Calculus

Download PDF by James J. Callahan: Advanced Calculus: A Geometric View

By James J. Callahan

ISBN-10: 1441973311

ISBN-13: 9781441973313

With a clean geometric process that comes with greater than 250 illustrations, this textbook units itself except all others in complex calculus. along with the classical capstones--the swap of variables formulation, implicit and inverse functionality theorems, the indispensable theorems of Gauss and Stokes--the textual content treats different vital themes in differential research, equivalent to Morse's lemma and the Poincaré lemma. the tips at the back of so much subject matters should be understood with simply or 3 variables. This invitations geometric visualization; the e-book contains smooth computational instruments to offer visualization actual strength. utilizing 2nd and 3D pictures, the e-book bargains new insights into primary components of the calculus of differentiable maps, reminiscent of the function of the by-product because the neighborhood linear approximation to a map and its function within the swap of variables formulation for a number of integrals. The geometric subject matter maintains with an research of the actual which means of the divergence and the curl at a degree of element now not present in different complex calculus books. complex Calculus: a geometrical View is a textbook for undergraduates and graduate scholars in arithmetic, the actual sciences, and economics. necessities are an advent to linear algebra and multivariable calculus. there's adequate fabric for a year-long direction on complicated calculus and for quite a few semester courses--including themes in geometry. It avoids duplicating the cloth of genuine research. The measured velocity of the e-book, with its huge examples and illustrations, make it in particular compatible for self sustaining study.

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Suppose x(t), a ≤ t ≤ b parametrizes +C, and r(u), c ≤ u ≤ d parametrizes −C. Then there is a parameter change t = h(u) with h′ ≤ 0, h(c) = b, h(d) = a. We have x′ (t) dt = u b → x(a) r(b) b Reversing orientation x′(t) → −C arc length of +C = Unit-speed parametrization Arc length of an unoriented path 18 1 Starting Points d c x′ (h(u)) h′ (u) du = d r′ (u) du = arc length of −C. c Thus, we can assign to the underlying unoriented path C an arc length equal to the common arc lengths of +C and −C.

C Thus, we can assign to the underlying unoriented path C an arc length equal to the common arc lengths of +C and −C. The path integral of a scalar function Mass of a wire Note that the integrand of every path integral we have considered—with the exception of the integral for arc length—has been a vector function. , the orientation) of the integration path. With a scalar function, there is no similar concern and, as we have seen, arc length is meaningful for an unoriented path. We now define the integral of a general scalar function over an unoriented path, illustrating the ideas by using mass density.

There is still no convenient antiderivative, but now make a change to polar coordinates, x = r cos θ , y = r sin θ . The new limits of integration are then 0 ≤ θ ≤ 2π , 0 ≤ r < ∞, and dx dy becomes r dr d θ . This is the key because it introduces a new 2 factor r into the integrand, and with this new factor, the integrand r e−r /2 does have 2 a simple antiderivative, namely −e−r /2 : I2 = 2π 0 ∞ 0 e−r 2 /2 r dr d θ = 2π 0 −e−r 2 /2 ∞ dθ = 0 2π 0 d θ = 2π . √ Hence I = 2π . In the exercises you are asked to show that Iµ ,σ = ∞ −∞ e−(x−µ ) 2 /2σ 2 √ dx = σ 2π , which implies Prob(a ≤ Xµ ,σ ≤ b) = 1 √ σ 2π b a e−(x−µ ) 2 /2σ 2 dx.

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Advanced Calculus: A Geometric View by James J. Callahan


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