Lecture 7: The wave equation: higher dimensional case

Size: px
Start display at page:

Download "Lecture 7: The wave equation: higher dimensional case"

Transcription

1 Lecture 7: The wave equation: higher dimensional case Some auxiliary facts The co area formula. Let be a bounded domain and be a function of class. Denote by : the level set of. Let be any integrable function. Then for almost every the set is a regular hypersurface and, where is the surface measure on. In fact, the exterior integral should be taken only over the interval of those for which is non empty. Example. When is the distance function we have Denote by : the dimensional ball centered at of radius. In this case is non empty only for 0 and the level sets are spheres. Then applying the co area formula gives. This formula is well known (the integral in polar coordinates). The divergence theorem. Let be an open subset with a piecewise smooth boundary and let be a continuously differentiable vector field defined in a neighborhood of. Then div Where is the unit outward normal vector field along the boundary.

2 Spherical means We return to t he wave equation in and consider the following initial value problem 0, 0,, 0,, 0 Define the spherical means, o r averages, of a function, by,,, where is the dimensional area of the dimensional sphere of radius. Here is the dimensional area of the unit sphere : For instance, 2 Γ 2. 2, 2, 4, 2. Similarly we define the averages of t he initial data:,,,, Remark. We recall that the volume of the unit ball and the area of the unit surface are related by This fact is, in principle, well known but we can derive this relation immediately from the co area formu la. Ind eed, from formula in Example we have for : Hence we obtain which yields the required relation for.

3 Lemma (Euler Poisson Darboux equation). Fix and let satisfy the initial value problem above. Then,, 0; and 0, 0, 0, 0,,,, 0,,, 0 Proof. The fact that,, is two times continuously differentiable in follows from standard facts on differentiability of an integral depending on parameters. The situation with parameter requires however a further analysis because the set itself depends on. Step : We show that,, is continuously differentiable in and,,,. We consider an auxiliary function and fo r 0 we have,,,,,, where \ is the spherical slab with the oriented boundary (equipped with outward normals):, see the picture below: In order to apply the divergence theorem to the latter integrals we observe the following identity. From Example above we know that unit vector field : coincides with the gradient of the distant function:. For the further convenience we notice that the divergence of this vector field is

4 div div. On the other hand, the unit normal v ector at is found as. In particular, we have for the scalar products:. Summarizing this and applyin g the divergence theorem we obtain:,, div, Applying the formula from Example again, we see that div, and it follows easily that the following limit exists lim ε div, Similarly the left limit is shown to exist, hence is (continuously) differentiable and div,. In order to transform the latter integral, we apply the above formula for div and recall ag ain that the unit normal coincides with : div,,,, div, Hence applying the divergence theorem we get,, On the other hand, Returning to the definition of,, we obtain,.,,,.

5 It follows also from the above argument that differentiability of is equivalent to that of integral,. But this integral is easily seen to be differentiable (see again formula in Example ) because the Laplacian, is a continuous function in all parameters. Step 2. We return to our formula for,, and apply to it our wave equation:,,,, Differentiation of the latter identity w.r.t. yields by the co area formula (see Example ),,, which implies the desired equation:,., The initial conditions, 0,, and, 0,, follow immediately from the definition. Casen=3 The idea how the above method w orks is the following simple relation, lim,, which makes it possible to connect any solution of the symmetrized equation to the corresponding solution of the initial wave equation. It turns out however that the method works differently for even and odd dimensions. Below we describe briefly the 3 dimensional case. We suppose th at, 0, solves the following initial value problem: We set additionally, 0,, 0,,,. Lemma 2. Function solves,, the following two dimensional initial value problem 0, 0, 0, 0,,,, 0,,,, 0 0, 0.

6 Proof. Indeed, in our case 3, hence 2 Applying the reflection principle (see formula on page 4 in Lectur e 7) we find for 0 :,, 2,, 2,. Since, lim,,, we find tha t,,, lim lim,,. Hence we arrive at the Kirchhoff formula, 2,, 2 4 4, Case n=2 We give only the final result, the so called Poisson formula. Namel y, in the above notation,, 4, where and 0.

