New PDF release: Calculus of Several Variables

By Serge Lang

ISBN-10: 1461210682

ISBN-13: 9781461210689

ISBN-10: 1461270014

ISBN-13: 9781461270010

The current direction on calculus of a number of variables is intended as a textual content, both for one semester following a primary direction in Calculus, or for a 12 months if the calculus series is so dependent. For a one-semester path, it doesn't matter what, one should still conceal the 1st 4 chapters, as much as the legislation of conservation of power, which supplies a stunning software of the chain rule in a actual context, and ties up the maths of this direction with usual fabric from classes on physics. Then there are approximately percentages: One is to hide Chapters V and VI on maxima and minima, quadratic types, serious issues, and Taylor's formulation. you'll be able to then end with bankruptcy IX on double integration to around off the one-term path. the opposite is to enter curve integrals, double integration, and Green's theorem, that's Chapters VII, VIII, IX, and X, §1. This varieties a coherent whole.

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X(t)+X'(t) Figure 4 We define the tangent line to a curve X at time t to be the line passing through X(t) in the direction of X'(t), provided that X'(t)"# o. Otherwise, we don't define a tangent line. We have therefore given two interpretations for X'(t): X'(t) is the velocity at time t; X'(t) is parallel to a tangent vector at time t. 54 DIFFERENTIATION OF VECTORS [II, §1] By abuse of language, we sometimes call X'(t) a tangent vector, although strictly speaking, we should refer to the located vector X(t)(X(t) + X'(t)) as the tangent vector.

We have therefore given two interpretations for X'(t): X'(t) is the velocity at time t; X'(t) is parallel to a tangent vector at time t. 54 DIFFERENTIATION OF VECTORS [II, §1] By abuse of language, we sometimes call X'(t) a tangent vector, although strictly speaking, we should refer to the located vector X(t)(X(t) + X'(t)) as the tangent vector. However, to write down this located vector each time is cumbersome. ) Example 6. Find a parametric equation of the tangent line to the curve X(t) = (sin t, cos t) at t = n/3.

Then the parametric representation of the line through P in the direction of A gives us x =2- t, y = 1 + 5t. Multiplying the first equation by 5 and adding yields 5x +Y= 11, which is the familiar equation of a line. This elimination of t shows that every pair (x, y) which satisfies the parametric representation (*) for some value of t also satisfies equation (**). Conversely, suppose we have a pair of numbers (x, y) satisfying (**). Let t = 2 - x. Then y = 11 - 5x = 11 - 5(2 - t) = 1 + 5t. [I, §5] PARAMETRIC LINES 35 Hence there exists some value of t which satisfies equation (*).

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Calculus of Several Variables by Serge Lang


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