DOCSLIB.ORG

Introduction to Differential Equations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Differential Equations Mechanics – A Brief Introduction to Alternative Formulations Concepts of primary interest: Newtonian Mechanics Lagrangian Mechanics Generalized Coordinates qi L Generalized Momenta pi qi Hamiltonian Mechanics Poisson Brackets Sample calculations: Box on an online Object rolling down an incline 1D particle motion Tools of the Trade Total Derivatives Newton’s ‘overdot’ notation Alternative Formulations of Mechanics – a very quick look The goal of this handout is to introduce two alternative formulations of mechanics that apply well to problems with conservative forces and normal constraint forces. Energy methods proved so fruitful in introductory mechanics that extensions must exist that essentially allow an energy viewpoint to replace the Newtonian force viewpoint. This energy viewpoint is a prerequisite for quantum mechanics so we will just bite the bullet, push the ‘I believe’ button and proceed. In some sense, the energy-view alternatives, the Lagrangian and the Hamiltonian formulations, can be derived or deduced from Newton’s Laws. No such logical development is to be presented here. In fact, the prescriptions in this handout have restricted validity. The whole truth is to found in your junior year mechanics course. Our point of departure is Newton’s Second Law: ma Fi V() r Fj. [MechAlt.1] i,, forceson j forces do the particle nowork The forces have been separated into those derived from potentials and those do to normal constraints, the classes of forces handled most easily in the Lagrangian formalism. The normal force on a box sliding down an incline and the tension force exerted by a string on a particle tethered to move around a circular path are examples of normal constraint forces. As they are directed at right Contact: [email protected] angles to the displacement, normal constraint forces due no work. Our example problems are to be a box sliding down a frictionless incline and an object rolling down an incline without slipping. The sliding box could be treated using the Cartesian coordinates x and y, but, as the box moves along a line, using two coordinates is over-kill. Instead a coordinate q is chosen – basically the distance from the top of the incline to the box as measured along a parallel to the inclined surface. The restriction to motion along a line means that the motion has a single degree of freedom. In the developments that follow care is to be taken to restrict the number of coordinates chosen to be equal to the degrees of freedom for the motion. A bead might be constrained to move along a curved wire requiring a generalized or unusual coordinate to describe its single-degree-of-freedom position. All coordinates are to be called generalized coordinates are to be represented by qi where the index i assumes j values, the number of degrees of freedom. The total time derivatives of the coordinates dq are called coordinate velocities: q i . i dt q y x Example I: A problem with a conservative force, the gravitational attraction due to the earth, and a normal constraint force that does no work and serves to prevent the box from moving into the space occupied by the incline. As the mass of the earth is huge compared to that of the box and the problem is restricted to altitudes near the earth’s surface, the potential energy can be represented as mgy. As illustrated, q is the distance that the box has moved parallel to the incline. It is the generalized coordinate. The kinetic energy for uniform translation can always be computed from its Cartesian form: 1122ˆˆ1 ˆˆˆˆ 22 T(xyz 22,mvm,x2iy )jzk()xiy jzk) m()xyz ( 9/3/2009 Physics Handout Series.Tank: Alternative Forms of Mechanics MechAlt- 2 Given the choice of coordinates, it follows that: (, the incline angle is a constant.) x = xo + q cos; y = yo - q sin xq cosy q sin Note that zand z are not needed here. Substituting, 1 2 Tqq(,) 2 mq and Vqq( , ) mgyo mgq sin 122212 Exercise: Show that 2mx() y z 2 mq for the example above. The Lagrangian process: The description that follows is specialized to simple cases. The term usually means that it is true for our applications, but may need modification for more general applications. I.) Choose a set of generalized coordinates (q1, …, qj) where j is the number of degrees of freedom. II.) Find expressions for the total kinetic energy T and potential energy V in terms of the generalized coordinates and there corresponding coordinate velocities, the qi . The natural variables for a Lagrangian description are