Get A Course in Multivariable Calculus and Analysis PDF

By Sudhir R. Ghorpade, Balmohan V. Limaye

ISBN-10: 1441916210

ISBN-13: 9781441916211

This self-contained textbook supplies a radical exposition of multivariable calculus. it may be considered as a sequel to the one-variable calculus textual content, A path in Calculus and genuine research, released within the similar sequence. The emphasis is on correlating common suggestions and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, while the final definition of the quantity of a high-quality is given utilizing triple integrals, the authors clarify why the shell and washing machine equipment of one-variable calculus for computing the quantity of a high-quality of revolution needs to provide an identical solution. additional, the e-book contains real analogues of easy leads to one-variable calculus, reminiscent of the suggest worth theorem and the basic theorem of calculus.

This ebook is individual from others at the topic: it examines themes no longer regularly lined, comparable to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature ideas for approximate review of double integrals, and conditional in addition to unconditional convergence of double sequence and incorrect double integrals. furthermore, the emphasis is on a geometrical method of such easy notions as neighborhood extremum and saddle point.

Each bankruptcy comprises certain proofs of suitable effects, in addition to quite a few examples and a large number of workouts of various levels of hassle, making the publication priceless to undergraduate and graduate scholars alike. there's additionally an informative part of "Notes and Comments’’ indicating a few novel positive aspects of the therapy of subject matters in that bankruptcy in addition to references to appropriate literature. the one 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)

Example text

Show that f : R2 → R defined by f (x, y) := x2 − y 2 has a saddle point at (0, 0). 28. Recall that for any a ∈ R, the integer part of a is denoted by [a]. Consider f : R2 → R defined by one of the following. In each case, determine whether f has the IVP on R2 . (i) f (x, y) := x + y, (ii) f (x, y) := [x] + [y], (iii) f (x, y) = x + [y]. 29. Let R := [0, 2] × [0, 2] and let f : R → R be defined by   if 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1, 0 f (x, y) := x − 1 if 1 < x ≤ 2 and 0 ≤ y ≤ 1,   1 if 0 ≤ x ≤ 2 and 1 < y ≤ 2.

These provide an alternative and useful way to represent points in the plane other than the origin. Recall that the polar coordinates of (x, y) ∈ R2 \ {(0, 0)} are given by (r, θ), where r and θ are real numbers determined by the equations x = r cos θ and y = r sin θ and the conditions r > 0 and θ ∈ (−π, π]. The precise relationship is stated below. 20 of ACICARA. 26. If x, y ∈ R are such that (x, y) = (0, 0), then r and θ defined by  x  cos−1 if y ≥ 0,   r and θ := r := x2 + y 2   − cos−1 x if y < 0, r satisfy the following properties: r, θ ∈ R, r > 0, θ ∈ (−π, π], x = r cos θ, and y = r sin θ.

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 , .

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A Course in Multivariable Calculus and Analysis (Undergraduate Texts in Mathematics) by Sudhir R. Ghorpade, Balmohan V. Limaye

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