Math 124B February 02, 2012

Size: px
Start display at page:

Download "Math 124B February 02, 2012"

Transcription

1 Math 24B Febuay 02, 202 Vikto Gigoyan 8 Laplace s equation: popeties We have aleady encounteed Laplace s equation in the context of stationay heat conduction and wave phenomena. Recall that in two spatial dimensions, the heat equation is u t k(u xx + u yy 0, which descibes the tempeatues of a two dimensional plate. Similaly, the vibations of a two dimensional membane ae descibed by the wave equation in two spatial dimensions, u tt c 2 (u xx + u yy 0. If one consides stationay heat and wave states, i.e. not changing with time, then u t u tt 0, and both the heat and wave equations educe to the stationay equation u xx + u yy 0. This is the two dimensional Laplace equation. Analogously, in thee dimension one has the equation u xx + u yy + u zz 0. Using the notation, we can ewite Laplace s equation in any dimension as u 0. ( The opeato is called Laplace s opeato, o Laplacian. To distinguish the Laplacian in diffeent dimensions, we will use the subscipt notation n, whee n stands fo the dimensions. The solutions of the Laplace equation ( ae called hamonic functions. The inhomogeneous Laplace s equation n u f(x, x 2,..., x n, is called Poisson s equation. Besides descibing stationay heat and wave phenomena, Laplace s and Poisson s equations come up in the study of electostatics, incompessible fluid flow, analytic functions theoy, Bownian motion, etc. Notice that in one dimension Laplace s equation is the OE u xx 0, so the only hamonic functions in one dimension ae the linea functions u(x A + Bx. An obvious distinction between Laplace s equation and the heat and wave equations is that the pocesses descibed by Laplace s equation do not involve dynamics o evolution of the initial data in time. Hence the natual poblem to study fo Laplace s equation is the bounday value poblem in some given domain. We will conside the equation u f in, (2 with eithe of the following conditions on the bounday of the domain. ( : u u h; (N : u n h; (R : n + au h, whee n is the oute nomal vecto to this bounday. When consideing heat and wave bounday value poblems, we saw that the bounday in one dimension consists of the endpoints of the inteval (a, b, in which the equation is being solved. In two dimensions the bounday will be a cuve, while in thee dimensions it is a suface, and we expect that the geomety of the bounday will play a ole in solving Laplace s equation. We will estict ou study of Laplace s equation to two and thee dimensions, and will occasionally use the vecto notation x (x, y, o x (x, y, z to denote a point in eithe two dimensional, o thee dimensional space. Lets us stat by fist discussing the popeties of Laplace s equation. 8. Maximum pinciple It tuns out that hamonic functions obey a maximum pinciple, which is simila to the maximum pinciple fo the heat equation. In what follows, an open bounded connected set is a set that does not contain any of its bounday points (open, entiely lies inside some ball centeed at the oigin (bounded,

2 and consists of one piece, i.e. any two points of the set can be connected by a cuve that entiely lies inside the set (connected. Maximum Pinciple. Let be a connected bounded open set (in eithe two o thee dimensional space, and u be a hamonic function in, which is also continuous on the closue of the set,. Then the maximum and minimum values of u ae attained on the bounday of. Moeove, the maximum and minimum values cannot be attained inside, unless u constant. Mathematically, this means that if u is a non-constant hamonic function in an open bounded connected set, then max{u} < max{u} max{u}, and min{u} min{u} < min {u}. If we think of hamonic functions as equilibium states in the heat conduction, then the maximum pinciple makes pefect sense, since if thee was a maximum tempeatue at an inteio point of the domain, thee would have been a heat flux fom this point to the points with lowe tempeatue, which would consequently decease the tempeatue of this point, making the state unsteady. The idea of the poof of the maximum pinciple is simila to the one used in poving the maximum pinciple fo the heat equation. We stat by noting that if, say in two dimensions, (x 0, y 0 is a maximum point, then both u xx (x 0, y 0 0 and u yy (x 0, y 0 0 by the second deivative test. But then u(x 0, y 0 u xx (x 0, y 0 + u yy (x 0, y 0 0. This would be a contadiction, if the inequality was stict, howeve second ode deivatives may be zeo at extema points. To eliminate this scenaio, we modify the function u, and conside the new function v(x u(x+ɛ x 2, whee ɛ > 0 is a small constant. But then we have 2 v 2 u + ɛ 2 (x 2 + y ɛ > 0 in, and similaly in thee dimensions. But 2 v v xx + v yy 0 at an inteio maximum point, theefoe v(x has no inteio maximum points in. Since v(x is continuous, it must attain its maximum value somewhee in the closed set, so the maximum must be attained at some point x 0. Then fo evey point x in, we have u(x v(x v(x 0 u(x 0 + ɛ x 0 2 max {u} + ɛl2, whee l is the lagest distance fom the oigin to the bounday of the (bounded set. Since ɛ was abitay, we can make it go to zeo, yielding u(x max{u}, fo evey x. So the maximum of u must be attained on the bounday. The poof fo the minimum is simila. The stonge statement that the maximum cannot be attained inside will be poved late via the mean value popety of hamonic functions. 8.2 Uniqueness of the iichlet poblem As was the case fo the heat equation, the maximum pinciple diectly implies uniqueness of the iichlet poblem fo Poisson s equation. Indeed, suppose that the iichlet poblem { u f in, u h on, whee is open bounded and connected, has two solutions u, u 2. Then thei diffeence, w u u 2, is hamonic, and has zeo iichlet data on the bounday. But by the maximum/minimum pinciple 2

