A Course in Multivariable Calculus and Analysis
Springer Science & Business Media, Dec 10, 2009 - Mathematics - 475 pages
Calculus of real-valued functions of several real variables, also known as m- tivariable calculus, is a rich and fascinating subject. On the one hand, it seeks to extend eminently useful and immensely successful notions in one-variable calculus such as limit, continuity, derivative, and integral to “higher dim- sions. ” On the other hand, the fact that there is much more room to move n about in the n-space R than on the real line R brings to the fore deeper geometric and topological notions that play a signi?cant role in the study of functions of two or more variables. Courses in multivariable calculus at an undergraduate level and even at an advanced level are often faced with the unenviable task of conveying the multifarious and multifaceted aspects of multivariable calculus to a student in the span of just about a semester or two. Ambitious courses and teachers would try to give some idea of the general Stokes’s theorem for di?erential forms on manifolds as a grand generalization of the fundamental theorem of calculus, and prove the change of variables formula in all its glory. They would also try to do justice to important results such as the implicit function theorem, which really have no counterpart in one-variable calculus. Most courses would require the student to develop a passing acquaintance with the theorems of Green, Gauss, and Stokes, never mind the tricky questions about orientability, simple connectedness, etc.
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absolutely convergent ACICARA affine transformation ak,e analogue Area(D bimonotonically bounded bivariation bounded function bounded subset bounded variation centroid closed and bounded conditional convergence Consider f continuous function convex Corollary cubature cubature rule deﬁned defined by f(x,y deﬁnition denote derivatives of f diﬀerentiable divergent double power series double sequence equal example exists f is continuous f is integrable follows Fubini's Theorem function f G K2 G N2 G R2 Hence Hessian form Implicit Function Theorem improper double integral integrable function integral of f interior point JJD f Lemma let f monotonically increasing Moreover nonnegative notion obtain one-variable calculus partial derivatives partition polynomial Proof R2 and let real numbers rectangle Riemann Condition Riemann integrable saddle point series of f Show that f subrectangles subset of R2 triangular region triple integral vector xi,yi xn,yn xo,yo