A first course in calculus by Serge Lang PDF

By Serge Lang

ISBN-10: 0387962018

ISBN-13: 9780387962016

This 5th version of Lang's booklet covers all of the issues normally taught within the first-year calculus series. Divided into 5 components, each one part of a primary path IN CALCULUS comprises examples and purposes on the subject of the subject lined. furthermore, the rear of the e-book comprises specified suggestions to a number of the workouts, letting them be used as worked-out examples -- one of many major advancements over earlier variants.

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Extra resources for A first course in calculus

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Each complex ODE defines the motion of a single robot in the complex--plane. */ // Author: Dr. *t); /* ddz[1] = ... 0); } int main() { /* declare variables */ // Do NOT use I as a loop variable! 4, showing the difference between a linear and a deformed (nonlinear) complex–valued ODEs. 2*I*Cos[5*t]; 38 2 Nonlinear Dynamics in the Complex Plane Fig. 3. 1i. Fig. 4. 2i cos(5t), with the same initial conditions as above. }},<>]}} In[4]:= ParametricPlot[Evaluate[{Re[z[t]], Im[z[t]]} /. 01]]; In[5]:= ParametricPlot[Evaluate[{Re[Derivative[1][z][t]], Im[Derivative[1][z][t]]} /.

We say that γ is homologous to 0 in U , and write γ ∼ 0, if W (γ, α) = 0 for every point α in the complement of U . If γ and η are closed paths in U and are homotopic, then they are homologous. If γ and η are closed paths in U and are close together, then they are homologous. Let γ 1 , . . , γ n be curves in an open set U ⊂ MC , and let m1 , . . , mn be n integers. A formal sum γ = m1 γ 1 + · · · + mn γ n = i=1 mi γ i is called a chain in U . The chain is called closed if it is a finite sum of closed paths.

The length L(γ) is defined to be the integral of the speed, L(γ) = b | γ(t)| ˙ dt. a If γ = γ 1 , γ 2 , . . , the sum of the integrals of f i over each curve γ i (i = 1, . . , n of the path γ. The length of a path is defined n as L(γ) = i=1 L(γ i ). Let f be continuous on an open set U ⊂ MC , and suppose that f has a primitive g, that is, g is holomorphic and g = f . Let α, β be two points in U , and let γ be a path in U joining α to β. Then γ f = g(β) − g(α); this integral is independent of the path and depends only on the beginning and end point of the path.

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A first course in calculus by Serge Lang

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