DOCSLIB.ORG

Stokes' Theorem on Smooth Manifolds

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stokes' Theorem on Smooth Manifolds Stokes’ Theorem on Smooth Manifolds Erik Jörgenfelt Spring 2016 Thesis, 15hp Bachelor of mathematics, 180hp Department of mathematics and mathematical statistics Abstract A proof of Stokes’ theorem on smooth manifolds is given, complete with prerequisite re- sults in tensor algebra and differential geometry. The essay assumes familiarity with multi- variable calculus and linear algebra, as well as a basic understanding of point-set topology. Stokes’ theorem is then applied to the conservation of energy-momentum in general relativity under the existence of so called Killing vectors. Sammanfattning Stokes sats för släta mångfalder bevisas, komplett med nödvändiga resultat från tensoral- gebran och differentialgeometrin. Uppsatsen förutsätter förtrogenhet med flervariabelanalys och linjär algebra, samt en grundläggande förståelse för allmän topologi. Stokes sats appli- ceras sedan till bevarande av energi-momentum i allmän relativitetsteori under existensen av so kallade Killingvektorer. Contents 1 Introduction 1 2 Preliminaries 3 2.1 Tensors.........................................3 2.2 Smooth Manifolds with Boundary..........................9 2.3 Vector Bundles..................................... 14 3 Partitions of Unity 17 4 The Tangent Bundle 25 5 Differential Forms and Stokes’ Theorem 35 5.1 Differential Forms and Integration.......................... 35 5.2 Stokes’ Theorem on Smooth Manifolds........................ 47 6 Conservation of Energy-Momentum 50 6.1 Regular Submanifolds................................. 51 6.2 The Tensor Bundles.................................. 51 6.3 Lie Derivative...................................... 53 6.4 Lorentz Manifolds................................... 54 6.5 Connection....................................... 56 6.6 Hodge Duality..................................... 59 6.7 Conservation of Energy-Momentum and Killing Vectors.............. 60 1 Introduction Stokes’ theorem on smooth manifolds subsumes three important theorems from standard calculus, unifying them and extending the application to any number of dimensions. It does this in the language of differential geometry. According to Katz[6] the unified theorem was first published in 1945 by Élie Cartan in [1], although several previous versions are also listed. It lends its name from the last of the subsumed theorems, which was published by George Stokes as a problem in the Smith’s Prize Exam at Cambridge in 1854[13]. According to Katz he had received a letter with the theorem by Lord Kelvin on July 2. Today we know this theorem as Stokes’ theorem, the Kelvin-Stokes theorem, or the curl theorem. The first published proof is however, again according to Katz, by Hermann Hankel in 1861[4]. The other subsumed theorems are today referred to as Gauss’ theorem or the divergence theorem, and Green’s theorem. Although Gauss did publish three special cases of the former[3], Katz lists Michael Ostrogradsky[10] as the first to state and prove the theorem in a lecture, not published until later, and publish it first in 1831[11]. Green’s theorem was stated first without proof by Augustin Cauchy[2] in 1846, but proved five years later by Bernhard Riemann[12]. Katz also notes that all the aforementioned authors, as well as others publishing related results, were interested in them for applications in physics. Our main theorem, stated below, among other things allows for native application in the formalism of space-time — a 4-dimensional Lorentz manifold — where it is frequently used to work with flux integrals and conservation theorems. In [5] Harrison produces a Stokes’ theorem for non-smooth chains, thus building on the work of Whitney[16], who used TheoremA to define integration over certain non-smooth domains. TheoremA (Stokes’ theorem on smooth manifolds). For any smooth (n − 1)-form ω with compact support on the oriented n-dimensional smooth manifold M with boundary ∂M, we have that Z Z dω = ω, M ∂M where ∂M is given the induced orientation. This essay is structured as follows. In Section2 we cover preliminary results in tensor al- gebra, and define