3 we have fo any point x, 0 min{w} w(x max{w} 0, so w(x u (x u 2 (x 0. We will give an altenative poof of the uniqueness using the enegy method in a subsequent lectue, which will also show that solutions to the Neumann poblem ae unique up to a constant. 8.3 Invaiance Laplace s equation is invaiant unde igid motions, which ae the tanslations, and otations. A tanslation is a tansfomation x x, which is given by x x + a fo some vecto a. In two dimensions this vecto equation is equivalent to x x + a, y y + b, and it is easy to see that u xx + u yy u x x + u y y 0. So if a function is hamonic in the vaiables (x, y, it must also be hamonic in the vaiables (x, y. This is the invaiance unde tanslations. Clealy this holds in highe dimensions as well. Fo the invaiance unde otations, we need to show that Laplace s equation emains the same in the vaiables x x cos α + y sin α y x sin α + y cos α, o [ x y ] ( cos α sin α sin α cos α [ x y whee α is the angle of otation. Using the chain ule, one can compute u x, u y, and then u xx, u yy in tems of the patial deivatives of u with espect to (x, y vaiables, and show that u xx + u yy (u x x + u y y (cos2 α + sin 2 α u x x + u y y. Thus, Laplace s opeato is invaiant unde otations in two dimensions. One can pove the invaiance unde otations in any dimension n 2, 3,... using the matix notation as follows. In any dimension n a otation is given by ], x Bx, o x k b ki x i, i whee B {b ij } is an othogonal matix, that is BB t B t B I, o b ki b li δk, l i whee δ l k, if k l, and δl k 0, if k l is the Konecke symbol. Using the chain ule, we can compute x i k x k x i x k b ki x k k To compute the second ode deivatives, we multiply the fist ode deivative opeato by itself. ( ( 2 2 b x 2 ki b i x li b k x ki b li. l x k x l k l k,l. 3

4 But then x 2 x 2 i i i k,l b ki b li 2 x k x l ( b ki b li k,l i 2 x k x l k,l δ l k 2 x k x l x. So Laplace s opeato is indeed invaiant unde otations. The otation invaiance also implies that Laplace s equation allows otationally invaiant solutions, that is, solutions that depend only on the adial vaiable x. We will call such solutions adial. 8.4 Radial solutions of Laplace s equation In ode to find adial solutions to Laplace s equation, we make a change to pola vaiables in two dimensions, and to spheical vaiables in thee dimensions. Notice that in this case the adial solution simply means that u(, θ u(, o u(, θ, φ u(, that is, the function depends on only one vaiable, and, as a consequence, the PE will educe to an OE. We fist make a change to pola vaiables in two dimensions, fo which the tansfomation fomulas ae { x cos θ y sin θ, with Jacobian matix (x, y (, θ ( x x θ Using x 2 + y 2, one can compute the patial deivatives x y y θ 2x 2 x 2 + y x 2 cos θ, y y Also, diffeentiating both sides of x cos θ with espect to x, we get θ y θ cos θ sin θ x x cos2 θ sin θ θ x sin θ. θ x cos2 θ sin θ ( cos θ sin θ sin θ cos θ can be computed similaly. So the Jacobian matix of the invese tansfomation is (, θ (x, y By the chain ule, we will have Using these, one can compute ( x y x cos θ sin θ 2 θ x θ y ( cos θ sin θ/ sin θ cos θ/. θ, and y sin θ + cos θ ( 2 ( x y θ. 2 And the Laplace equation can be witten in pola vaiables as u + u + 2 u θθ 0. θ. sin θ. Fo a adial solution u(, θ u(, the last tem in the above equation will vanish, yielding the equation u + u 0, o u + u 0,. 4

5 which is an OE, as expected. The last equation can be witten as (u 0, whee we used the integating facto exp( d. Integating the last equation gives Integating once moe gives the solution u c, o u c. u( c log + c 2. isegading the constant solution c 2, we see that the function log log x is hamonic in two dimensions. To find adial solutions in thee dimensions, we need to make a change to spheical vaiables, which is given by the tansfomations x 2 + y 2 + z 2 s 2 + z 2 s x 2 + y 2 x s cos φ z cos θ y s sin φ s sin θ. Thus, the tansfomation to spheical vaiables can be thought of as the pai of successive tansfomations (x, y, z (s, φ, z (, θ, φ. Using the above computation in two dimensions, we have that u zz + u ss u + u + 2 u θθ, and u xx + u yy u ss + s u s + s 2 u φφ. Adding these two identities, and canceling the tem u ss on both sides, we get 3 u u + u + s u s + 2 u θθ + s 2 u φφ. (3 We can also compute u s u s u s + u θ θ s + u φ φ s u s + u cos θ θ. Then eplacing u s in (3 by the above expession, and substituting s sin θ fo all occuences of s, we obtain Laplace s equation in the spheical vaiables in thee dimensions u + 2 u + [ u 2 θθ + cot θu θ + ] sin 2 θ u φφ 0. Fo a adial solution u(, θ, φ u(, the entie squae backets tem will vanish, so Laplace s equation will educe to the OE u + 2 u 0. 5