the q’s, the qs ' and the time t. jj LLL dL dqik dq dt qqi k t ik11 That is: The natural variables show up nicely in the differential of the function. This introduction does not include cases with time dependent forces so all functions are to be written as forms depending on the q’s and qs ' . When partial derivatives are indicated, each qi is to be considered to be an independent variable even independent of its conjugate qi. III) Form the Lagrangian (function) as: LqqqqTqqqqVqqqq(11 , ;...;j ,jj ) (11 , ;...; ,j ) (11 , ;...;j ,j ) . [MechAlt.2] IV.) For each coordinate, compute the conjugate generalized momentum. L p [MechAlt.3]. qi qi Note that the process calls for a partial derivative with respect to qi . The variables qi and qi are to be considered as independent variables as these partial derivatives are computed. V.) Compute the equations of motion for each coordinate degree of freedom. 9/3/2009 Physics Handout Series.Tank: Alternative Forms of Mechanics MechAlt- 3 dL L d jj where qqi k [MechAlt.4] dt qii q dtik11 q i qk t In this last expression, a total time derivative is to be computed. Note the mixing of the d d conventional /dt and the Newtonian dot to represent total time derivatives. The /dt form is used for emphasis. Example I: Box on the incline: I.) There is one degree of freedom and the single coordinate q, the distance moved down the incline is chosen to be the coordinate. 1 2 II.) It has been shown that: Tqq(,) 2 mq and Vqq( , ) mgyo mgq sin . 1 2 III.) Lqq(,) = Tqq(,) Vqq (,) 2 mq mgyo mgqsin . L IV.) p mq q q dL d L V.) mq mq m gsin q g sin . The box accelerates down dt q dt qi along the incline with a linear acceleration g sin as expected. Example II: An symmetric object of radius R with moment if inertia I rolls down the same incline without slipping. As before there is one degree of freedom, and the net angle of rotation is to be chosen as the one generalized coordinate. In terms of this coordinate, the center of mass coordinates of the rolling object are: X = Xo + q cos; Y = Yo - q sin and its components of velocity are: Xq cos and Yq sin where q = R. The kinetic energy decomposes into the translational energy of the center of mass and the kinetic energy of rotation about the center of mass. 1122 TMXYI2[]2CM . The potential energy depends on the center of mass position only. V = M g Y. Example II: Rotationally symmetric object rolling w/o slipping on the incline the distance that the center of mass has moved parallel to the incline is q = R . I.) There is one degree of freedom; we choose the single coordinate , the net angle of rotation. 1122 2 II.) It follows that: TMRI(,) 22CM andVVMgR(,) 0 sin . 9/3/2009 Physics Handout Series.Tank: Alternative Forms of Mechanics MechAlt- 4 1122 2 III.) LMRIVMgR(,) 220 sin . L IV.) p MR2 I . Note that the momentum conjugate to the angle has the form of an angular momentum, but it is not just I , the angular momentum due to the rotation about the center of mass. The contribution MR2 is the angular momentum associated with the center of mass motion relative to the instantaneous point of contact (or any other point of the surface of the incline). The angular momentum of a rigid rotating body decomposes into the angular momentum associated with the center of mass motion plus that due to the body’s rotation about the center of mass. Adopting a single coordinate to describe the rotation and translation requires that the momentum conjugate to represents the combination of the coupled motions. 2 dL d 22L gMRsin V.) MR I MR I MgRsin . 2 dt dt R MR I 2 2 1 If the object is a uniform cylinder, I = ½ M R and 3 gR sin . Center of Mass Theorems: Exercise: Repeat the object rolling down the incline problem, but choose q, the displacement of the center of mass along a line parallel to the incline. Find expressions for the kinetic energy T(q, q ) and the generalized momentum pq. Discuss the result for pq given that it represents the total mass- motion and that the rotation and translation are coupled. The Hamiltonian Process: The Hamiltonian picture is built on the Lagrangian picture by a process L involving the conjugate momenta computed as: p . The Hamiltonian (function) is defined to qi qi j jj LLL be Hqpt(,ii ,) pq kk Lqqt (,,) ii . As dL dqik dq dt , qqi k t k 1 ik11 jjjj LLL dH(, qii p ,) t d pkk q dL(,,) q ii q t pkk dq q k dp k qqt dqi dq k dt . kki111i k1k L As ppq i , i qi jjj LL dH(, qii p ,) t pkdqkqk dp kqt dqi pm dqm dt ki11i m1 jj LL dH(, qii p ,) t q k dp k qt dqi dt k11i i As they appear as the differentials, qi, pi and t are the natural variables for the hamiltonian.