smooth manifolds with boundary and vector bundles. In Section3 we show the existence of a partition of unity subordinate to any open cover of a smooth manifold. In Section4 we construct the tangent bundle over a smooth manifold. In Section5 we construct smooth differential forms, define oriented manifolds, and define integration over oriented smooth manifolds with boundary. We then prove TheoremA. In Section6 we apply TheoremA in the context of general relativity to prove TheoremB: TheoremB. Let S be a Lorentz manifold with the Levi-Civita connection, with dim(S) = n, which we will refer to as spacetime, endowed with a stress-energy tensor, T . If ξ is a Killing i i j vector then J := Tj ξ is a conserved flux, i.e. I i J dΣi ≡ 0, Σ for any sufficiently nice closed hypersurface Σ. This essay is based on [7], with some inspiration from [14] and [8]. The proof of TheoremB is inspired by [8]. We prove several supporting results that were left without proof in [7], among 1 these Proposition 2.20, Proposition 2.27, and Proposition 5.1; and state and prove Proposition 5.16, which was not included in [7]. Additionally, our definition of smooth manifolds makes the stronger assumption of second countability over paracompactness, which [7] uses. Second countability is necessary in some other applications such as Sard’s theorem. We thus proceed to show, in Proposition 3.3, that our definition implies paracompactness. Furthermore, we make no mention of what Lee refers to as TpMphys, or roughly “the tangent space according to physicists,” defined via transformation laws. Instead we use what Lee refers to as “the kinematic tangent space,” or TpMkin, here referred to as simply TpM, defined via curves, to define the tangent space at the boundary (Definition 4.15), and in Lemma 5.18. This essay also expands on any arguments in [7] that the author found abstruse; either because they presumed a greater familiarity with the subject than can be expected at this level, or because Lee intentionally left parts of the argument to the reader. Finally, the focus on TheoremA allows the material to be restructured to provide a (hopefully) clearer line of reasoning. 2 Preliminaries 2.1 Tensors Here we study alternating covariant tensors. Tensors can be thought of as generalisations of vectors, in as much as it is linear in each argument, and indeed all vectors are also tensors. We will not concern ourselves with this fact here, but rather construct our tensors from vectors and dual vectors, the latter of which are elements of the dual of a vector space. We therefore first define the dual of a vector space, before introducing consolidated tensors. Although we quickly move on to consider only alternating covariant tensors, the dual space will appear again in Section4 in the form of the cotangent space. In Section6 we briefly touch on more general tensors again. Alternating covariant tensors will form the basis which we build our definition of differential forms (Section5) on. As such they are crucial to our development of TheoremA. The exterior product of two alternating covariant tensors is defined and explored to sufficient depth to provide a basis for each space of alternating covariant tensors, and at the end this allows us to give an alternate definition of the determinant of a linear transformation on a vector space and show that it is equivalent to the matrix definition. The last result not only provides a satisfying link between these results and ones known from elementary linear algebra, but will also be used in Section5 when we investigate oriented manifolds. We begin with the definition of the dual of a vector space, and the dual of any basis: Definition 2.1. The dual space of a vector space V is the set of all linear maps ω : V → R. ∗ We denote it by V , and call its elements dual vectors. If (e1, . , en) is a basis of V , and 1 n ∗ i i 1 n ω , . , ω are elements of V such that ω (ej) = δj, then ω , . , ω is called the dual basis of (e1, . , en). That ω1, . , ωn is a basis of V ∗ follows from the linearity of its elements, and we will immediately move on to define (consolidated) tensors over vector spaces: Definition 2.2. A (consolidated) tensor on a real valued vector space V is a multilinear