Wave Equations: Explicit Formulas In this lecture we derive the representation formulas for the wave equation in the whole space:

Wave Equations: Explicit Formulas In this lecture we derive the representation formulas for the wave equation in the whole space: Math 57 Fall 009 Lecture 7 Sep. 8, 009) Wave Equations: Explicit Formulas In this lecture we derive the representation formulas for the wave equation in the whole space: u u t t u = 0, R n 0, ) ; u x,

More information

10.8 Further Applications of the Divergence Theorem

10.8 Further Applications of the Divergence Theorem 10.8 Further Applications of the Divergence Theorem Let D be an open subset of R 3, and let f : D R. Recall that the Laplacian of f, denoted by 2 f, is defined by We say that f is harmonic on D provided

More information

THE WAVE EQUATION. d = 1: D Alembert s formula We begin with the initial value problem in 1 space dimension { u = utt u xx = 0, in (0, ) R, (2)

THE WAVE EQUATION. d = 1: D Alembert s formula We begin with the initial value problem in 1 space dimension { u = utt u xx = 0, in (0, ) R, (2) THE WAVE EQUATION () The free wave equation takes the form u := ( t x )u = 0, u : R t R d x R In the literature, the operator := t x is called the D Alembertian on R +d. Later we shall also consider the

More information

Wave Equations Explicit Formulas In this lecture we derive the representation formulas for the wave equation in the whole space:

Wave Equations Explicit Formulas In this lecture we derive the representation formulas for the wave equation in the whole space: Nov. 07 Wave Equations Explicit Formulas In this lecture we derive the representation formulas for the wave equation in the whole space: u u t t u = 0, R n 0, ) ; u x, 0) = g x), u t x, 0) = h x). ) It

More information

4 Divergence theorem and its consequences

4 Divergence theorem and its consequences Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........

More information

Laplace s Equation. Chapter Mean Value Formulas

Laplace s Equation. Chapter Mean Value Formulas Chapter 1 Laplace s Equation Let be an open set in R n. A function u C 2 () is called harmonic in if it satisfies Laplace s equation n (1.1) u := D ii u = 0 in. i=1 A function u C 2 () is called subharmonic

More information

Module 16 : Line Integrals, Conservative fields Green's Theorem and applications. Lecture 48 : Green's Theorem [Section 48.

Module 16 : Line Integrals, Conservative fields Green's Theorem and applications. Lecture 48 : Green's Theorem [Section 48. Module 16 : Line Integrals, Conservative fields Green's Theorem and applications Lecture 48 : Green's Theorem [Section 48.1] Objectives In this section you will learn the following : Green's theorem which

More information

2 2 + x =

2 2 + x = Lecture 30: Power series A Power Series is a series of the form c n = c 0 + c 1 x + c x + c 3 x 3 +... where x is a variable, the c n s are constants called the coefficients of the series. n = 1 + x +

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

More information

1.1. Fields Partial derivatives

1.1. Fields Partial derivatives 1.1. Fields A field associates a physical quantity with a position A field can be also time dependent, for example. The simplest case is a scalar field, where given physical quantity can be described by

More information

Notation. 0,1,2,, 1 with addition and multiplication modulo

Notation. 0,1,2,, 1 with addition and multiplication modulo Notation Q,, The set of all natural numbers 1,2,3, The set of all integers The set of all rational numbers The set of all real numbers The group of permutations of distinct symbols 0,1,2,,1 with addition

More information

14 Divergence, flux, Laplacian

14 Divergence, flux, Laplacian Tel Aviv University, 204/5 Analysis-III,IV 240 4 Divergence, flux, Laplacian 4a What is the problem................ 240 4b Integral of derivative (again)........... 242 4c Divergence and flux.................

More information

Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay. Poisson s and Laplace s Equations

Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay. Poisson s and Laplace s Equations Poisson s and Laplace s Equations Lecture 13: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We will spend some time in looking at the mathematical foundations of electrostatics.

