions. 42. Given any n ≥ 0, let φn : [0, 1] → R be defined by φn (0) := 0 and φn (x) := xn sin(1/x) for 0 < x ≤ 1.

It turns out that the properties of (xn , yn ) can be completely understood in terms of the properties of the sequences (xn ) and (yn ) in R. 1. Given a sequence (xn , yn ) in R2 , we have the following. (i) If (xn , yn ) is convergent, then it has a unique limit. (ii) (xn , yn ) is bounded ⇐⇒ both (xn ) and (yn ) are bounded. (iii) (xn , yn ) is convergent ⇐⇒ both (xn ) and (yn ) are convergent. In fact, for (x0 , y0 ) ∈ R2 , we have (xn , yn ) → (x0 , y0 ) ⇐⇒ xn → x0 and yn → y0 . Proof. Each of (i), (ii), and (iii) is immediate from the definitions.