map ∗ ∗ t : V1 × ... × Vr × V1 × ... × Vs → R, ∗ ∗ where all Vi = V and all Vj = V . We say that t is contravariant of dimension r and covariant r of dimension s, and write t ∈ Ts (V ). In fact, once we have defined the tensor product and shown its associativity — a property we should expect from a product operator — we will only be dealing with covariant tensors, i.e. 0 tensors that are contravariant of dimension zero, denoted Ts (V ) or just Ts(V ). As an example of a covariant tensor, consider the familiar inner product (or “dot-product”) on Rn. It takes two vectors as arguments, is linear in each argument, and returns a real number — a two-dimensional, covariant tensor. The tensor product combines two tensors to produce a third. In principle it adjoins the two tensors. As such it can easily be seen to have an identity element, but is obviously not commutative. Definition 2.3. For tensors t ∈ T r1 (V ) and t ∈ T r2 (V ), we define the tensor product t ⊗t ∈ 1 s1 2 s2 1 2 T r1+r2 by s1+s2 1 r1+r2 t1 ⊗ t2(θ , . , θ ,v1, . vs1+s2 ) := 1 r1 r1+1 r1+r2 t1(θ , . , θ , v1, . , vs1 )t2(θ , . , θ , vs1+1, . , vs1+s2 ). 3 Proposition 2.4. Let t ∈ T r1 (V ), t ∈ T r2 (V ), t ∈ T r3 (V ). Then 1 s1 2 s2 3 s3 (t1 ⊗ t2) ⊗ t3 = t1 ⊗ (t2 ⊗ t3). ri ,...,ri Proof. Applying both sides to (α1, . , αr1+r2+r3 , v , . , v ) and taking t 1 k to mean 1 s1+s2+s3 sj1 ,...,sj` r r t α i1 , .
Load more

Recommended publications
  • The Directional Derivative the Derivative of a Real Valued Function (Scalar field) with Respect to a Vector
    Math 253 The Directional Derivative The derivative of a real valued function (scalar field) with respect to a vector. f(x + h) f(x) What is the vector space analog to the usual derivative in one variable? f 0(x) = lim − ? h 0 h ! x -2 0 2 5 Suppose f is a real valued function (a mapping f : Rn R). ! (e.g. f(x; y) = x2 y2). − 0 Unlike the case with plane figures, functions grow in a variety of z ways at each point on a surface (or n{dimensional structure). We'll be interested in examining the way (f) changes as we move -5 from a point X~ to a nearby point in a particular direction. -2 0 y 2 Figure 1: f(x; y) = x2 y2 − The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point . z -5 -2 0 y 2 Figure 2: f(x; y) = x2 y2 − y If we give the direction by using a second vector, say ~u, then for →u X→ + h any real number h, the vector X~ + h~u represents a change in X→ position from X~ along a line through X~ parallel to ~u. →u →u h x Figure 3: Change in position along line parallel to ~u The Directional Derivative x -2 0 2 5 We'll be interested in examining the way (f) changes as we move ~ in a particular direction 0 from a point X to a nearby point .
    [Show full text]
  • Lecture 11 Line Integrals of Vector-Valued Functions (Cont'd)
    Lecture 11 Line integrals of vector-valued functions (cont’d) In the previous lecture, we considered the following physical situation: A force, F(x), which is not necessarily constant in space, is acting on a mass m, as the mass moves along a curve C from point P to point Q as shown in the diagram below. z Q P m F(x(t)) x(t) C y x The goal is to compute the total amount of work W done by the force. Clearly the constant force/straight line displacement formula, W = F · d , (1) where F is force and d is the displacement vector, does not apply here. But the fundamental idea, in the “Spirit of Calculus,” is to break up the motion into tiny pieces over which we can use Eq. (1 as an approximation over small pieces of the curve. We then “sum up,” i.e., integrate, over all contributions to obtain W . The total work is the line integral of the vector field F over the curve C: W = F · dx . (2) ZC Here, we summarize the main steps involved in the computation of this line integral: 1. Step 1: Parametrize the curve C We assume that the curve C can be parametrized, i.e., x(t)= g(t) = (x(t),y(t), z(t)), t ∈ [a, b], (3) so that g(a) is point P and g(b) is point Q. From this parametrization we can compute the velocity vector, ′ ′ ′ ′ v(t)= g (t) = (x (t),y (t), z (t)) . (4) 1 2. Step 2: Compute field vector F(g(t)) over curve C F(g(t)) = F(x(t),y(t), z(t)) (5) = (F1(x(t),y(t), z(t), F2(x(t),y(t), z(t), F3(x(t),y(t), z(t))) , t ∈ [a, b] .