More information

The Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III)

The Divergence Theorem Stokes Theorem Applications of Vector Calculus. Calculus. Vector Calculus (III) Calculus Vector Calculus (III) Outline 1 The Divergence Theorem 2 Stokes Theorem 3 Applications of Vector Calculus The Divergence Theorem (I) Recall that at the end of section 12.5, we had rewritten Green

More information

for surface integrals, which helps to represent flux across a closed surface

for surface integrals, which helps to represent flux across a closed surface Module 17 : Surfaces, Surface Area, Surface integrals, Divergence Theorem and applications Lecture 51 : Divergence theorem [Section 51.1] Objectives In this section you will learn the following : Divergence

More information

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3)

u( x) = g( y) ds y ( 1 ) U solves u = 0 in U; u = 0 on U. ( 3) M ath 5 2 7 Fall 2 0 0 9 L ecture 4 ( S ep. 6, 2 0 0 9 ) Properties and Estimates of Laplace s and Poisson s Equations In our last lecture we derived the formulas for the solutions of Poisson s equation

More information

MULTIPOLE EXPANSIONS IN THE PLANE

MULTIPOLE EXPANSIONS IN THE PLANE MULTIPOLE EXPANSIONS IN THE PLANE TSOGTGEREL GANTUMUR Contents 1. Electrostatics and gravitation 1 2. The Laplace equation 2 3. Multipole expansions 5 1. Electrostatics and gravitation Newton s law of

More information

PDEs in Image Processing, Tutorials

PDEs in Image Processing, Tutorials PDEs in Image Processing, Tutorials Markus Grasmair Vienna, Winter Term 2010 2011 Direct Methods Let X be a topological space and R: X R {+ } some functional. following definitions: The mapping R is lower

More information

Math Divergence and Curl

Math Divergence and Curl Math 23 - Divergence and Curl Peter A. Perry University of Kentucky November 3, 28 Homework Work on Stewart problems for 6.5: - (odd), 2, 3-7 (odd), 2, 23, 25 Finish Homework D2 due tonight Begin Homework

More information

Math 342 Partial Differential Equations «Viktor Grigoryan

Math 342 Partial Differential Equations «Viktor Grigoryan Math 342 Partial ifferential Equations «Viktor Grigoryan 3 Green s first identity Having studied Laplace s equation in regions with simple geometry, we now start developing some tools, which will lead

More information

Introduction to Real Analysis Alternative Chapter 1

Introduction to Real Analysis Alternative Chapter 1 Christopher Heil Introduction to Real Analysis Alternative Chapter 1 A Primer on Norms and Banach Spaces Last Updated: March 10, 2018 c 2018 by Christopher Heil Chapter 1 A Primer on Norms and Banach Spaces

More information

(x x 0 ) 2 + (y y 0 ) 2 = ε 2, (2.11)

(x x 0 ) 2 + (y y 0 ) 2 = ε 2, (2.11) 2.2 Limits and continuity In order to introduce the concepts of limit and continuity for functions of more than one variable we need first to generalise the concept of neighbourhood of a point from R to

More information

Spezielle Funktionen 4 Skalare Kugelfunktionen

Spezielle Funktionen 4 Skalare Kugelfunktionen Spezielle Funktionen 4 Skalare Kugelfunktionen M. Gutting 28. Mai 2015 4.1 Scalar Spherical Harmonics We start by introducing some basic spherical notation. S 2 = {ξ R 3 ξ = 1} is the unit sphere in R

More information

Power Series. Part 1. J. Gonzalez-Zugasti, University of Massachusetts - Lowell

Power Series. Part 1. J. Gonzalez-Zugasti, University of Massachusetts - Lowell Power Series Part 1 1 Power Series Suppose x is a variable and c k & a are constants. A power series about x = 0 is c k x k A power series about x = a is c k x a k a = center of the power series c k =

More information

Harmonic Polynomials and Dirichlet-Type Problems. 1. Derivatives of x 2 n

Harmonic Polynomials and Dirichlet-Type Problems. 1. Derivatives of x 2 n Harmonic Polynomials and Dirichlet-Type Problems Sheldon Axler and Wade Ramey 30 May 1995 Abstract. We take a new approach to harmonic polynomials via differentiation. Surprisingly powerful results about

More information

The Minimal Element Theorem

The Minimal Element Theorem The Minimal Element Theorem The CMC Dynamics Theorem deals with describing all of the periodic or repeated geometric behavior of a properly embedded CMC surface with bounded second fundamental form in

More information

IMA Preprint Series # 2143

IMA Preprint Series # 2143 A BASIC INEQUALITY FOR THE STOKES OPERATOR RELATED TO THE NAVIER BOUNDARY CONDITION By Luan Thach Hoang IMA Preprint Series # 2143 ( November 2006 ) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS UNIVERSITY

More information

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:

The Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply

More information

1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps 1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

