Download Algebre: Chapitre 10.Algebre homologique by N. Bourbaki PDF

By N. Bourbaki

Les Éléments de mathématique de Nicolas Bourbaki ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements.

Ce dixième chapitre du Livre d Algèbre, deuxième Livre du traité, pose les bases du calcul homologique.

Ce quantity est a été publié en 1980.

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Extra info for Algebre: Chapitre 10.Algebre homologique

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35) with a1 , b1 , and c1 to be determined. 1. The dot product of two unit vectors is the cosine of the angle between them. 12 shows that the angle between e ˆρ and e ˆy is π/2 − ϕ. 35) by e ˆx · e ˆρ = a1 + 0 + 0 = a1 ⇒ a1 = cos ϕ, e =cos ϕ ˆy · e ˆρ = 0 + b1 + 0 = b1 ⇒ b1 = sin ϕ, e =sin ϕ ˆz · e ˆρ = 0 + 0 + c1 = c1 ⇒ c1 = 0. e =0 32 Coordinate Systems and Vectors Therefore, ˆρ = e ˆx cos ϕ + e ˆy sin ϕ. 5): ⎛ ⎞ ˆy ˆz ˆx e e e ˆϕ = e ˆz × e ˆρ = det ⎝ 0 ˆy cos ϕ. e 0 1 ⎠ = −ˆ ex sin ϕ + e cos ϕ sin ϕ 0 Thus, ˆρ = e ˆx cos ϕ + e ˆy sin ϕ, e ˆy cos ϕ, ˆ ex sin ϕ + e eϕ = −ˆ ˆz = e ˆz .

To find the component equations in a coordinate system, one needs to pick a fixed point (say the origin), a set of unit vectors at that point (usually the unit vectors along the axes of some coordinate system), and substitute the components of rk along those unit vectors to find the components of r along the unit vectors. 1 Fields and Potentials The distributive property of the dot product and the fact that the unit vectors of the bases in all coordinate systems are mutually perpendicular can be used to derive the following: a · b = ax b x + ay b y + az b z a · b = aρ b ρ + aϕ b ϕ + az b z (Cartesian), (cylindrical), a · b = ar b r + aθ b θ + aϕ b ϕ (spherical).

Physical laws are always coordinate-independent. For example, when we write F = ma both F and a could be expressed in terms of Cartesian, spherical, cylindrical, or any other convenient coordinate system. This independence allows us the freedom to choose the coordinate systems most convenient for the problem at hand. For example, it is extremely difficult to solve the planetary motions in Cartesian coordinates, while the use of spherical coordinates facilitates the solution of the problem tremendously.

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