    [Show full text]
  • Notes on Partial Differential Equations John K. Hunter
    Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4.
    [Show full text]
  • MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009)
    MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009) Required Topics • Multivariable functions • 3-D space o Distance o Equations of Spheres o Graphing Planes Spheres Miscellaneous function graphs Sections and level curves (contour diagrams) Level surfaces • Planes o Equations of planes, in different forms o Equation of plane from three points o Linear approximation and tangent planes • Partial Derivatives o Compute using the definition of partial derivative o Compute using differentiation rules o Interpret o Approximate, given contour diagram, table of values, or sketch of sections o Signs from real-world description o Compute a gradient vector • Vectors o Arithmetic on vectors, pictorially and by components o Dot product compute using components v v v v v ⋅ w = v w cosθ v v v v u ⋅ v = 0⇔ u ⊥ v (for nonzero vectors) o Cross product Direction and length compute using components and determinant • Directional derivatives o estimate from contour diagram o compute using the lim definition t→0 v v o v compute using fu = ∇f ⋅ u ( u a unit vector) • Gradient v o v geometry (3 facts, and understand how they follow from fu = ∇f ⋅ u ) Gradient points in direction of fastest increase Length of gradient is the directional derivative in that direction Gradient is perpendicular to level set o given contour diagram, draw a gradient vector • Chain rules • Higher-order partials o compute o mixed partials are equal (under certain conditions) • Optimization o Locate and
    [Show full text]
  • Engineering Analysis 2 : Multivariate Functions
    Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Engineering Analysis 2 : Multivariate functions P. Rees, O. Kryvchenkova and P.D. Ledger, [email protected] College of Engineering, Swansea University, UK PDL (CoE) SS 2017 1/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Outline 1 Multivariate functions 2 Partial Differentiation 3 Higher Order Partial Derivatives 4 Total Differentiation 5 Line Integrals 6 Surface Integrals PDL (CoE) SS 2017 2/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction Recall from EG189: analysis of a function of single variable: Differentiation d f (x) d x Expresses the rate of change of a function f with respect to x. Integration Z f (x) d x Reverses the operation of differentiation and can used to work out areas under curves, and as we have just seen to solve ODEs. We’ve seen in vectors we often need to deal with physical fields, that depend on more than one variable. We may be interested in rates of change to each coordinate (e.g. x; y; z), time t, but sometimes also other physical fields as well. PDL (CoE) SS 2017 3/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 1: the area A of a rectangular plate of width x and breadth y can be calculated A = xy The variables x and y are independent of each other. In this case, the dependent variable A is a function of the two independent variables x and y as A = f (x; y) or A ≡ A(x; y) PDL (CoE) SS 2017 4/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 2: the volume of a rectangular plate is given by V = xyz where the thickness of the plate is in z-direction.