6 Multiplying this equation by 2, we can wite it as ( 2 u 0, 2 whee we used the integating facto exp( d. Integating this equation gives Integating yet again, we obtain the solution 2 u c, o u c 2. u( c + c 2. So the function / / x is hamonic in thee dimensions. Notice that both / and log functions ae not defined at the oigin 0, but they will be hamonic on any domain which does not contain the oigin. We will see in subsequent lectues that these functions in the context of Laplace s equation play a ole simila to that of the heat kenel in the context of the heat equation. 8.5 Conclusion In this lectue we studied the maximum pinciple fo Laplace s equation, which tivially implies the uniqueness of solutions to the iichlet poblem fo Poisson s equation. We also saw that Laplace s equation is invaiant unde tanslations and otations. The last fact accounted fo existence of adial solutions, which ae solutions that ae invaiant unde otations, and hence depend only on the adial vaiable. Making a change to pola vaiables in two dimensions, and spheical vaiables in thee dimensions, we wee able to find adial hamonic functions by solving the OEs satisfied by these functions. We will see in a late lectue that these adial hamonic functions play a cucial ole in finding the solution to the iichlet poblem fo Laplace s and Poisson s equations. 6

15 Solving the Laplace equation by Fourier method

15 Solving the Laplace equation by Fourier method 5 Solving the Laplace equation by Fouie method I aleady intoduced two o thee dimensional heat equation, when I deived it, ecall that it taes the fom u t = α 2 u + F, (5.) whee u: [0, ) D R, D R is the

More information

MATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates

MATH 417 Homework 3 Instructor: D. Cabrera Due June 30. sin θ v x = v r cos θ v θ r. (b) Then use the Cauchy-Riemann equations in polar coordinates MATH 417 Homewok 3 Instucto: D. Cabea Due June 30 1. Let a function f(z) = u + iv be diffeentiable at z 0. (a) Use the Chain Rule and the fomulas x = cosθ and y = to show that u x = u cosθ u θ, v x = v

More information

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2

THE LAPLACE EQUATION. The Laplace (or potential) equation is the equation. u = 0. = 2 x 2. x y 2 in R 2 THE LAPLACE EQUATION The Laplace (o potential) equation is the equation whee is the Laplace opeato = 2 x 2 u = 0. in R = 2 x 2 + 2 y 2 in R 2 = 2 x 2 + 2 y 2 + 2 z 2 in R 3 The solutions u of the Laplace

More information

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3.

As is natural, our Aerospace Structures will be described in a Euclidean three-dimensional space R 3. Appendix A Vecto Algeba As is natual, ou Aeospace Stuctues will be descibed in a Euclidean thee-dimensional space R 3. A.1 Vectos A vecto is used to epesent quantities that have both magnitude and diection.

More information

8 Separation of Variables in Other Coordinate Systems

8 Separation of Variables in Other Coordinate Systems 8 Sepaation of Vaiables in Othe Coodinate Systems Fo the method of sepaation of vaiables to succeed you need to be able to expess the poblem at hand in a coodinate system in which the physical boundaies

More information

EM Boundary Value Problems

EM Boundary Value Problems EM Bounday Value Poblems 10/ 9 11/ By Ilekta chistidi & Lee, Seung-Hyun A. Geneal Desciption : Maxwell Equations & Loentz Foce We want to find the equations of motion of chaged paticles. The way to do

More information

Lecture 7: Angular Momentum, Hydrogen Atom

Lecture 7: Angular Momentum, Hydrogen Atom Lectue 7: Angula Momentum, Hydogen Atom Vecto Quantization of Angula Momentum and Nomalization of 3D Rigid Roto wavefunctions Conside l, so L 2 2 2. Thus, we have L 2. Thee ae thee possibilities fo L z

More information

Geometry of the homogeneous and isotropic spaces

Geometry of the homogeneous and isotropic spaces Geomety of the homogeneous and isotopic spaces H. Sonoda Septembe 2000; last evised Octobe 2009 Abstact We summaize the aspects of the geomety of the homogeneous and isotopic spaces which ae most elevant

More information

-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r.

-Δ u = λ u. u(x,y) = u 1. (x) u 2. (y) u(r,θ) = R(r) Θ(θ) Δu = 2 u + 2 u. r = x 2 + y 2. tan(θ) = y/x. r cos(θ) = cos(θ) r. The Laplace opeato in pola coodinates We now conside the Laplace opeato with Diichlet bounday conditions on a cicula egion Ω {(x,y) x + y A }. Ou goal is to compute eigenvalues and eigenfunctions of the

More information

MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form

MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE. We consider second order constant coefficient scalar linear PDEs on R n. These have the form MATH 220: SECOND ORDER CONSTANT COEFFICIENT PDE ANDRAS VASY We conside second ode constant coefficient scala linea PDEs on R n. These have the fom Lu = f L = a ij xi xj + b i xi + c i whee a ij b i and

More information

On the integration of the equations of hydrodynamics

On the integration of the equations of hydrodynamics Uebe die Integation de hydodynamischen Gleichungen J f eine u angew Math 56 (859) -0 On the integation of the equations of hydodynamics (By A Clebsch at Calsuhe) Tanslated by D H Delphenich In a pevious

More information

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk

Appendix A. Appendices. A.1 ɛ ijk and cross products. Vector Operations: δ ij and ɛ ijk Appendix A Appendices A1 ɛ and coss poducts A11 Vecto Opeations: δ ij and ɛ These ae some notes on the use of the antisymmetic symbol ɛ fo expessing coss poducts This is an extemely poweful tool fo manipulating

More information

Green s Identities and Green s Functions

Green s Identities and Green s Functions LECTURE 7 Geen s Identities and Geen s Functions Let us ecall The ivegence Theoem in n-dimensions Theoem 7 Let F : R n R n be a vecto field ove R n that is of class C on some closed, connected, simply

