By Sudhir R. Ghorpade, Balmohan V. Limaye

This self-contained textbook offers 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 research, released within the comparable sequence. The emphasis is on correlating normal thoughts and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the overall definition of the amount of a superior is given utilizing triple integrals, the authors clarify why the shell and washing machine tools of one-variable calculus for computing the amount of an outstanding of revolution needs to supply an analogous solution. additional, the ebook comprises real analogues of uncomplicated leads to one-variable calculus, akin to the suggest worth theorem and the basic theorem of calculus.

This booklet is distinctive from others at the topic: it examines subject matters no longer mostly lined, akin to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature principles for approximate assessment of double integrals, and conditional in addition to unconditional convergence of double sequence and unsuitable double integrals. furthermore, the emphasis is on a geometrical method of such easy notions as neighborhood extremum and saddle point.

Each bankruptcy includes precise proofs of proper effects, besides quite a few examples and a large selection of workouts of various levels of hassle, making the publication invaluable to undergraduate and graduate scholars alike. there's additionally an informative part of "Notes and Comments’’ indicating a few novel beneficial properties of the therapy of themes in that bankruptcy in addition to references to suitable literature. the single prerequisite for this article is a path in one-variable calculus.

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

**Example text**

Changing the inequalities ≤ and ≥ to strict inequalities < and >, respectively, in 1 and 2 above, we obtain the notions of strictly convex and strictly concave functions. Geometrically speaking, convex functions are those whose graph lies below the triangle in the plane determined by three points on the graph. More precisely, if D ⊆ R2 is convex and not a line segment in R2 , then f : D → R is convex on D if and only if for any noncollinear points Pi := (xi , yi ), i = 1, 2, 3, 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.

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.

First, functions defined on N with values in R2 , and second, functions defined on N2 with values in R. As we shall see, for developing the notions of continuity and limit of a function of two variables, only the former is relevant, and it is studied in this chapter. The study of the latter will be taken up in Chapter 7. This chapter is organized as follows. 1 below and their fundamental properties, including the Bolzano– Weierstrass Theorem and the Cauchy Criterion, are derived from the corresponding results for sequences in R.