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 actual Analysis*, released within the related sequence. The emphasis is on correlating basic innovations and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, while 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 amount of a great of revolution needs to supply an analogous solution. extra, the ebook contains real analogues of uncomplicated leads to one-variable calculus, equivalent to the suggest worth theorem and the elemental theorem of calculus.

This booklet is exceptional from others at the topic: it examines themes now not mostly coated, similar 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 mistaken double integrals. additionally, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle element.

Each bankruptcy includes distinctive proofs of suitable effects, besides quite a few examples and a large choice of routines of various levels of hassle, making the e-book worthy to undergraduate and graduate scholars alike. there's additionally an informative portion of "Notes and Comments’’ indicating a few novel gains of the remedy of themes 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**

**Sample text**

An ) = 0 for all (a1 , . . , an ) ∈ E1 × · · · × En , then show that p(x1 , . . , xn ) must be the zero polynomial, that is, all its coefficients are zero. Now suppose f is a polynomial function on a subset D of Rn , that is, suppose there is a polynomial p(x1 , . . , xn ) in n variables with coefficients in R such that f (a1 , . . , an ) = p(a1 , . . , 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, .

Suppose Γ lies in D, that is, (x(t), y(t)) ∈ D for all t ∈ [α, β], and Γ passes through (x0 , y0 ), that is, (x0 , y0 ) = (x(t0 ), y(t0 )) for some t0 ∈ (α, β). Given any f : D → R, the function F : [α, β] → R given by F (t) := f (x(t), y(t)) is sometimes referred to as the restriction of f to the path Γ . We shall say that f has a local maximum at (x0 , y0 ) along Γ if F has a local maximum at t0 . Likewise, we say that f has a local minimum at (x0 , y0 ) along Γ if F has a local minimum at t0 .

5. Show that if I and J are intervals in R, then the complement of their product, that is, the set D := R2 \ (I × J) = {(x, y) ∈ R2 : (x, y) ∈ I × J}, is path-connected. Further, show that if I and J are nonempty bounded intervals in R, then D is not convex. 6. Let r, s ∈ R with 0 < r < s, and let c ∈ Rn . Show that the sets Bs (c) \ Br (c) and Ss (c) \ Sr (c) are path-connected, but not convex. 7. Let k ∈ N and P1 , . . , Pk ∈ Rn . A convex combination of P1 , . . , Pk is an element in Rn of the form λ1 P1 + · · · + λk Pk , where λ1 , .