mathematical methods in science and engineering pdf

... polar coordinates (r, 6,Q,) are defined x, as x y z = rsinxsinBcosQ, = rsinXsinBsinq5 = rsinxcos6 w = r cos x 220 COORDINATES AND TENSORS i) Write the line element ds2 = dx2 + dy2 + dz2 + ... be obtained by using dZa a; = - dxp (10. 277 ) In Minkowski spacetime the distance between two infinitesimally close points (events) can be written... Abelian 226 CONTINUOUS GROUPS AND REPRESENTATIONS ... however, the form of the tensor equation remains the same. One of the early uses for tensors in physics was in searching and expressing the coordinate independent properties of crystals. However,

Ngày tải lên: 13/08/2014, 09:21

... kinetic energy: T h e binomial formula is probably one of the most widely used formulas in science and engineering An important application of the binomial formula was given by Einstein... ... INFINITE PRODUCTS Infinite products are closely related to infinite series Most of the known functions can be written as infinite products, which are also useful in calculating some of... INFINITE ... f(0)l-... be seen by taking Mi = /ail Sz in the M-test 15.8.2 Continuity In a power series, since every term, that is, u,(x) = unlcn, is a continuous function and since in the interval -S 5 2

Ngày tải lên: 13/08/2014, 09:21

... transfer equation Consider the cooling of an infinitely long cylinder heated to an initial with the boundary condition and the initial condition then find C, so that the initial condition T(p,O) = f ... of the second kind Note that N,(z) and Jm(z) are linearly independent even for the integer values of m Hence it is common practice to take N,(z) and Jm(z) as the two linearly independent solutions ... second kind are defined as Chebyshev Polynomials of the Second Kind Un(x) = sin(nX), n = O,1,2 , (5.39) where x = cosx Chebyshev polynomials of the first and second kinds are linearly independent,

Ngày tải lên: 13/08/2014, 09:21

... uniformly in all subintervals free of points of discontinuity At the points of discontinuity this series r e p resents (as in the Fourier series) the arithmetic mean of the right- and the left-hand ... eigenvalues In other words, {urn(.)} spans the infinite dimensional vector space of square integrable functions The inner product (analog of dot product) in Hilbert space is defined as Trang 6and the ... constant independent of z but dependent on 1 and m We are interested in differential equations the coefficients of which may have singularities only at the end points of our interval Square inte

Ngày tải lên: 13/08/2014, 09:21

... cos+cos - cos8 sin +sin cos - cos 6sin +cos sin 8 sin - sin + (10. 56) (10.57) ++ + + cos+ sin cos8cos+sin sin cos 6cos +cos -sin 8cos q5 - sin + ++ The inverse of... -sin4 1 - sin $ sin4cos$ cos~cos$ ... 178 COORDINATES AND TENSORS which again proves that infinitesimal rotations commute 10.7 CARTESIAN TENSORS Certain physical properties... A(t)r(O) (10.72) Differentiating and using Equation ... COORDINATES In threedimensional Euclidean space a Cartesian coordinate system can be constructed by choosing three mutually orthogonal straight lines A point is defined by giving its coordinates,

Ngày tải lên: 13/08/2014, 09:21

... Defining a unit normal vector 1- (/3 + ia) sin w cos w - i y sin w The transformation (11.125) - v = u(w)v Trang 8240 CONTINUOUS GROUPS AND REPRESENTATIONS induces the following change in ... axis pointing in the fi direction by On In quantum mechanics we adhere to the right-handed screw convention, that is, when we curl the fingers of our right hand about the axis of rotation and in ... Coordinates we need to express R as rotations entirely in terms of the original coordinate in expressing the operator R in terms of the ( z ~ , yn, zn) coordinates Action of R on the coordinates

