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

By Sudhir R. Ghorpade, Balmohan V. Limaye

ISBN-10: 1441916202

ISBN-13: 9781441916204

This self-contained textbook provides an intensive 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 normal suggestions and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, while the final definition of the amount of an outstanding is given utilizing triple integrals, the authors clarify why the shell and washing machine equipment of one-variable calculus for computing the quantity of an effective of revolution needs to provide an identical resolution. additional, the publication contains real analogues of easy leads to one-variable calculus, equivalent to the suggest price theorem and the elemental theorem of calculus.

This ebook is distinct from others at the topic: it examines themes now not commonly lined, comparable 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 mistaken double integrals. additionally, the emphasis is on a geometrical method of such easy notions as neighborhood extremum and saddle aspect.

Each bankruptcy comprises particular proofs of suitable effects, besides quite a few examples and a large choice of workouts of various levels of hassle, making the ebook valuable to undergraduate and graduate scholars alike. there's additionally an informative component to "Notes and Comments’’ indicating a few novel beneficial properties of the remedy of themes 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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Additional info for A Course in Multivariable Calculus and Analysis

Example text

Proof. Let a, b ∈ f (I × J). Then a = f (x1 , y1 ) and b = f (x2 , y2 ) for some (x1 , y1 ), (x2 , y2 ) ∈ I × J. If r ∈ Ia,b , then by the IVP of f on I × J, there is (x0 , y0 ) ∈ I(x1 ,y1 ),(x2 ,y2 ) such that f (x0 , y0 ) = r. Since I(x1 ,y1 ),(x2 ,y2 ) ⊆ I × J, we see that Ia,b ⊆ f (I × J). This proves that f (I × J) is an interval. 24. 23 is not true. In fact, in contrast to one-variable calculus, the converse is not true even for monotonic functions. ) For example, consider I = J = [0, 1] and f : I × J → R defined by f (x, y) := [x] + y, where [x] denotes the integer part of x.

Since E ⊆ I × J, it follows that f has the IVP on I × J. 3 Cylindrical and Spherical Coordinates For points in R2 , one has the familiar notion of polar coordinates. 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.

Show that Q1 , Q2 , Q3 lie on a plane in R3 given by z = Ax + By + C for uniquely determined A, B, C ∈ R. (Hint: Those familiar with determinants may note that P1 , P2 , P3 are noncollinear if and only if the 3×3 matrix having (xi , yi , 1) as its ith row, for i = 1, 2, 3, has a nonzero determinant. ) (ii) Show that f is convex on D if and only if for any noncollinear points P1 , P2 , P3 in D and any (x, y) in the triangle with P1 , P2 , P3 as its vertices, we have f (x, y) ≤ g(x, y), where z = g(x, y) is the equation of the plane passing through (xi , yi , f (xi , yi )) for i = 1, 2, 3.

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

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