More information

ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi

ENGI 4430 Non-Cartesian Coordinates Page xi Fy j Fzk from Cartesian coordinates z to another orthonormal coordinate system u, v, ˆ i ˆ ˆi ENGI 44 Non-Catesian Coodinates Page 7-7. Conesions between Coodinate Systems In geneal, the conesion of a ecto F F xi Fy j Fzk fom Catesian coodinates x, y, z to anothe othonomal coodinate system u,,

More information

Vector d is a linear vector function of vector d when the following relationships hold:

Vector d is a linear vector function of vector d when the following relationships hold: Appendix 4 Dyadic Analysis DEFINITION ecto d is a linea vecto function of vecto d when the following elationships hold: d x = a xxd x + a xy d y + a xz d z d y = a yxd x + a yy d y + a yz d z d z = a zxd

More information

1D2G - Numerical solution of the neutron diffusion equation

1D2G - Numerical solution of the neutron diffusion equation DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both

More information

=0, (x, y) Ω (10.1) Depending on the nature of these boundary conditions, forced, natural or mixed type, the elliptic problems are classified as

=0, (x, y) Ω (10.1) Depending on the nature of these boundary conditions, forced, natural or mixed type, the elliptic problems are classified as Chapte 1 Elliptic Equations 1.1 Intoduction The mathematical modeling of steady state o equilibium phenomena geneally esult in to elliptic equations. The best example is the steady diffusion of heat in

More information

Homework # 3 Solution Key

Homework # 3 Solution Key PHYSICS 631: Geneal Relativity Homewok # 3 Solution Key 1. You e on you hono not to do this one by hand. I ealize you can use a compute o simply look it up. Please don t. In a flat space, the metic in

More information

Compactly Supported Radial Basis Functions

Compactly Supported Radial Basis Functions Chapte 4 Compactly Suppoted Radial Basis Functions As we saw ealie, compactly suppoted functions Φ that ae tuly stictly conditionally positive definite of ode m > do not exist The compact suppot automatically

More information

Mechanics Physics 151

Mechanics Physics 151 Mechanics Physics 151 Lectue 5 Cental Foce Poblem (Chapte 3) What We Did Last Time Intoduced Hamilton s Pinciple Action integal is stationay fo the actual path Deived Lagange s Equations Used calculus

More information

On a quantity that is analogous to potential and a theorem that relates to it

On a quantity that is analogous to potential and a theorem that relates to it Su une quantité analogue au potential et su un théoème y elatif C R Acad Sci 7 (87) 34-39 On a quantity that is analogous to potential and a theoem that elates to it By R CLAUSIUS Tanslated by D H Delphenich

More information

c n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0.

c n ψ n (r)e ient/ h (2) where E n = 1 mc 2 α 2 Z 2 ψ(r) = c n ψ n (r) = c n = ψn(r)ψ(r)d 3 x e 2r/a0 1 πa e 3r/a0 r 2 dr c 1 2 = 2 9 /3 6 = 0. Poblem {a} Fo t : Ψ(, t ψ(e iet/ h ( whee E mc α (α /7 ψ( e /a πa Hee we have used the gound state wavefunction fo Z. Fo t, Ψ(, t can be witten as a supeposition of Z hydogenic wavefunctions ψ n (: Ψ(,

More information

What Form of Gravitation Ensures Weakened Kepler s Third Law?

What Form of Gravitation Ensures Weakened Kepler s Third Law? Bulletin of Aichi Univ. of Education, 6(Natual Sciences, pp. - 6, Mach, 03 What Fom of Gavitation Ensues Weakened Keple s Thid Law? Kenzi ODANI Depatment of Mathematics Education, Aichi Univesity of Education,

More information

Stress, Cauchy s equation and the Navier-Stokes equations

Stress, Cauchy s equation and the Navier-Stokes equations Chapte 3 Stess, Cauchy s equation and the Navie-Stokes equations 3. The concept of taction/stess Conside the volume of fluid shown in the left half of Fig. 3.. The volume of fluid is subjected to distibuted

More information

Do not turn over until you are told to do so by the Invigilator.

Do not turn over until you are told to do so by the Invigilator. UNIVERSITY OF EAST ANGLIA School of Mathematics Main Seies UG Examination 2015 16 FLUID DYNAMICS WITH ADVANCED TOPICS MTH-MD59 Time allowed: 3 Hous Attempt QUESTIONS 1 and 2, and THREE othe questions.

More information

2 Governing Equations

2 Governing Equations 2 Govening Equations This chapte develops the govening equations of motion fo a homogeneous isotopic elastic solid, using the linea thee-dimensional theoy of elasticity in cylindical coodinates. At fist,

More information

(read nabla or del) is defined by, k. (9.7.1*)

(read nabla or del) is defined by, k. (9.7.1*) 9.7 Gadient of a scala field. Diectional deivative Some of the vecto fields in applications can be obtained fom scala fields. This is vey advantageous because scala fields can be handled moe easily. The

More information

Lecture 23. Representation of the Dirac delta function in other coordinate systems

Lecture 23. Representation of the Dirac delta function in other coordinate systems Lectue 23 Repesentation of the Diac delta function in othe coodinate systems In a geneal sense, one can wite, ( ) = (x x ) (y y ) (z z ) = (u u ) (v v ) (w w ) J Whee J epesents the Jacobian of the tansfomation.

More information

d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c

d 2 x 0a d d =0. Relative to an arbitrary (accelerating frame) specified by x a = x a (x 0b ), the latter becomes: d 2 x a d 2 + a dx b dx c Chapte 6 Geneal Relativity 6.1 Towads the Einstein equations Thee ae seveal ways of motivating the Einstein equations. The most natual is pehaps though consideations involving the Equivalence Pinciple.