More information

n 2 xi = x i. x i 2. r r ; i r 2 + V ( r) V ( r) = 0 r > 0. ( 1 1 ) a r n 1 ( 1 2) V( r) = b ln r + c n = 2 b r n 2 + c n 3 ( 1 3)

n 2 xi = x i. x i 2. r r ; i r 2 + V ( r) V ( r) = 0 r > 0. ( 1 1 ) a r n 1 ( 1 2) V( r) = b ln r + c n = 2 b r n 2 + c n 3 ( 1 3) Sep. 7 The L aplace/ P oisson Equations: Explicit Formulas In this lecture we study the properties of the Laplace equation and the Poisson equation with Dirichlet boundary conditions through explicit representations

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

Functional Analysis Exercise Class

Functional Analysis Exercise Class Functional Analysis Exercise Class Wee November 30 Dec 4: Deadline to hand in the homewor: your exercise class on wee December 7 11 Exercises with solutions Recall that every normed space X can be isometrically

More information

X.9 Revisited, Cauchy s Formula for Contours

X.9 Revisited, Cauchy s Formula for Contours X.9 Revisited, Cauchy s Formula for Contours Let G C, G open. Let f be holomorphic on G. Let Γ be a simple contour with range(γ) G and int(γ) G. Then, for all z 0 int(γ), f (z 0 ) = 1 f (z) dz 2πi Γ z

More information

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives.

PRACTICE PROBLEMS. Please let me know if you find any mistakes in the text so that i can fix them. 1. Mixed partial derivatives. PRACTICE PROBLEMS Please let me know if you find any mistakes in the text so that i can fix them. 1.1. Let Show that f is C 1 and yet How is that possible? 1. Mixed partial derivatives f(x, y) = {xy x

More information

v( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0.

v( x) u( y) dy for any r > 0, B r ( x) Ω, or equivalently u( w) ds for any r > 0, B r ( x) Ω, or ( not really) equivalently if v exists, v 0. Sep. 26 The Perron Method In this lecture we show that one can show existence of solutions using maximum principle alone.. The Perron method. Recall in the last lecture we have shown the existence of solutions

More information

MATH 332: Vector Analysis Summer 2005 Homework

MATH 332: Vector Analysis Summer 2005 Homework MATH 332, (Vector Analysis), Summer 2005: Homework 1 Instructor: Ivan Avramidi MATH 332: Vector Analysis Summer 2005 Homework Set 1. (Scalar Product, Equation of a Plane, Vector Product) Sections: 1.9,

More information

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction

SHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms

More information

9 Sequences of Functions

9 Sequences of Functions 9 Sequences of Functions 9.1 Pointwise convergence and uniform convergence Let D R d, and let f n : D R be functions (n N). We may think of the functions f 1, f 2, f 3,... as forming a sequence of functions.

More information

The Hodge Star Operator

The Hodge Star Operator The Hodge Star Operator Rich Schwartz April 22, 2015 1 Basic Definitions We ll start out by defining the Hodge star operator as a map from k (R n ) to n k (R n ). Here k (R n ) denotes the vector space

More information

(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively);

(x k ) sequence in F, lim x k = x x F. If F : R n R is a function, level sets and sublevel sets of F are any sets of the form (respectively); STABILITY OF EQUILIBRIA AND LIAPUNOV FUNCTIONS. By topological properties in general we mean qualitative geometric properties (of subsets of R n or of functions in R n ), that is, those that don t depend

More information

COMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES. 1. Introduction

COMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES. 1. Introduction COMPACT STABLE HYPERSURFACES WITH FREE BOUNDARY IN CONVEX SOLID CONES WITH HOMOGENEOUS DENSITIES ANTONIO CAÑETE AND CÉSAR ROSALES Abstract. We consider a smooth Euclidean solid cone endowed with a smooth

More information

GREEN S IDENTITIES AND GREEN S FUNCTIONS

GREEN S IDENTITIES AND GREEN S FUNCTIONS GREEN S IENTITIES AN GREEN S FUNCTIONS Green s first identity First, recall the following theorem. Theorem: (ivergence Theorem) Let be a bounded solid region with a piecewise C 1 boundary surface. Let

More information

Differential Operators and the Divergence Theorem

Differential Operators and the Divergence Theorem 1 of 6 1/15/2007 6:31 PM Differential Operators and the Divergence Theorem One of the most important and useful mathematical constructs is the "del operator", usually denoted by the symbol Ñ (which is