Load more

Recommended publications
  • Lecture 11 Line Integrals of Vector-Valued Functions (Cont'd)
    Lecture 11 Line integrals of vector-valued functions (cont’d) In the previous lecture, we considered the following physical situation: A force, F(x), which is not necessarily constant in space, is acting on a mass m, as the mass moves along a curve C from point P to point Q as shown in the diagram below. z Q P m F(x(t)) x(t) C y x The goal is to compute the total amount of work W done by the force. Clearly the constant force/straight line displacement formula, W = F · d , (1) where F is force and d is the displacement vector, does not apply here. But the fundamental idea, in the “Spirit of Calculus,” is to break up the motion into tiny pieces over which we can use Eq. (1 as an approximation over small pieces of the curve. We then “sum up,” i.e., integrate, over all contributions to obtain W . The total work is the line integral of the vector field F over the curve C: W = F · dx . (2) ZC Here, we summarize the main steps involved in the computation of this line integral: 1. Step 1: Parametrize the curve C We assume that the curve C can be parametrized, i.e., x(t)= g(t) = (x(t),y(t), z(t)), t ∈ [a, b], (3) so that g(a) is point P and g(b) is point Q. From this parametrization we can compute the velocity vector, ′ ′ ′ ′ v(t)= g (t) = (x (t),y (t), z (t)) . (4) 1 2. Step 2: Compute field vector F(g(t)) over curve C F(g(t)) = F(x(t),y(t), z(t)) (5) = (F1(x(t),y(t), z(t), F2(x(t),y(t), z(t), F3(x(t),y(t), z(t))) , t ∈ [a, b] .
    [Show full text]
  • Notes on Partial Differential Equations John K. Hunter
    Notes on Partial Differential Equations John K. Hunter Department of Mathematics, University of California at Davis Contents Chapter 1. Preliminaries 1 1.1. Euclidean space 1 1.2. Spaces of continuous functions 1 1.3. H¨olderspaces 2 1.4. Lp spaces 3 1.5. Compactness 6 1.6. Averages 7 1.7. Convolutions 7 1.8. Derivatives and multi-index notation 8 1.9. Mollifiers 10 1.10. Boundaries of open sets 12 1.11. Change of variables 16 1.12. Divergence theorem 16 Chapter 2. Laplace's equation 19 2.1. Mean value theorem 20 2.2. Derivative estimates and analyticity 23 2.3. Maximum principle 26 2.4. Harnack's inequality 31 2.5. Green's identities 32 2.6. Fundamental solution 33 2.7. The Newtonian potential 34 2.8. Singular integral operators 43 Chapter 3. Sobolev spaces 47 3.1. Weak derivatives 47 3.2. Examples 47 3.3. Distributions 50 3.4. Properties of weak derivatives 53 3.5. Sobolev spaces 56 3.6. Approximation of Sobolev functions 57 3.7. Sobolev embedding: p < n 57 3.8. Sobolev embedding: p > n 66 3.9. Boundary values of Sobolev functions 69 3.10. Compactness results 71 3.11. Sobolev functions on Ω ⊂ Rn 73 3.A. Lipschitz functions 75 3.B. Absolutely continuous functions 76 3.C. Functions of bounded variation 78 3.D. Borel measures on R 80 v vi CONTENTS 3.E. Radon measures on R 82 3.F. Lebesgue-Stieltjes measures 83 3.G. Integration 84 3.H. Summary 86 Chapter 4.