More information

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum

2. Electrostatics. Dr. Rakhesh Singh Kshetrimayum 8/11/ Electromagnetic Field Theory by R. S. Kshetrimayum 2. Electostatics D. Rakhesh Singh Kshetimayum 1 2.1 Intoduction In this chapte, we will study how to find the electostatic fields fo vaious cases? fo symmetic known chage distibution fo un-symmetic known

More information

B da = 0. Q E da = ε. E da = E dv

B da = 0. Q E da = ε. E da = E dv lectomagnetic Theo Pof Ruiz, UNC Asheville, doctophs on YouTube Chapte Notes The Maxwell quations in Diffeential Fom 1 The Maxwell quations in Diffeential Fom We will now tansfom the integal fom of the

More information

is the instantaneous position vector of any grid point or fluid

is the instantaneous position vector of any grid point or fluid Absolute inetial, elative inetial and non-inetial coodinates fo a moving but non-defoming contol volume Tao Xing, Pablo Caica, and Fed Sten bjective Deive and coelate the govening equations of motion in

More information

MODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ...

MODULE 5a and 5b (Stewart, Sections 12.2, 12.3) INTRO: In MATH 1114 vectors were written either as rows (a1, a2,..., an) or as columns a 1 a. ... MODULE 5a and 5b (Stewat, Sections 2.2, 2.3) INTRO: In MATH 4 vectos wee witten eithe as ows (a, a2,..., an) o as columns a a 2... a n and the set of all such vectos of fixed length n was called the vecto

More information

FOURIER-BESSEL SERIES AND BOUNDARY VALUE PROBLEMS IN CYLINDRICAL COORDINATES

FOURIER-BESSEL SERIES AND BOUNDARY VALUE PROBLEMS IN CYLINDRICAL COORDINATES FOURIER-BESSE SERIES AND BOUNDARY VAUE PROBEMS IN CYINDRICA COORDINATES The paametic Bessel s equation appeas in connection with the aplace opeato in pola coodinates. The method of sepaation of vaiables

More information

POISSON S EQUATION 2 V 0

POISSON S EQUATION 2 V 0 POISSON S EQUATION We have seen how to solve the equation but geneally we have V V4k We now look at a vey geneal way of attacking this poblem though Geen s Functions. It tuns out that this poblem has applications

More information

Scattering in Three Dimensions

Scattering in Three Dimensions Scatteing in Thee Dimensions Scatteing expeiments ae an impotant souce of infomation about quantum systems, anging in enegy fom vey low enegy chemical eactions to the highest possible enegies at the LHC.

More information

DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS

DOING PHYSICS WITH MATLAB COMPUTATIONAL OPTICS DOING PHYIC WITH MTLB COMPUTTIONL OPTIC FOUNDTION OF CLR DIFFRCTION THEORY Ian Coope chool of Physics, Univesity of ydney ian.coope@sydney.edu.au DOWNLOD DIRECTORY FOR MTLB CRIPT View document: Numeical

More information

Lecture 8 - Gauss s Law

Lecture 8 - Gauss s Law Lectue 8 - Gauss s Law A Puzzle... Example Calculate the potential enegy, pe ion, fo an infinite 1D ionic cystal with sepaation a; that is, a ow of equally spaced chages of magnitude e and altenating sign.

More information

Rigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018

Rigid Body Dynamics 2. CSE169: Computer Animation Instructor: Steve Rotenberg UCSD, Winter 2018 Rigid Body Dynamics 2 CSE169: Compute Animation nstucto: Steve Rotenbeg UCSD, Winte 2018 Coss Poduct & Hat Opeato Deivative of a Rotating Vecto Let s say that vecto is otating aound the oigin, maintaining

More information

x x2 2 B A ) v(0, t) = 0 and v(l, t) = 0. L 2. This is a familiar heat equation initial/boundary-value problem and has solution

x x2 2 B A ) v(0, t) = 0 and v(l, t) = 0. L 2. This is a familiar heat equation initial/boundary-value problem and has solution Hints to homewok 7 8.2.d. The poblem is u t ku xx + k ux fx u t A u t B. It has a souce tem and inhomogeneous bounday conditions but none of them depend on t. So as in example 3 of the notes we should

More information

RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION

RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION RADIALLY SYMMETRIC SOLUTIONS TO THE GRAPHIC WILLMORE SURFACE EQUATION JINGYI CHEN AND YUXIANG LI Abstact. We show that a smooth adially symmetic solution u to the gaphic Willmoe suface equation is eithe

More information

f(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4

f(k) e p 2 (k) e iax 2 (k a) r 2 e a x a a 2 + k 2 e a2 x 1 2 H(x) ik p (k) 4 r 3 cos Y 2 = 4 Fouie tansfom pais: f(x) 1 f(k) e p 2 (k) p e iax 2 (k a) 2 e a x a a 2 + k 2 e a2 x 1 2, a > 0 a p k2 /4a2 e 2 1 H(x) ik p 2 + 2 (k) The fist few Y m Y 0 0 = Y 0 1 = Y ±1 1 = l : 1 Y2 0 = 4 3 ±1 cos Y

More information

Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009

Physics 111 Lecture 5 (Walker: 3.3-6) Vectors & Vector Math Motion Vectors Sept. 11, 2009 Physics 111 Lectue 5 (Walke: 3.3-6) Vectos & Vecto Math Motion Vectos Sept. 11, 2009 Quiz Monday - Chap. 2 1 Resolving a vecto into x-component & y- component: Pola Coodinates Catesian Coodinates x y =