More information

18 Green s function for the Poisson equation

18 Green s function for the Poisson equation 8 Green s function for the Poisson equation Now we have some experience working with Green s functions in dimension, therefore, we are ready to see how Green s functions can be obtained in dimensions 2

More information

BIHARMONIC WAVE MAPS INTO SPHERES

BIHARMONIC WAVE MAPS INTO SPHERES BIHARMONIC WAVE MAPS INTO SPHERES SEBASTIAN HERR, TOBIAS LAMM, AND ROLAND SCHNAUBELT Abstract. A global weak solution of the biharmonic wave map equation in the energy space for spherical targets is constructed.

More information

Dielectrics - III. Lecture 22: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay

Dielectrics - III. Lecture 22: Electromagnetic Theory. Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Dielectrics - III Lecture 22: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We continue with our discussion of dielectric medium. Example : Dielectric Sphere in a uniform

More information

LECTURE-15 : LOGARITHMS AND COMPLEX POWERS

LECTURE-15 : LOGARITHMS AND COMPLEX POWERS LECTURE-5 : LOGARITHMS AND COMPLEX POWERS VED V. DATAR The purpose of this lecture is twofold - first, to characterize domains on which a holomorphic logarithm can be defined, and second, to show that

More information

Math 5BI: Problem Set 9 Integral Theorems of Vector Calculus

Math 5BI: Problem Set 9 Integral Theorems of Vector Calculus Math 5BI: Problem et 9 Integral Theorems of Vector Calculus June 2, 2010 A. ivergence and Curl The gradient operator = i + y j + z k operates not only on scalar-valued functions f, yielding the gradient

More information

u xx + u yy = 0. (5.1)

u xx + u yy = 0. (5.1) Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function

More information

APPLICATIONS OF DIFFERENTIABILITY IN R n.

APPLICATIONS OF DIFFERENTIABILITY IN R n. APPLICATIONS OF DIFFERENTIABILITY IN R n. MATANIA BEN-ARTZI April 2015 Functions here are defined on a subset T R n and take values in R m, where m can be smaller, equal or greater than n. The (open) ball

More information

Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach

Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach Extremal Solutions of Differential Inclusions via Baire Category: a Dual Approach Alberto Bressan Department of Mathematics, Penn State University University Park, Pa 1682, USA e-mail: bressan@mathpsuedu

More information

Math The Laplacian. 1 Green s Identities, Fundamental Solution

Math The Laplacian. 1 Green s Identities, Fundamental Solution Math. 209 The Laplacian Green s Identities, Fundamental Solution Let be a bounded open set in R n, n 2, with smooth boundary. The fact that the boundary is smooth means that at each point x the external

More information

2 A Model, Harmonic Map, Problem

2 A Model, Harmonic Map, Problem ELLIPTIC SYSTEMS JOHN E. HUTCHINSON Department of Mathematics School of Mathematical Sciences, A.N.U. 1 Introduction Elliptic equations model the behaviour of scalar quantities u, such as temperature or

More information

LECTURE 3: SMOOTH FUNCTIONS

LECTURE 3: SMOOTH FUNCTIONS LECTURE 3: SMOOTH FUNCTIONS Let M be a smooth manifold. 1. Smooth Functions Definition 1.1. We say a function f : M R is smooth if for any chart {ϕ α, U α, V α } in A that defines the smooth structure

More information

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement.

Math 421, Homework #6 Solutions. (1) Let E R n Show that = (E c ) o, i.e. the complement of the closure is the interior of the complement. Math 421, Homework #6 Solutions (1) Let E R n Show that (Ē) c = (E c ) o, i.e. the complement of the closure is the interior of the complement. 1 Proof. Before giving the proof we recall characterizations

More information

UL XM522 Mutivariable Integral Calculus

UL XM522 Mutivariable Integral Calculus 1 UL XM522 Mutivariable Integral Calculus Instructor: Margarita Kanarsky 2 3 Vector fields: Examples: inverse-square fields the vector field for the gravitational force 4 The Gradient Field: 5 The Divergence

More information

MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions.

MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. MATH 409 Advanced Calculus I Lecture 10: Continuity. Properties of continuous functions. Continuity Definition. Given a set E R, a function f : E R, and a point c E, the function f is continuous at c if

More information

Green s Functions and Distributions

Green s Functions and Distributions CHAPTER 9 Green s Functions and Distributions 9.1. Boundary Value Problems We would like to study, and solve if possible, boundary value problems such as the following: (1.1) u = f in U u = g on U, where

More information

Stationary states for the aggregation equation with power law attractive-repulsive potentials

Stationary states for the aggregation equation with power law attractive-repulsive potentials Stationary states for the aggregation equation with power law attractive-repulsive potentials Daniel Balagué Joint work with J.A. Carrillo, T. Laurent and G. Raoul Universitat Autònoma de Barcelona BIRS

More information

Fixed Points & Fatou Components

Fixed Points & Fatou Components Definitions 1-3 are from [3]. Definition 1 - A sequence of functions {f n } n, f n : A B is said to diverge locally uniformly from B if for every compact K A A and K B B, there is an n 0 such that f n

More information

Gauss s Law & Potential

Gauss s Law & Potential Gauss s Law & Potential Lecture 7: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay Flux of an Electric Field : In this lecture we introduce Gauss s law which happens to

More information

Gradient, Divergence and Curl in Curvilinear Coordinates

Gradient, Divergence and Curl in Curvilinear Coordinates Gradient, Divergence and Curl in Curvilinear Coordinates Although cartesian orthogonal coordinates are very intuitive and easy to use, it is often found more convenient to work with other coordinate systems.

More information

arxiv: v3 [math.dg] 19 Jun 2017

arxiv: v3 [math.dg] 19 Jun 2017 THE GEOMETRY OF THE WIGNER CAUSTIC AND AFFINE EQUIDISTANTS OF PLANAR CURVES arxiv:160505361v3 [mathdg] 19 Jun 017 WOJCIECH DOMITRZ, MICHA L ZWIERZYŃSKI Abstract In this paper we study global properties

More information

Partial Differential Equations

Partial Differential Equations Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,

More information

14 Higher order forms; divergence theorem

14 Higher order forms; divergence theorem Tel Aviv University, 2013/14 Analysis-III,IV 221 14 Higher order forms; divergence theorem 14a Forms of order three................ 221 14b Divergence theorem in three dimensions.... 225 14c Order four,

More information

The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative,

The fundamental theorem of calculus for definite integration helped us to compute If has an anti-derivative, Module 16 : Line Integrals, Conservative fields Green's Theorem and applications Lecture 47 : Fundamental Theorems of Calculus for Line integrals [Section 47.1] Objectives In this section you will learn

More information

ON THE REAL MILNOR FIBRE OF SOME MAPS FROM R n TO R 2

ON THE REAL MILNOR FIBRE OF SOME MAPS FROM R n TO R 2 ON THE REAL MILNOR FIBRE OF SOME MAPS FROM R n TO R 2 NICOLAS DUTERTRE Abstract. We consider a real analytic map-germ (f, g) : (R n, 0) (R 2, 0). Under some conditions, we establish degree formulas for

More information

Lecture notes: Introduction to Partial Differential Equations

Lecture notes: Introduction to Partial Differential Equations Lecture notes: Introduction to Partial Differential Equations Sergei V. Shabanov Department of Mathematics, University of Florida, Gainesville, FL 32611 USA CHAPTER 1 Classification of Partial Differential

More information

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Fall 2011 Professor: Jared Speck MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Fall 20 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations

More information

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck

MATH COURSE NOTES - CLASS MEETING # Introduction to PDEs, Spring 2018 Professor: Jared Speck MATH 8.52 COURSE NOTES - CLASS MEETING # 6 8.52 Introduction to PDEs, Spring 208 Professor: Jared Speck Class Meeting # 6: Laplace s and Poisson s Equations We will now study the Laplace and Poisson equations

More information

Mathematics for Economists

Mathematics for Economists Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb. 2017 1 / 34 Definitions and Examples

More information

Lecture 8. Engineering applications

Lecture 8. Engineering applications Lecture 8 Engineering applications In Lecture 6 we saw one classic example of the application of vector calculus to Maxwell s equation. In this lecture we explore a few more examples from fluid mechanics

More information

MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW

MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW MEAN CURVATURE FLOW OF ENTIRE GRAPHS EVOLVING AWAY FROM THE HEAT FLOW GREGORY DRUGAN AND XUAN HIEN NGUYEN Abstract. We present two initial graphs over the entire R n, n 2 for which the mean curvature flow

