Download e-book for kindle: A Course in Multivariable Calculus and Analysis by Sudhir R. Ghorpade, Balmohan V. Limaye

By Sudhir R. Ghorpade, Balmohan V. Limaye

ISBN-10: 1441916210

ISBN-13: 9781441916211

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 path in Calculus and genuine research, released within the comparable sequence. The emphasis is on correlating common innovations and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the overall 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 fantastic of revolution needs to supply an analogous resolution. additional, the booklet contains actual analogues of simple ends up in one-variable calculus, resembling the suggest worth theorem and the elemental theorem of calculus.

This booklet is unusual from others at the topic: it examines themes no longer quite often coated, reminiscent of 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 wrong double integrals. in addition, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle point.

Each bankruptcy comprises designated proofs of correct effects, besides a number of examples and a large selection of workouts of various levels of hassle, making the booklet important to undergraduate and graduate scholars alike. there's additionally an informative component of "Notes and Comments’’ indicating a few novel positive factors of the remedy of issues in that bankruptcy in addition to references to appropriate literature. the one 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 (Undergraduate Texts in Mathematics)

Example text

Xn ) is uniquely determined by the function f . 37. Let (x0 , y0 ) and (x1 , y1 ) be any two points in R2 and let Γ be a path joining them, that is, Γ is the path given by (x(t), y(t)), t ∈ [α, β], where x, y : [α, β] → R are continuous functions with (x(α), y(α)) = (x0 , y0 ) and (x(β), y(β)) = (x1 , y1 ). Prove that the image of Γ , that is, the set {(x(t), y(t)) : t ∈ [α, β]}, is a closed and path-connected subset of R2 . 38. Let I and J be nonempty intervals in R and let f : I × J → R be any function that is bimonotonically increasing as well as bimonotonically decreasing.

Y y Γ1 Γ x Γ2 Γ3 x Fig. 8. A nonregular path Γ and regular paths Γ1 , Γ2 , Γ3 ; note that Γ1 and Γ2 as well as Γ1 and Γ3 intersect transversally, but Γ2 and Γ3 do not. For example, the path Γ given by (t2 , t3 ), t ∈ [−1, 1], is not a regular path. ) On the other hand, consider the paths Γ1 , Γ2 , and Γ3 given by (t, −t), (t, t), and (2t + t2 , 2t − t2 ), respectively, with t ∈ [−1, 1] in each case. Each of these paths is regular and passes through the origin (0, 0). The tangent vectors to Γ1 , Γ2 , and Γ3 at the origin are (1, −1), (1, 1), and (2, 2), respectively.

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].

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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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