More information

1 Notes on Order Statistics

1 Notes on Order Statistics 1 Notes on Ode Statistics Fo X a andom vecto in R n with distibution F, and π S n, define X π by and F π by X π (X π(1),..., X π(n) ) F π (x 1,..., x n ) F (x π 1 (1),..., x π 1 (n)); then the distibution

More information

Supplementary material for the paper Platonic Scattering Cancellation for Bending Waves on a Thin Plate. Abstract

Supplementary material for the paper Platonic Scattering Cancellation for Bending Waves on a Thin Plate. Abstract Supplementay mateial fo the pape Platonic Scatteing Cancellation fo Bending Waves on a Thin Plate M. Fahat, 1 P.-Y. Chen, 2 H. Bağcı, 1 S. Enoch, 3 S. Guenneau, 3 and A. Alù 2 1 Division of Compute, Electical,

More information

S7: Classical mechanics problem set 2

S7: Classical mechanics problem set 2 J. Magoian MT 9, boowing fom J. J. Binney s 6 couse S7: Classical mechanics poblem set. Show that if the Hamiltonian is indepdent of a genealized co-odinate q, then the conjugate momentum p is a constant

More information

Physics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!

Physics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!! Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time

More information

ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS

ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS ON THE TWO-BODY PROBLEM IN QUANTUM MECHANICS L. MICU Hoia Hulubei National Institute fo Physics and Nuclea Engineeing, P.O. Box MG-6, RO-0775 Buchaest-Maguele, Romania, E-mail: lmicu@theoy.nipne.o (Received

More information

Right-handed screw dislocation in an isotropic solid

Right-handed screw dislocation in an isotropic solid Dislocation Mechanics Elastic Popeties of Isolated Dislocations Ou study of dislocations to this point has focused on thei geomety and thei ole in accommodating plastic defomation though thei motion. We

More information

arxiv: v1 [physics.pop-ph] 3 Jun 2013

arxiv: v1 [physics.pop-ph] 3 Jun 2013 A note on the electostatic enegy of two point chages axiv:1306.0401v1 [physics.pop-ph] 3 Jun 013 A C Tot Instituto de Física Univesidade Fedeal do io de Janeio Caixa Postal 68.58; CEP 1941-97 io de Janeio,

More information

3.6 Applied Optimization

3.6 Applied Optimization .6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the

More information

Brief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis

Brief summary of functional analysis APPM 5440 Fall 2014 Applied Analysis Bief summay of functional analysis APPM 5440 Fall 014 Applied Analysis Stephen Becke, stephen.becke@coloado.edu Standad theoems. When necessay, I used Royden s and Keyzsig s books as a efeence. Vesion

More information

Physics 235 Chapter 5. Chapter 5 Gravitation

Physics 235 Chapter 5. Chapter 5 Gravitation Chapte 5 Gavitation In this Chapte we will eview the popeties of the gavitational foce. The gavitational foce has been discussed in geat detail in you intoductoy physics couses, and we will pimaily focus

More information

On the radial derivative of the delta distribution

On the radial derivative of the delta distribution On the adial deivative of the delta distibution Fed Backx, Fank Sommen & Jasson Vindas Depatment of Mathematical Analysis, Faculty of Engineeing and Achitectue, Ghent Univesity Depatment of Mathematics,

More information

PHYS 301 HOMEWORK #10 (Optional HW)

PHYS 301 HOMEWORK #10 (Optional HW) PHYS 301 HOMEWORK #10 (Optional HW) 1. Conside the Legende diffeential equation : 1 - x 2 y'' - 2xy' + m m + 1 y = 0 Make the substitution x = cos q and show the Legende equation tansfoms into d 2 y 2

More information

I. CONSTRUCTION OF THE GREEN S FUNCTION

I. CONSTRUCTION OF THE GREEN S FUNCTION I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,

More information

Exceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs

Exceptional regular singular points of second-order ODEs. 1. Solving second-order ODEs (May 14, 2011 Exceptional egula singula points of second-ode ODEs Paul Gaett gaett@math.umn.edu http://www.math.umn.edu/ gaett/ 1. Solving second-ode ODEs 2. Examples 3. Convegence Fobenius method fo solving

More information

In the previous section we considered problems where the

In the previous section we considered problems where the 5.4 Hydodynamically Fully Developed and Themally Developing Lamina Flow In the pevious section we consideed poblems whee the velocity and tempeatue pofile wee fully developed, so that the heat tansfe coefficient

More information

A Relativistic Electron in a Coulomb Potential

A Relativistic Electron in a Coulomb Potential A Relativistic Electon in a Coulomb Potential Alfed Whitehead Physics 518, Fall 009 The Poblem Solve the Diac Equation fo an electon in a Coulomb potential. Identify the conseved quantum numbes. Specify

More information

Appendix B The Relativistic Transformation of Forces

Appendix B The Relativistic Transformation of Forces Appendix B The Relativistic Tansfomation of oces B. The ou-foce We intoduced the idea of foces in Chapte 3 whee we saw that the change in the fou-momentum pe unit time is given by the expession d d w x

More information

Graphs of Sine and Cosine Functions

Graphs of Sine and Cosine Functions Gaphs of Sine and Cosine Functions In pevious sections, we defined the tigonometic o cicula functions in tems of the movement of a point aound the cicumfeence of a unit cicle, o the angle fomed by the