    [Show full text]
  • MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009)
    MULTIVARIABLE CALCULUS (MATH 212 ) COMMON TOPIC LIST (Approved by the Occidental College Math Department on August 28, 2009) Required Topics • Multivariable functions • 3-D space o Distance o Equations of Spheres o Graphing Planes Spheres Miscellaneous function graphs Sections and level curves (contour diagrams) Level surfaces • Planes o Equations of planes, in different forms o Equation of plane from three points o Linear approximation and tangent planes • Partial Derivatives o Compute using the definition of partial derivative o Compute using differentiation rules o Interpret o Approximate, given contour diagram, table of values, or sketch of sections o Signs from real-world description o Compute a gradient vector • Vectors o Arithmetic on vectors, pictorially and by components o Dot product compute using components v v v v v ⋅ w = v w cosθ v v v v u ⋅ v = 0⇔ u ⊥ v (for nonzero vectors) o Cross product Direction and length compute using components and determinant • Directional derivatives o estimate from contour diagram o compute using the lim definition t→0 v v o v compute using fu = ∇f ⋅ u ( u a unit vector) • Gradient v o v geometry (3 facts, and understand how they follow from fu = ∇f ⋅ u ) Gradient points in direction of fastest increase Length of gradient is the directional derivative in that direction Gradient is perpendicular to level set o given contour diagram, draw a gradient vector • Chain rules • Higher-order partials o compute o mixed partials are equal (under certain conditions) • Optimization o Locate and
    [Show full text]
  • Engineering Analysis 2 : Multivariate Functions
    Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Engineering Analysis 2 : Multivariate functions P. Rees, O. Kryvchenkova and P.D. Ledger, [email protected] College of Engineering, Swansea University, UK PDL (CoE) SS 2017 1/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Outline 1 Multivariate functions 2 Partial Differentiation 3 Higher Order Partial Derivatives 4 Total Differentiation 5 Line Integrals 6 Surface Integrals PDL (CoE) SS 2017 2/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction Recall from EG189: analysis of a function of single variable: Differentiation d f (x) d x Expresses the rate of change of a function f with respect to x. Integration Z f (x) d x Reverses the operation of differentiation and can used to work out areas under curves, and as we have just seen to solve ODEs. We’ve seen in vectors we often need to deal with physical fields, that depend on more than one variable. We may be interested in rates of change to each coordinate (e.g. x; y; z), time t, but sometimes also other physical fields as well. PDL (CoE) SS 2017 3/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 1: the area A of a rectangular plate of width x and breadth y can be calculated A = xy The variables x and y are independent of each other. In this case, the dependent variable A is a function of the two independent variables x and y as A = f (x; y) or A ≡ A(x; y) PDL (CoE) SS 2017 4/ 67 Multivariate functions Partial Differentiation Higher Order Partial Derivatives Total Differentiation Line Integrals Surface Integrals Introduction (Continue ...) Example 2: the volume of a rectangular plate is given by V = xyz where the thickness of the plate is in z-direction.
    [Show full text]
  • Relativistic Dynamics
    Chapter 4 Relativistic dynamics We have seen in the previous lectures that our relativity postulates suggest that the most efficient (lazy but smart) approach to relativistic physics is in terms of 4-vectors, and that velocities never exceed c in magnitude. In this chapter we will see how this 4-vector approach works for dynamics, i.e., for the interplay between motion and forces. A particle subject to forces will undergo non-inertial motion. According to Newton, there is a simple (3-vector) relation between force and acceleration, f~ = m~a; (4.0.1) where acceleration is the second time derivative of position, d~v d2~x ~a = = : (4.0.2) dt dt2 There is just one problem with these relations | they are wrong! Newtonian dynamics is a good approximation when velocities are very small compared to c, but outside of this regime the relation (4.0.1) is simply incorrect. In particular, these relations are inconsistent with our relativity postu- lates. To see this, it is sufficient to note that Newton's equations (4.0.1) and (4.0.2) predict that a particle subject to a constant force (and initially at rest) will acquire a velocity which can become arbitrarily large, Z t ~ d~v 0 f ~v(t) = 0 dt = t ! 1 as t ! 1 . (4.0.3) 0 dt m This flatly contradicts the prediction of special relativity (and causality) that no signal can propagate faster than c. Our task is to understand how to formulate the dynamics of non-inertial particles in a manner which is consistent with our relativity postulates (and then verify that it matches observation, including in the non-relativistic regime).