Ngày tải lên: 13/08/2014, 09:21

... evaluating some definite integrals 3 Infinite series, infinite products,... (x,y) in the z-plane to another point (u, in the w-plane, v) which implies that curves and domains in the z-plane ... the point is mapped to w = fie'" = -6 (12 .99 ) in the w-plane However the coordinates (12 .97 ) and (12 .98 ) represent the same point in the z-plane In other words, a single point in ... useful in finding solutions of Laplace equation in two dimensions 2 The method of analytic continuation is very useful in finding solutions of differential equations and evaluating some

Ngày tải lên: 13/08/2014, 09:21

... field lines inside... CLASSIFICATION OF SINGULAR POINTS 13.4 347 CLASSIFICATION OF SINGULAR POINTS Using Laurent the series we can classify singular points of a function Definition I Isolated singular ... < 0, a, = 0 and a-lrnl then # 0, is called a singular point of order m Definition I11 Essential singular point: If m is infinity, then singular point a is called an essential Definition IV Simple ... lines u = c1 and u = c2 (12.125) in the w-plane (Fig. 12.14). The problem is now reduced to finding the equipotentials and the electric field lines between two infinitely long MAPPINGS

Ngày tải lên: 13/08/2014, 09:21

... evaluating definite integrals and finding sums of infinite series In this chapter, we introduce some of the basic properties of fractional calculus along with some mathematical techniques and their ... a and then continue integration on the other side from an arbitrarily close point, (a + S), to infinity, that is, define the integral I as Trang 14t' fig 13.18 Contour G for the Cauchy principal ... u is inside our contour Using the residue theorem we obtain (13.185) In this case we again have two choices for the detour around the singular point on the real axis Again the Cauchy principal

Ngày tải lên: 13/08/2014, 09:21

... (14.97) and choose the integer n as q - n < 0 We now evaluate the differintegral inside the square brackets using formula (14.71) as Combining this with the results in Equations (14.96) and (14.98) ... interpret the left-handed definition] in general the boundary or the initial conditions determine which definition is t o be used It is also possible to give a left-handed version of the Griinwald ... an integer satisfying O < Q < m < Q + 1 (14.161) For the j values that make ( p - Q + j + 1) positive, the gamma function in the denominator is finite, and the corresponding terms in

Ngày tải lên: 13/08/2014, 09:21

... function in the infinite interval (-CO,CO) We now consider a nonperiodic function in the infinite interval (-o,co) Physically this corresponds to expressing an arbitrary signal in terms of sine and ... transform is defined as and it is usually encountered in potential energy calculations in cylindrical coordinates Another useful integral transform is the Mellin transform: (16.14) The Mellin transform ... transform is defined as ( 16.12) Trang 7DERIVATION OF THE FOURIER INTEGRAL 479 and it is very useful in finding solutions of systems of ordinary differential equations by converting them into a system

Ngày tải lên: 13/08/2014, 09:21

... variational problems encountered in physics and engineering are expressed in terms of an integral: (17.1) where y(x) is the desired function and f is a known function depending on y, its derivative ... in engineering problems we encounter functionals given as (17.66) where y(") stands for the nth order derivative, the independent variable x takes values in the closed interval [a,b], and ... equation of a straight line, where the integration constants a and b are to be determined from the coordinates of the end points The shortest paths between two points in a given geometry are

Ngày tải lên: 13/08/2014, 09:21

... Xj and take its complex conjugate as (18.97) Multiplying Equation (18.96) by Xjy;(z) and Equation (18.97) by Xiyi(z), and integrating over x in the interval [a, b] we obtain two ... chapter, we introduce the basic features of both the time-dependent and the timeindependent Green’s functions, which have found a wide range of applications in science and engineering. .. ... tool in solving differential equations They are also very useful in transforming differential equations into integral equations, which are preferred in certain cases like... 19.108) Using