More information

2. Plane Elasticity Problems

2. Plane Elasticity Problems S0 Solid Mechanics Fall 009. Plane lasticity Poblems Main Refeence: Theoy of lasticity by S.P. Timoshenko and J.N. Goodie McGaw-Hill New Yok. Chaptes 3..1 The plane-stess poblem A thin sheet of an isotopic

More information

New problems in universal algebraic geometry illustrated by boolean equations

New problems in universal algebraic geometry illustrated by boolean equations New poblems in univesal algebaic geomety illustated by boolean equations axiv:1611.00152v2 [math.ra] 25 Nov 2016 Atem N. Shevlyakov Novembe 28, 2016 Abstact We discuss new poblems in univesal algebaic

More information

DonnishJournals

DonnishJournals DonnishJounals 041-1189 Donnish Jounal of Educational Reseach and Reviews. Vol 1(1) pp. 01-017 Novembe, 014. http:///dje Copyight 014 Donnish Jounals Oiginal Reseach Pape Vecto Analysis Using MAXIMA Savaş

More information

Section 8.2 Polar Coordinates

Section 8.2 Polar Coordinates Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,

More information

< 1. max x B(0,1) f. ν ds(y) Use Poisson s formula for the ball to prove. (r x ) x y n ds(y) (x B0 (0, r)). 1 nα(n)r n 1

< 1. max x B(0,1) f. ν ds(y) Use Poisson s formula for the ball to prove. (r x ) x y n ds(y) (x B0 (0, r)). 1 nα(n)r n 1 7 On the othe hand, u x solves { u n in U u on U, so which implies that x G(x, y) x n ng(x, y) < n (x B(, )). Theefoe u(x) fo all x B(, ). G ν ds(y) + max g G x ν ds(y) + C( max g + max f ) f(y)g(x, y)

More information

PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.

PROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr. POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and

More information

Vectors, Vector Calculus, and Coordinate Systems

Vectors, Vector Calculus, and Coordinate Systems ! Revised Apil 11, 2017 1:48 PM! 1 Vectos, Vecto Calculus, and Coodinate Systems David Randall Physical laws and coodinate systems Fo the pesent discussion, we define a coodinate system as a tool fo descibing

More information

Chemical Engineering 412

Chemical Engineering 412 Chemical Engineeing 41 Intoductoy Nuclea Engineeing Lectue 16 Nuclea eacto Theoy III Neuton Tanspot 1 One-goup eacto Equation Mono-enegetic neutons (Neuton Balance) DD φφ aa φφ + ss 1 vv vv is neuton speed

More information

An Exact Solution of Navier Stokes Equation

An Exact Solution of Navier Stokes Equation An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in

More information

3.1 Random variables

3.1 Random variables 3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated

More information

Transformation of the Navier-Stokes Equations in Curvilinear Coordinate Systems with Maple

Transformation of the Navier-Stokes Equations in Curvilinear Coordinate Systems with Maple Global Jounal of Pue and Applied Mathematics. ISSN 0973-1768 Volume 12, Numbe 4 2016, pp. 3315 3325 Reseach India Publications http://www.ipublication.com/gjpam.htm Tansfomation of the Navie-Stokes Equations

More information

3. Electromagnetic Waves II

3. Electromagnetic Waves II Lectue 3 - Electomagnetic Waves II 9 3. Electomagnetic Waves II Last time, we discussed the following. 1. The popagation of an EM wave though a macoscopic media: We discussed how the wave inteacts with

More information

Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere

Absorption Rate into a Small Sphere for a Diffusing Particle Confined in a Large Sphere Applied Mathematics, 06, 7, 709-70 Published Online Apil 06 in SciRes. http://www.scip.og/jounal/am http://dx.doi.og/0.46/am.06.77065 Absoption Rate into a Small Sphee fo a Diffusing Paticle Confined in

More information

OSCILLATIONS AND GRAVITATION

OSCILLATIONS AND GRAVITATION 1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,

More information

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx.

( ) [ ] [ ] [ ] δf φ = F φ+δφ F. xdx. 9. LAGRANGIAN OF THE ELECTROMAGNETIC FIELD In the pevious section the Lagangian and Hamiltonian of an ensemble of point paticles was developed. This appoach is based on a qt. This discete fomulation can

More information

Quantum theory of angular momentum

Quantum theory of angular momentum Quantum theoy of angula momentum Igo Mazets igo.mazets+e141@tuwien.ac.at (Atominstitut TU Wien, Stadionallee 2, 1020 Wien Time: Fiday, 13:00 14:30 Place: Feihaus, Sem.R. DA gün 06B (exception date 18 Nov.:

More information

TheWaveandHelmholtzEquations

TheWaveandHelmholtzEquations TheWaveandHelmholtzEquations Ramani Duaiswami The Univesity of Mayland, College Pak Febuay 3, 2006 Abstact CMSC828D notes (adapted fom mateial witten with Nail Gumeov). Wok in pogess 1 Acoustic Waves 1.1

More information

Conformal transformations + Schwarzschild

Conformal transformations + Schwarzschild Intoduction to Geneal Relativity Solutions of homewok assignments 5 Confomal tansfomations + Schwazschild 1. To pove the identity, let s conside the fom of the Chistoffel symbols in tems of the metic tenso

More information

The Schrödinger Equation in Three Dimensions

The Schrödinger Equation in Three Dimensions The Schödinge Equation in Thee Dimensions Paticle in a Rigid Thee-Dimensional Box (Catesian Coodinates) To illustate the solution of the time-independent Schödinge equation (TISE) in thee dimensions, we

