By Sudhir R. Ghorpade, Balmohan V. Limaye
This self-contained textbook offers an intensive exposition of multivariable calculus. it may be considered 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 ideas 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 a superior of revolution needs to supply an analogous resolution. extra, the booklet comprises actual analogues of uncomplicated ends up in one-variable calculus, equivalent to the suggest worth theorem and the basic theorem of calculus.
This ebook is distinct from others at the topic: it examines subject matters now not as a rule lined, comparable to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature ideas for approximate assessment of double integrals, and conditional in addition to unconditional convergence of double sequence and fallacious double integrals. in addition, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle element.
Each bankruptcy comprises exact proofs of correct effects, besides a variety of examples and a large number of routines of various levels of hassle, making the booklet invaluable to undergraduate and graduate scholars alike. there's additionally an informative component to "Notes and Comments’’ indicating a few novel positive factors of the remedy of subject matters in that bankruptcy in addition to references to proper literature. the single 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
43. Let φ : [a, b] → R and ψ : [c, d] → R be functions of one variable. Define f, g : [a, b] × [c, d] → R by f (x, y) := φ(x) + ψ(y) and g(x, y) := φ(x)ψ(y). (i) Show that f is of bounded variation on [a, b] × [c, d] if and only if φ is of bounded variation on [a, b] and ψ is of bounded variation on [c, d]. (ii) Assume that φ and ψ are not identically zero, that is, φ(x0 ) = 0 and ψ(y0 ) = 0 for some x0 ∈ [a, b] and y0 ∈ [c, d]. Show that g is of bounded variation on [a, b] × [c, d] if and only if φ is of bounded variation on [a, b] and ψ is of bounded variation on [c, d].
In the 3-space R3 , there are at least two important and useful representations of points, and these are known as cylindrical coordinates and spherical coordinates. Of these, the former is a straightforward extension of the notion of polar coordinates, and we describe it first. 3 Cylindrical and Spherical Coordinates 31 Cylindrical Coordinates The cylindrical coordinates of a point that is not on the z-axis, that is, a point (x, y, z) in R3 for which (x, y) = (0, 0), are defined to be the triple (r, θ, z), where (r, θ) are the polar coordinates of (x, y).
Denote by Tf the set of finite double sums of the form n m i=1 j=1 |f (xi , yj ) + f (xi−1 , yj−1 ) − f (xi , yj−1 ) − f (xi−1 , yj )| , where n, m ∈ N and (x0 , y0 ), . . , (xn , ym ) are any points in R2 satisfying a = x0 ≤ x1 ≤ · · · ≤ xn−1 ≤ xn = b and c = y0 ≤ y1 ≤ · · · ≤ ym−1 ≤ ym = d. If the set Tf is bounded above, then f is said to be of bounded bivariation. In this case, we denote the supremum of Tf by W (f ), and call it the total bivariation of f on [a, b] × [c, d]. 19 (ii)). 15. Let f, g : [a, b] × [c, d] → R and r ∈ R.
A Course in Multivariable Calculus and Analysis by Sudhir R. Ghorpade, Balmohan V. Limaye