Mathematical Principles of Theoretical Physics
Chapter 3 Mathematical Foundations The aim of this chapter is to provide mathematical foundations for the remaining part of the ...
108 CHAPTER 3. MATHEMATICAL FOUNDATIONS Section3.3proves the orthogonal decomposition theorem for tensors onRiemannian manifolds ...
3.1 Basic Concepts curves and two-dimensional surfaces. It is, however, difficult for us to tell whether the three- dimensional ...
110 CHAPTER 3. MATHEMATICAL FOUNDATIONS is a constant metric, and the metric (3.1.4) of the sphere (3.1.3), i.e. (3.1.6) ( g 11 ...
3.1. BASIC CONCEPTS 111 Hence we have ̃gijd ̃xidx ̃j=(d ̃x^1 ,···,d ̃xn)( ̃gij) dx ̃^1 .. . dx ̃n (3.1.10) =(dx^ ...
112 CHAPTER 3. MATHEMATICAL FOUNDATIONS be the embedding function. Consider the vector product ofnvectors inRn: [ ∂~r ∂x^1 ,···, ...
3.1. BASIC CONCEPTS 113 the inner product ofXandYis defined by (3.1.18) 〈X,Y〉=gij(p)XiYj. It is clear that the inner product (3. ...
114 CHAPTER 3. MATHEMATICAL FOUNDATIONS Example 3.1.The motion of the air or seawater can be ideally considered asa fluid motion ...
3.1. BASIC CONCEPTS 115 1) For a real (complex) scalar fieldφ, the associated vector bundle isM⊗pR^1 (M⊗pC): φ:M→M⊗pR^1 (M⊗pC). ...
116 CHAPTER 3. MATHEMATICAL FOUNDATIONS 2) Electromagnetism:U( 1 )gauge and fermion fields, (3.1.27) Aμ:M→T∗M, Ψ:M→M⊗pC^4. 3) We ...
3.1. BASIC CONCEPTS 117 As discussed in Subsection2.1.5, each symmetry possesses three ingredients: space (manifold), transforma ...
118 CHAPTER 3. MATHEMATICAL FOUNDATIONS The above two ways to define general tensors are equivalent. However, in the second fash ...
3.1. BASIC CONCEPTS 119 By the definition of spinors in Subsection2.2.6, under the Lorentz transformation of (3.1.42), we have ( ...
120 CHAPTER 3. MATHEMATICAL FOUNDATIONS However, ifTpdepends onp∈M, then we have (3.1.52) ̃∂μ(TpF) =Tp ̃∂μF+ ̃∂μTpF, which viola ...
3.1. BASIC CONCEPTS 121 By (3.1.55),A=I. The connection ofSU(N)group is the gauge fieldsGaμ: Γμ= { igGaμτa| {τa}N (^2) − 1 a= 1 ...
122 CHAPTER 3. MATHEMATICAL FOUNDATIONS It is known that (3.1.64) FkGk=a scalar field, andD(FkGk) =∂(FkGk). We infer then from ( ...
3.2 Analysis on Riemannian Manifolds. 3.2 Analysis on Riemannian Manifolds 3.2.1 Sobolev spaces of tensor fields The gravitation ...
124 CHAPTER 3. MATHEMATICAL FOUNDATIONS In this case,φis called thek-th weak derivative ofuj, denoted by (3.2.3) φ=∂kuj. The spa ...
3.2. ANALYSIS ON RIEMANNIAN MANIFOLDS 125 The spacesWk,p(M⊗pEN)are called Sobolev spaces, and the norms are defined by (3.2.9) | ...
126 CHAPTER 3. MATHEMATICAL FOUNDATIONS and[v]αis the H ̈older modulus: [v]α= sup x,y∈M,x 6 =y |v(x)−v(y)| |x−y|α , 0 <α< ...
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