More information

10.2 Parametric Calculus

10.2 Parametric Calculus 10. Paametic Calculus Let s now tun ou attention to figuing out how to do all that good calculus stuff with a paametically defined function. As a woking eample, let s conside the cuve taced out by a point

More information

Double-angle & power-reduction identities. Elementary Functions. Double-angle & power-reduction identities. Double-angle & power-reduction identities

Double-angle & power-reduction identities. Elementary Functions. Double-angle & power-reduction identities. Double-angle & power-reduction identities Double-angle & powe-eduction identities Pat 5, Tigonomety Lectue 5a, Double Angle and Powe Reduction Fomulas In the pevious pesentation we developed fomulas fo cos( β) and sin( β) These fomulas lead natually

More information

PES 3950/PHYS 6950: Homework Assignment 6

PES 3950/PHYS 6950: Homework Assignment 6 PES 3950/PHYS 6950: Homewok Assignment 6 Handed out: Monday Apil 7 Due in: Wednesday May 6, at the stat of class at 3:05 pm shap Show all woking and easoning to eceive full points. Question 1 [5 points]

More information

Math 2263 Solutions for Spring 2003 Final Exam

Math 2263 Solutions for Spring 2003 Final Exam Math 6 Solutions fo Sping Final Exam ) A staightfowad appoach to finding the tangent plane to a suface at a point ( x, y, z ) would be to expess the cuve as an explicit function z = f ( x, y ), calculate

More information

Euclidean Figures and Solids without Incircles or Inspheres

Euclidean Figures and Solids without Incircles or Inspheres Foum Geometicoum Volume 16 (2016) 291 298. FOUM GEOM ISSN 1534-1178 Euclidean Figues and Solids without Incicles o Insphees Dimitis M. Chistodoulou bstact. ll classical convex plana Euclidean figues that

More information

Lecture 10. Vertical coordinates General vertical coordinate

Lecture 10. Vertical coordinates General vertical coordinate Lectue 10 Vetical coodinates We have exclusively used height as the vetical coodinate but thee ae altenative vetical coodinates in use in ocean models, most notably the teainfollowing coodinate models

More information

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface

transformation Earth V-curve (meridian) λ Conical projection. u,v curves on the datum surface projected as U,V curves on the projection surface . CONICAL PROJECTIONS In elementay texts on map pojections, the pojection sufaces ae often descibed as developable sufaces, such as the cylinde (cylindical pojections) and the cone (conical pojections),

More information

The inviscid limit of incompressible fluid flow in an annulus

The inviscid limit of incompressible fluid flow in an annulus The inviscid limit of incompessible fluid flow in an annulus Saa Fietze, Robet Geity, and Tiago Picon ugust 8, 2006 bstact Incompessible, ciculaly symmetic fluid flow in a two-dimensional annulus = {x

More information

Physics 2A Chapter 10 - Moment of Inertia Fall 2018

Physics 2A Chapter 10 - Moment of Inertia Fall 2018 Physics Chapte 0 - oment of netia Fall 08 The moment of inetia of a otating object is a measue of its otational inetia in the same way that the mass of an object is a measue of its inetia fo linea motion.

More information

The Schwarzschild Solution

The Schwarzschild Solution The Schwazschild Solution Johannes Schmude 1 Depatment of Physics Swansea Univesity, Swansea, SA2 8PP, United Kingdom Decembe 6, 2007 1 pyjs@swansea.ac.uk Intoduction We use the following conventions:

More information

Vectors, Vector Calculus, and Coordinate Systems

Vectors, Vector Calculus, and Coordinate Systems Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado 80523. Scalas and vectos Any

More information

5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS

5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS 5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,

More information

Chapter 5 Linear Equations: Basic Theory and Practice

Chapter 5 Linear Equations: Basic Theory and Practice Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and

More information

Chapter 2: Introduction to Implicit Equations

Chapter 2: Introduction to Implicit Equations Habeman MTH 11 Section V: Paametic and Implicit Equations Chapte : Intoduction to Implicit Equations When we descibe cuves on the coodinate plane with algebaic equations, we can define the elationship

More information

Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012

Stanford University CS259Q: Quantum Computing Handout 8 Luca Trevisan October 18, 2012 Stanfod Univesity CS59Q: Quantum Computing Handout 8 Luca Tevisan Octobe 8, 0 Lectue 8 In which we use the quantum Fouie tansfom to solve the peiod-finding poblem. The Peiod Finding Poblem Let f : {0,...,

More information

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},

ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0}, ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability

More information

The Precession of Mercury s Perihelion

The Precession of Mercury s Perihelion The Pecession of Mecuy s Peihelion Owen Biesel Januay 25, 2008 Contents 1 Intoduction 2 2 The Classical olution 2 3 Classical Calculation of the Peiod 4 4 The Relativistic olution 5 5 Remaks 9 1 1 Intoduction

More information

I( x) t e. is the total mean free path in the medium, [cm] tis the total cross section in the medium, [cm ] A M

I( x) t e. is the total mean free path in the medium, [cm] tis the total cross section in the medium, [cm ] A M t I ( x) I e x x t Ie (1) whee: 1 t is the total mean fee path in the medium, [cm] N t t -1 tis the total coss section in the medium, [cm ] A M 3 is the density of the medium [gm/cm ] v 3 N= is the nuclea

More information

1 Spherical multipole moments

1 Spherical multipole moments Jackson notes 9 Spheical multipole moments Suppose we have a chage distibution ρ (x) wheeallofthechageiscontained within a spheical egion of adius R, as shown in the diagam. Then thee is no chage in the

More information