    [Show full text]
  • The Divergence As the Rate of Change in Area Or Volume
    The divergence as the rate of change in area or volume Here we give a very brief sketch of the following fact, which should be familiar to you from multivariable calculus: If a region D of the plane moves with velocity given by the vector field V (x; y) = (f(x; y); g(x; y)); then the instantaneous rate of change in the area of D is the double integral of the divergence of V over D: ZZ ZZ rate of change in area = r · V dA = fx + gy dA: D D (An analogous result is true in three dimensions, with volume replacing area.) To see why this should be so, imagine that we have cut D up into a lot of little pieces Pi, each of which is rectangular with width Wi and height Hi, so that the area of Pi is Ai ≡ HiWi. If we follow the vector field for a short time ∆t, each piece Pi should be “roughly rectangular”. Its width would be changed by the the amount of horizontal stretching that is induced by the vector field, which is difference in the horizontal displacements of its two sides. Thisin turn is approximated by the product of three terms: the rate of change in the horizontal @f component of velocity per unit of displacement ( @x ), the horizontal distance across the box (Wi), and the time step ∆t. Thus ! @f ∆W ≈ W ∆t: i @x i See the figure below; the partial derivative measures how quickly f changes as you move horizontally, and Wi is the horizontal distance across along Pi, so the product fxWi measures the difference in the horizontal components of V at the two ends of Wi.
    [Show full text]
  • Partial Derivatives
    Partial Derivatives Partial Derivatives Just as derivatives can be used to explore the properties of functions of 1 vari- able, so also derivatives can be used to explore functions of 2 variables. In this section, we begin that exploration by introducing the concept of a partial derivative of a function of 2 variables. In particular, we de…ne the partial derivative of f (x; y) with respect to x to be f (x + h; y) f (x; y) fx (x; y) = lim h 0 h ! when the limit exists. That is, we compute the derivative of f (x; y) as if x is the variable and all other variables are held constant. To facilitate the computation of partial derivatives, we de…ne the operator @ = “The partial derivative with respect to x” @x Alternatively, other notations for the partial derivative of f with respect to x are @f @ f (x; y) = = f (x; y) = @ f (x; y) x @x @x x 2 2 EXAMPLE 1 Evaluate fx when f (x; y) = x y + y : Solution: To do so, we write @ @ @ f (x; y) = x2y + y2 = x2y + y2 x @x @x @x and then we evaluate the derivative as if y is a constant. In partic- ular, @ 2 @ 2 fx (x; y) = y x + y = y 2x + 0 @x @x That is, y factors to the front since it is considered constant with respect to x: Likewise, y2 is considered constant with respect to x; so that its derivative with respect to x is 0. Thus, fx (x; y) = 2xy: Likewise, the partial derivative of f (x; y) with respect to y is de…ned f (x; y + h) f (x; y) fy (x; y) = lim h 0 h ! 1 when the limit exists.
    [Show full text]
  • Multivariable Calculus Workbook Developed By: Jerry Morris, Sonoma State University
    Multivariable Calculus Workbook Developed by: Jerry Morris, Sonoma State University A Companion to Multivariable Calculus McCallum, Hughes-Hallett, et. al. c Wiley, 2007 Note to Students: (Please Read) This workbook contains examples and exercises that will be referred to regularly during class. Please purchase or print out the rest of the workbook before our next class and bring it to class with you every day. 1. To Purchase the Workbook. Go to the Sonoma State University campus bookstore, where the workbook is available for purchase. The copying charge will probably be between $10.00 and $20.00. 2. To Print Out the Workbook. Go to the Canvas page for our course and click on the link \Math 261 Workbook", which will open the file containing the workbook as a .pdf file. BE FOREWARNED THAT THERE ARE LOTS OF PICTURES AND MATH FONTS IN THE WORKBOOK, SO SOME PRINTERS MAY NOT ACCURATELY PRINT PORTIONS OF THE WORKBOOK. If you do choose to try to print it, please leave yourself enough time to purchase the workbook before our next class in case your printing attempt is unsuccessful. 2 Sonoma State University Table of Contents Course Introduction and Expectations .............................................................3 Preliminary Review Problems ........................................................................4 Chapter 12 { Functions of Several Variables Section 12.1-12.3 { Functions, Graphs, and Contours . 5 Section 12.4 { Linear Functions (Planes) . 11 Section 12.5 { Functions of Three Variables . 14 Chapter 13 { Vectors Section 13.1 & 13.2 { Vectors . 18 Section 13.3 & 13.4 { Dot and Cross Products . 21 Chapter 14 { Derivatives of Multivariable Functions Section 14.1 & 14.2 { Partial Derivatives .