More information

BOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1. (k > 1)

BOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1. (k > 1) GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 585-6 BOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1 (k > 1) S. TOPURIA Abstract. Boundary

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

More information

Math 225B: Differential Geometry, Final

Math 225B: Differential Geometry, Final Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of

More information

Topics in Stochastic Geometry. Lecture 4 The Boolean model

Topics in Stochastic Geometry. Lecture 4 The Boolean model Institut für Stochastik Karlsruher Institut für Technologie Topics in Stochastic Geometry Lecture 4 The Boolean model Lectures presented at the Department of Mathematical Sciences University of Bath May

More information

9.4 Vector and Scalar Fields; Derivatives

9.4 Vector and Scalar Fields; Derivatives 9.4 Vector and Scalar Fields; Derivatives Vector fields A vector field v is a vector-valued function defined on some domain of R 2 or R 3. Thus if D is a subset of R 3, a vector field v with domain D associates

More information

Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems.

Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems. S. R. Zinka zinka@vit.ac.in School of Electronics Engineering Vellore Institute of Technology July 16, 2013 Outline 1 Vectors 2 Coordinate Systems 3 VC - Differential Elements 4 VC - Differential Operators

More information

LECTURE 11: TRANSVERSALITY

LECTURE 11: TRANSVERSALITY LECTURE 11: TRANSVERSALITY Let f : M N be a smooth map. In the past three lectures, we are mainly studying the image of f, especially when f is an embedding. Today we would like to study the pre-image

More information

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1.

S chauder Theory. x 2. = log( x 1 + x 2 ) + 1 ( x 1 + x 2 ) 2. ( 5) x 1 + x 2 x 1 + x 2. 2 = 2 x 1. x 1 x 2. 1 x 1. Sep. 1 9 Intuitively, the solution u to the Poisson equation S chauder Theory u = f 1 should have better regularity than the right hand side f. In particular one expects u to be twice more differentiable

More information

Pogorelov Klingenberg theorem for manifolds homeomorphic to R n (translated from [4])

Pogorelov Klingenberg theorem for manifolds homeomorphic to R n (translated from [4]) Pogorelov Klingenberg theorem for manifolds homeomorphic to R n (translated from [4]) Vladimir Sharafutdinov January 2006, Seattle 1 Introduction The paper is devoted to the proof of the following Theorem

More information

f(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da

f(p i )Area(T i ) F ( r(u, w) ) (r u r w ) da MAH 55 Flux integrals Fall 16 1. Review 1.1. Surface integrals. Let be a surface in R. Let f : R be a function defined on. efine f ds = f(p i Area( i lim mesh(p as a limit of Riemann sums over sampled-partitions.

More information

GENERALIZED ISOPERIMETRIC INEQUALITIES FOR EXTRINSIC BALLS IN MINIMAL SUBMANIFOLDS. Steen Markvorsen and Vicente Palmer*

GENERALIZED ISOPERIMETRIC INEQUALITIES FOR EXTRINSIC BALLS IN MINIMAL SUBMANIFOLDS. Steen Markvorsen and Vicente Palmer* GENEALIZED ISOPEIMETIC INEQUALITIES FO EXTINSIC BALLS IN MINIMAL SUBMANIFOLDS Steen Markvorsen and Vicente Palmer* Abstract. The volume of an extrinsic ball in a minimal submanifold has a well defined

More information

Perron method for the Dirichlet problem.

Perron method for the Dirichlet problem. Introduzione alle equazioni alle derivate parziali, Laurea Magistrale in Matematica Perron method for the Dirichlet problem. We approach the question of existence of solution u C (Ω) C(Ω) of the Dirichlet

More information

An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018

More information

Lecture 04. Curl and Divergence

Lecture 04. Curl and Divergence Lecture 04 Curl and Divergence UCF Curl of Vector Field (1) F c d l F C Curl (or rotor) of a vector field a n curlf F d l lim c s s 0 F s a n C a n : normal direction of s follow right-hand rule UCF Curl

More information

Exercise 1 (Formula for connection 1-forms) Using the first structure equation, show that

Exercise 1 (Formula for connection 1-forms) Using the first structure equation, show that 1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral

