By James J. Callahan

With a clean geometric procedure that includes greater than 250 illustrations, this textbook units itself except all others in complex calculus. along with the classical capstones--the switch of variables formulation, implicit and inverse functionality theorems, the critical theorems of Gauss and Stokes--the textual content treats different vital issues in differential research, corresponding to Morse's lemma and the Poincaré lemma. the guidelines at the back of such a lot subject matters might be understood with simply or 3 variables. This invitations geometric visualization; the publication contains sleek computational instruments to offer visualization genuine strength. utilizing 2nd and 3D pictures, the ebook deals new insights into basic components of the calculus of differentiable maps, equivalent to the function of the spinoff 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 maintains with an research of the actual which means of the divergence and the curl at a degree of element no longer 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 sufficient fabric for a year-long direction on complex calculus and for quite a few semester courses--including issues in geometry. It avoids duplicating the fabric of actual research. The measured speed of the publication, with its broad examples and illustrations, make it particularly compatible for self reliant research.

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**Sample text**

Sketch the graph of z = g µ ,σ (x) for µ − 3σ ≤ x ≤ µ + 3σ . Do this first with µ = 5 and σ = 2 and then symbolically with general values for µ and σ . √ b. Without repeating the argument in the text that showed I = 2π , show that ∞ −∞ √ gµ ,σ (x) dx = σ 2π . You can do this by making an appropriate change of variable √that converts this integral to one you can evaluate knowing only that I = 2π . 38. a. Sketch together in the same coordinate plane the graphs of y = f0,σ (x) with σ = 12 , σ = 1, and σ = 3.

Example 4 ⊔ ⊓ The theorem implies that the value of the scalar path integral is independent of the parametrization of C; even oppositely oriented parametrizations give the same value. 2 Work and path integrals 19 C yz ds when C is√ the helix parametrized as x(t) = (cos(t), sin(t),t), 0 ≤ t ≤ 2π . Because x′ (t) = 2 and yz = t sint, we have C 2π yz ds = 0 √ √ t sin(t) 2 dt = −2π 2. By setting u = 2π − t, and thus t = 2π − u, we get the opposite parametrization r(u) = x(2π − u): r(u) = (cos(2π − u), sin(2π − u), 2π − u) = (cos u, − sin u, 2π − u).

B. Each of the following limits exists; determine the location of each as a point in the (x, y)-plane: lim (x(t), y(t)) t→+∞ lim (x(t), y(t)) t→−∞ c. , independent of t) that is consistent with the sketch of the curve you made in part (a). What is the curve and how does α relate to it? 24. Determine the work done by the force field F in moving a particle along the oriented curve C, where: a. F = (x, 3y), C : (τ 2 , τ 3 ), 1 ≤ τ ≤ 2. b. F = (−y, x), C : semicircle of radius 2 at origin, counterclockwise from (2, 0) to (−2, 0).