Ngày tải lên: 13/08/2014, 09:21

... However, in electrostatics we usually deal with cases in which we are interested in finding the potential of a charge distribution in the presence Trang 12G(?,?‘) and Equation (19.234) with ẳ), and ... surface enclosing the volume V with the outward unit normal ii, we obtain Interchanging the primed and the unprimed variables and assuming that the Green’s function is symmetric in anticipation ... Using this in Equation (19.271) we obtain the propagator, G1(7'),7')", T ~ , T o ) , that takes us from TO to 7 2 in a single step in terms of the propagators, that take us from 70 to 7 1 and

Ngày tải lên: 13/08/2014, 09:21

... also leads to many interesting applications in field theory. In this Chapter we introduce the basic features of this technique, which has many interesting existing applications and tremendous potential ... FUNCTIONS AND PATH INTEGRALS 20.7.2 Schrodinger Equation in the Presence of Interactions In the presence of interactions the Schrdinger equation is given as a*(x,t) ~- at Making the transformation... ... t] denotes all continuous paths starting from (20,t o ) and ending a t ( 2 ,t) Before we discuss techniques of evaluating path integrals, we METHODS OF CALCULATING PATH INTEGRALS 647 should

Ngày tải lên: 13/08/2014, 09:21

... the infimum of E, denoted by infE, inf x∈E x, or inf{ x | x∈E} In cases where the set E is not bounded below, the infimum is defined as infE = −∞. Example 1.5.3 (Supremum and infimum)By inspection, ... 1.9 Determine the following numbers, and decide in each case whether“infimum” can be replaced by “minimum”: 1.10 Determine the following numbers, and decide in each case whether“lim inf” resp ... collect some basic inequalities concerning integrals. Three types of integrals will appear in this book:(i) The (Riemann) integral, concerning integration of bounded and piecewise continuous functions

Ngày tải lên: 20/10/2021, 21:35

... including type I and III interferons, lactoferrin, β-defensins, and nitric oxide, and also in the production of cytokines and chemokines, which recruit inflammatory cells and influence adaptive ... corroborated the findings of the original two US cohort studies, which both showed an increase in mortality with an increase in fine PM and sulfate The increase in mortality was mostly related to increased ... of the pollutant and from increasing airway susceptibility to other challenges, including respiratory virus infection NO2 causes a cascade of events, beginning with injury and inflammation of

Ngày tải lên: 09/08/2014, 16:21

... MATLAB IN SCIENCE AND ENGINEERING Edited by Tadeusz Michałowski Trang 2Applications of MATLAB in Science and Engineering Edited by Tadeusz Michałowski Published by InTech Janeza Trdine 9, 51000 ... areas include mathematics, physics, chemistry and chemical engineering, mechanical engineering, biological and medical sciences, communication and control systems, digital signal, image and video ... obtained from orders@intechweb.org Applications of MATLAB in Science and Engineering, Edited by Tadeusz Michałowski p cm ISBN 978-953-307-708-6 Trang 3free online editions of InTech Books and

Ngày tải lên: 09/08/2014, 16:21

... condition, quantifying the proteins e.g by measuring the staining intensity of a silver staining, resulting in single-channel data For multi-channel data, protein mixtures stemming from different ... changed Graphics windows building up fast in Windows were appearing comparably slow when run under Unix ever since, suggesting that it is now being developed in/for Windows and merely ported ... numerical solving functions Next we set the initial values and change someparameters: initial = zeros(2,1); initial = SetInitialValue(initial, switchAND, ’x’, 0.6); initial = SetInitialValue(initial,

Ngày tải lên: 09/08/2014, 16:21

... MATHEMATICAL METHODS IN SCIENCE AND ENGINEERING MATHEMATICS AND MIND 5 1.3 MATHEMATICS AND MIND Almost, everywhere mathematics is a very useful and powerful language in expressing ... part of the learning process. In a vast area like mathematical methods in science and engineering, there is always room for new approaches, new applications, and new topics. In fact, the number ... naturally comes from the inner efficiency of our brain. Research on subjects like brain stimulators, hard wiring of our brain, and mind reading machines are all aiming at a faster and much more efficient...

Ngày tải lên: 22/03/2014, 13:20