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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Put up yr word: First released March second 1979

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Introducing calculus on the easy point, this article covers hyperreal numbers and hyperreal line, non-stop capabilities, quintessential and differential calculus, primary theorem, countless sequences and sequence, endless polynomials, topology of the genuine line, and traditional calculus and sequences of capabilities. basically highschool arithmetic wanted.

From 1979 variation

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**Extra resources for A first course in calculus**

**Example text**

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.

### A first course in calculus by Serge Lang

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