    [Show full text]
  • Notes for Math 136: Review of Calculus
    Notes for Math 136: Review of Calculus Gyu Eun Lee These notes were written for my personal use for teaching purposes for the course Math 136 running in the Spring of 2016 at UCLA. They are also intended for use as a general calculus reference for this course. In these notes we will briefly review the results of the calculus of several variables most frequently used in partial differential equations. The selection of topics is inspired by the list of topics given by Strauss at the end of section 1.1, but we will not cover everything. Certain topics are covered in the Appendix to the textbook, and we will not deal with them here. The results of single-variable calculus are assumed to be familiar. Practicality, not rigor, is the aim. This document will be updated throughout the quarter as we encounter material involving additional topics. We will focus on functions of two real variables, u = u(x;t). Most definitions and results listed will have generalizations to higher numbers of variables, and we hope the generalizations will be reasonably straightforward to the reader if they become necessary. Occasionally (notably for the chain rule in its many forms) it will be convenient to work with functions of an arbitrary number of variables. We will be rather loose about the issues of differentiability and integrability: unless otherwise stated, we will assume all derivatives and integrals exist, and if we require it we will assume derivatives are continuous up to the required order. (This is also generally the assumption in Strauss.) 1 Differential calculus of several variables 1.1 Partial derivatives Given a scalar function u = u(x;t) of two variables, we may hold one of the variables constant and regard it as a function of one variable only: vx(t) = u(x;t) = wt(x): Then the partial derivatives of u with respect of x and with respect to t are defined as the familiar derivatives from single variable calculus: ¶u dv ¶u dw = x ; = t : ¶t dt ¶x dx c 2016, Gyu Eun Lee.
    [Show full text]
  • Multivariable and Vector Calculus
    Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space.
    [Show full text]
  • High Energy and Smoothness Asymptotic Expansion of the Scattering Amplitude
    HIGH ENERGY AND SMOOTHNESS ASYMPTOTIC EXPANSION OF THE SCATTERING AMPLITUDE D.Yafaev Department of Mathematics, University Rennes-1, Campus Beaulieu, 35042, Rennes, France (e-mail : [email protected]) Abstract We find an explicit expression for the kernel of the scattering matrix for the Schr¨odinger operator containing at high energies all terms of power order. It turns out that the same expression gives a complete description of the diagonal singular- ities of the kernel in the angular variables. The formula obtained is in some sense universal since it applies both to short- and long-range electric as well as magnetic potentials. 1. INTRODUCTION d−1 d−1 1. High energy asymptotics of the scattering matrix S(λ): L2(S ) → L2(S ) for the d Schr¨odinger operator H = −∆+V in the space H = L2(R ), d ≥ 2, with a real short-range potential (bounded and satisfying the condition V (x) = O(|x|−ρ), ρ > 1, as |x| → ∞) is given by the Born approximation. To describe it, let us introduce the operator Γ0(λ), −1/2 (d−2)/2 ˆ 1/2 d−1 (Γ0(λ)f)(ω) = 2 k f(kω), k = λ ∈ R+ = (0, ∞), ω ∈ S , (1.1) of the restriction (up to the numerical factor) of the Fourier transform fˆ of a function f to −1 −1 the sphere of radius k. Set R0(z) = (−∆ − z) , R(z) = (H − z) . By the Sobolev trace −r d−1 theorem and the limiting absorption principle the operators Γ0(λ)hxi : H → L2(S ) and hxi−rR(λ + i0)hxi−r : H → H are correctly defined as bounded operators for any r > 1/2 and their norms are estimated by λ−1/4 and λ−1/2, respectively.
    [Show full text]
  • Tensor Calculus and Differential Geometry
    Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane .
    [Show full text]