By Sudhir R. Ghorpade, Balmohan V. Limaye
This self-contained textbook offers a radical exposition of multivariable calculus. it may be seen as a sequel to the one-variable calculus textual content, A path in Calculus and actual research, released within the similar sequence. The emphasis is on correlating normal recommendations and result of multivariable calculus with their opposite numbers in one-variable calculus. for instance, whilst the final definition of the quantity of a high-quality 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 pretty good of revolution needs to provide an identical resolution. additional, the publication contains real analogues of simple ends up in one-variable calculus, similar to the suggest worth theorem and the basic theorem of calculus.
This publication is special from others at the topic: it examines issues no longer commonly coated, similar to monotonicity, bimonotonicity, and convexity, including their relation to partial differentiation, cubature ideas for approximate review of double integrals, and conditional in addition to unconditional convergence of double sequence and incorrect double integrals. in addition, the emphasis is on a geometrical method of such simple notions as neighborhood extremum and saddle point.
Each bankruptcy includes specified proofs of correct effects, in addition to a variety of examples and a large number of routines of various levels of trouble, making the publication necessary to undergraduate and graduate scholars alike. there's additionally an informative component of "Notes and Comments’’ indicating a few novel positive aspects of the remedy of subject matters 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)
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]. (iii) Show that f is always of bounded bivariation on [a, b] × [c, d]. (iv) Assume that φ and ψ are not constant functions, that is, φ(x∗ ) = φ(x∗ ) and ψ(y∗ ) = ψ(y ∗ ) for some x∗ , x∗ ∈ [a, b] with x∗ < x∗ and y∗ , y ∗ ∈ [c, d] with y∗ < y ∗ .
If f : [a, b] × [c, d] → R is of bounded variation, then for any (x, y) ∈ [a, b] × [c, d], the restriction f |[a,x]×[c,y] is of bounded variation and V f |[a,x]×[c,y] + |f (b, d) − f (x, y)| ≤ V (f ). Proof. Given any n ∈ N and any (x0 , y0 ), (x1 , y1 ), . . , (xn , yn ) ∈ R2 satisfying (a, c) = (x0 , y0 ) ≤ (x1 , y1 ) ≤ · · · ≤ (xn , yn ) = (x, y), we have n i=1 |f (xi , yi ) − f (xi−1 , yi−1 )| + |f (b, d) − f (x, y)| ≤ V (f ). Hence f |[a,x]×[c,y] is of bounded variation and its total variation is at most V (f ) − |f (b, d) − f (x, y)|.
If it is, then find an 36 1 Vectors and Functions upper bound. Also determine whether f is bounded below. If it is, then find a lower bound. Further, determine whether f attains its upper bound or lower bound. (i) D := S1 (0, 0) and f (x, y) := x2 + y 2 − 1, (ii) D := Sπ (0, 0) and f (x, y) := sin(xy), (iii) D := Sπ/4 (0, 0) and f (x, y) := tan(x + y). 13. Let I, J be nonempty intervals in R. Given any φ : I → R and ψ : J → R, define f, g : I × J → R by f (x, y) := φ(x) + ψ(y) and g(x, y) := φ(x)ψ(y).