By James J. Callahan

ISBN-10: 1441973311

ISBN-13: 9781441973313

With a clean geometric strategy that includes greater than 250 illustrations, this textbook units itself except all others in complex calculus. in addition to the classical capstones--the swap of variables formulation, implicit and inverse functionality theorems, the crucial theorems of Gauss and Stokes--the textual content treats different very important subject matters in differential research, comparable to Morse's lemma and the Poincaré lemma. the tips at the back of such a lot subject matters should be understood with simply or 3 variables. This invitations geometric visualization; the publication comprises sleek computational instruments to provide visualization genuine strength. utilizing second and 3D images, the publication deals new insights into basic parts of the calculus of differentiable maps, comparable to the function of the spinoff because the neighborhood linear approximation to a map and its position within the swap of variables formulation for a number of integrals. The geometric subject maintains with an research of the actual that means of the divergence and the curl at a degree of aspect no longer present in different complicated calculus books. complicated Calculus: a geometrical View is a textbook for undergraduates and graduate scholars in arithmetic, the actual sciences, and economics. must haves are an creation to linear algebra and multivariable calculus. there's sufficient 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 vast examples and illustrations, make it in particular compatible for autonomous study.

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**Sample text**

Dxn ), and take Pi to be the ith component of F, so Component form of a path integral F(x) = (P1 (x1 , . . , xn ), . . , Pn (x1 , . . , xn )), dW = F · dx = P1 dx1 + · · · + Pn dxn . Suppose x(t) = (ϕ1 (t), . . , ϕn (t)), a ≤ t ≤ b, parametrizes the oriented curve C; then dx = (ϕ1′ (t), . . , ϕn′ (t)) dt and W= = dW = C b a C F · dx = C P1 dx1 + · · · + Pn dxn [P1 (ϕ1 (t), . . , ϕn (t)) ϕ1′ (t) + · · · + Pn(ϕ1 (t), . . , ϕn (t)) ϕn′ (t)] dt. The final expression is an ordinary integral; it gives us a way to compute the path integral by means of the n pullback substitutions x1 = ϕ1 (t), .

If we adopt the informal practice of regarding an integral as an infinite sum of “infinitesimal” terms, then the integrand in the work integral is the infinitesimal amount of work dW done along an infinitesimal segment dx = (dx, dy): “Infinitesimal” work dW = F · dx = (P, Q) · (dx, dy) = P dx + Q dy. From this point of view, the expressions P dx and Q dy are the horizontal and vertical components of the infinitesimal work dW . Moving to Rn , we set x = (x1 , . . , xn ), dx = (dx1 , . . , dxn ), and take Pi to be the ith component of F, so Component form of a path integral F(x) = (P1 (x1 , .

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).

### Advanced Calculus: A Geometric View by James J. Callahan

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