More information

Chapter 4. Electrostatic Fields in Matter

Chapter 4. Electrostatic Fields in Matter Chapter 4. Electrostatic Fields in Matter 4.1. Polarization 4.2. The Field of a Polarized Object 4.3. The Electric Displacement 4.4. Linear Dielectrics 4.5. Energy in dielectric systems 4.6. Forces on

More information

A f = A f (x)dx, 55 M F ds = M F,T ds, 204 M F N dv n 1, 199 !, 197. M M F,N ds = M F ds, 199 (Δ,')! = '(Δ)!, 187

A f = A f (x)dx, 55 M F ds = M F,T ds, 204 M F N dv n 1, 199 !, 197. M M F,N ds = M F ds, 199 (Δ,')! = '(Δ)!, 187 References 1. T.M. Apostol; Mathematical Analysis, 2nd edition, Addison-Wesley Publishing Co., Reading, Mass. London Don Mills, Ont., 1974. 2. T.M. Apostol; Calculus Vol. 2: Multi-variable Calculus and

More information

Physics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves

Physics 3312 Lecture 9 February 13, LAST TIME: Finished mirrors and aberrations, more on plane waves Physics 331 Lecture 9 February 13, 019 LAST TIME: Finished mirrors and aberrations, more on plane waves Recall, Represents a plane wave having a propagation vector k that propagates in any direction with

More information

송석호 ( 물리학과 )

송석호 ( 물리학과 ) http://optics.hanyang.ac.kr/~shsong 송석호 ( 물리학과 ) Introduction to Electrodynamics, David J. Griffiths Review: 1. Vector analysis 2. Electrostatics 3. Special techniques 4. Electric fields in mater 5. Magnetostatics

More information

History (Minimal Dynamics Theorem, Meeks-Perez-Ros 2005, CMC Dynamics Theorem, Meeks-Tinaglia 2008) Giuseppe Tinaglia

History (Minimal Dynamics Theorem, Meeks-Perez-Ros 2005, CMC Dynamics Theorem, Meeks-Tinaglia 2008) Giuseppe Tinaglia History (Minimal Dynamics Theorem, Meeks-Perez-Ros 2005, CMC Dynamics Theorem, Meeks-Tinaglia 2008) Briefly stated, these Dynamics Theorems deal with describing all of the periodic or repeated geometric

More information

Week 2 Notes, Math 865, Tanveer

Week 2 Notes, Math 865, Tanveer Week 2 Notes, Math 865, Tanveer 1. Incompressible constant density equations in different forms Recall we derived the Navier-Stokes equation for incompressible constant density, i.e. homogeneous flows:

More information

Math 147, Homework 5 Solutions Due: May 15, 2012

Math 147, Homework 5 Solutions Due: May 15, 2012 Math 147, Homework 5 Solutions Due: May 15, 2012 1 Let f : R 3 R 6 and φ : R 3 R 3 be the smooth maps defined by: f(x, y, z) = (x 2, y 2, z 2, xy, xz, yz) and φ(x, y, z) = ( x, y, z) (a) Show that f is

More information

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008

Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Department of Mathematics, IIT Bombay End-Semester Examination, MA 105 Autumn-2008 Code: C-031 Date and time: 17 Nov, 2008, 9:30 A.M. - 12:30 P.M. Maximum Marks: 45 Important Instructions: 1. The question

More information

Mid Term-1 : Practice problems

Mid Term-1 : Practice problems Mid Term-1 : Practice problems These problems are meant only to provide practice; they do not necessarily reflect the difficulty level of the problems in the exam. The actual exam problems are likely to

More information

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0

1 )(y 0) {1}. Thus, the total count of points in (F 1 (y)) is equal to deg y0 1. Classification of 1-manifolds Theorem 1.1. Let M be a connected 1 manifold. Then M is diffeomorphic either to [0, 1], [0, 1), (0, 1), or S 1. We know that none of these four manifolds are not diffeomorphic

More information

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1

On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1 On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that

More information

a i,1 a i,j a i,m be vectors with positive entries. The linear programming problem associated to A, b and c is to find all vectors sa b

a i,1 a i,j a i,m be vectors with positive entries. The linear programming problem associated to A, b and c is to find all vectors sa b LINEAR PROGRAMMING PROBLEMS MATH 210 NOTES 1. Statement of linear programming problems Suppose n, m 1 are integers and that A = a 1,1... a 1,m a i,1 a i,j a i,m a n,1... a n,m is an n m matrix of positive

More information