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

By Sudhir R. Ghorpade, Balmohan V. Limaye

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 genuine research, released within the related sequence. The emphasis is on correlating basic thoughts and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, while the overall definition of the quantity 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 quantity of a great of revolution needs to supply an identical resolution. extra, the ebook comprises actual analogues of simple ends up in one-variable calculus, comparable to the suggest price theorem and the basic theorem of calculus.

This e-book is special from others at the topic: it examines issues no longer quite often coated, similar 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 wrong double integrals. additionally, the emphasis is on a geometrical method of such uncomplicated notions as neighborhood extremum and saddle point.

Each bankruptcy includes exact proofs of appropriate effects, in addition to quite a few examples and a large choice of workouts of various levels of trouble, making the e-book worthy to undergraduate and graduate scholars alike. there's additionally an informative element of "Notes and Comments’’ indicating a few novel good points of the remedy of themes in that bankruptcy in addition to references to appropriate literature. the single prerequisite for this article is a direction in one-variable calculus.

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Example text

Clearly, f is monotonic as well as bimonotonic. Moreover, f (I × J) = [0, 2] is an interval in R. Note, however, that the real number 43 lies between f (0, 0) = 0 and f (1, 12 ) = 32 , but 43 is not 30 1 Vectors and Functions the value of f at any point on the 2-interval I(0,0),(1, 12 ) = [0, 1] × [0, 12 ]. Indeed, the image of this 2-interval is [0, 12 ] ∪ [1, 23 ], which is not an interval in R. 23 yields a characterization of the IVP. 25. Let D ⊆ R2 and let f : D → R be a function. Then for any 2-interval I × J ⊆ D, f has the IVP on I × J ⇐⇒ f (E) is an interval in R for every 2-interval E ⊆ I × J.

Xn , ym ), where m = 1, y0 = 0, and y1 = 1, we see that f would be of bounded variation. 10, f would have to be bounded, which is not the case. 15. y (0, 1) (1, 1) f (x, y) = 1 f (x, y) = 0 (0, 0) (1, 0) x Fig. 7. 19 (iii) and the points (xi , yi ) of the rectangle [0, 1] × [0, 1] that straddle the diagonal line y = 1 − x. (iii) Consider f : [0, 1] × [0, 1] → R defined by f (x, y) := 0 if x + y ≤ 1, 1 if x + y > 1. Then f is monotonically increasing in [0, 1] × [0, 1]. Indeed, given any (x1 , y1 ), (x2 , y2 ) ∈ [0, 1] × [0, 1] with (x1 , y1 ) ≤ (x2 , y2 ), we have x1 + y1 ≤ x2 + y2 , and hence x1 + y1 > 1 implies x2 + y2 > 1.

The tangent vectors to Γ1 , Γ2 , and Γ3 at the origin are (1, −1), (1, 1), and (2, 2), respectively. Hence Γ1 and Γ2 intersect transversally at (0, 0); also, Γ1 and Γ3 intersect transversally at (0, 0), but Γ2 and Γ3 do not intersect transversally at (0, 0). ) Let D ⊆ R2 and let (x0 , y0 ) be an interior point of D. We say that a function f : D → R has 28 1 Vectors and Functions 1. a local maximum at (x0 , y0 ) if there is δ > 0 such that Sδ (x0 , y0 ) ⊆ D and f (x, y) ≤ f (x0 , y0 ) for all (x, y) ∈ Sδ (x0 , y0 ), 2.

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