    [Show full text]
  • OCC D 5 Gen5d Eee 1305 1A E
    this cover and their final version of the extended essay to is are not is chose to write about applications of differential calculus because she found a great interest in it during her IB Math class. She wishes she had time to complete a deeper analysis of her topic; however, her busy schedule made it difficult so she is somewhat disappointed with the outcome of her essay. It was a pleasure meeting with when she was able to and her understanding of her topic was evident during our viva voce. I, too, wish she had more time to complete a more thorough investigation. Overall, however, I believe she did well and am satisfied with her essay. must not use Examiner 1 Examiner 2 Examiner 3 A research 2 2 D B introduction 2 2 c 4 4 D 4 4 E reasoned 4 4 D F and evaluation 4 4 G use of 4 4 D H conclusion 2 2 formal 4 4 abstract 2 2 holistic 4 4 Mathematics Extended Essay An Investigation of the Various Practical Uses of Differential Calculus in Geometry, Biology, Economics, and Physics Candidate Number: 2031 Words 1 Abstract Calculus is a field of math dedicated to analyzing and interpreting behavioral changes in terms of a dependent variable in respect to changes in an independent variable. The versatility of differential calculus and the derivative function is discussed and highlighted in regards to its applications to various other fields such as geometry, biology, economics, and physics. First, a background on derivatives is provided in regards to their origin and evolution, especially as apparent in the transformation of their notations so as to include various individuals and ways of denoting derivative properties.
    [Show full text]
  • Time-Derivative Models of Pavlovian Reinforcement Richard S
    Approximately as appeared in: Learning and Computational Neuroscience: Foundations of Adaptive Networks, M. Gabriel and J. Moore, Eds., pp. 497–537. MIT Press, 1990. Chapter 12 Time-Derivative Models of Pavlovian Reinforcement Richard S. Sutton Andrew G. Barto This chapter presents a model of classical conditioning called the temporal- difference (TD) model. The TD model was originally developed as a neuron- like unit for use in adaptive networks (Sutton and Barto 1987; Sutton 1984; Barto, Sutton and Anderson 1983). In this paper, however, we analyze it from the point of view of animal learning theory. Our intended audience is both animal learning researchers interested in computational theories of behavior and machine learning researchers interested in how their learning algorithms relate to, and may be constrained by, animal learning studies. For an exposition of the TD model from an engineering point of view, see Chapter 13 of this volume. We focus on what we see as the primary theoretical contribution to animal learning theory of the TD and related models: the hypothesis that reinforcement in classical conditioning is the time derivative of a compos- ite association combining innate (US) and acquired (CS) associations. We call models based on some variant of this hypothesis time-derivative mod- els, examples of which are the models by Klopf (1988), Sutton and Barto (1981a), Moore et al (1986), Hawkins and Kandel (1984), Gelperin, Hop- field and Tank (1985), Tesauro (1987), and Kosko (1986); we examine several of these models in relation to the TD model. We also briefly ex- plore relationships with animal learning theories of reinforcement, including Mowrer’s drive-induction theory (Mowrer 1960) and the Rescorla-Wagner model (Rescorla and Wagner 1972).
    [Show full text]
  • Partial Derivatives
    Partial Derivatives Partial Derivatives Just as derivatives can be used to explore the properties of functions of 1 vari- able, so also derivatives can be used to explore functions of 2 variables. In this section, we begin that exploration by introducing the concept of a partial derivative of a function of 2 variables. In particular, we de…ne the partial derivative of f (x; y) with respect to x to be f (x + h; y) f (x; y) fx (x; y) = lim h 0 h ! when the limit exists. That is, we compute the derivative of f (x; y) as if x is the variable and all other variables are held constant. To facilitate the computation of partial derivatives, we de…ne the operator @ = “The partial derivative with respect to x” @x Alternatively, other notations for the partial derivative of f with respect to x are @f @ f (x; y) = = f (x; y) = @ f (x; y) x @x @x x 2 2 EXAMPLE 1 Evaluate fx when f (x; y) = x y + y : Solution: To do so, we write @ @ @ f (x; y) = x2y + y2 = x2y + y2 x @x @x @x and then we evaluate the derivative as if y is a constant. In partic- ular, @ 2 @ 2 fx (x; y) = y x + y = y 2x + 0 @x @x That is, y factors to the front since it is considered constant with respect to x: Likewise, y2 is considered constant with respect to x; so that its derivative with respect to x is 0. Thus, fx (x; y) = 2xy: Likewise, the partial derivative of f (x; y) with respect to y is de…ned f (x; y + h) f (x; y) fy (x; y) = lim h 0 h ! 1 when the limit exists.
    [Show full text]
  • Partial Differential Equations a Partial Difierential Equation Is a Relationship Among Partial Derivatives of a Function
    In contrast, ordinary differential equations have only one independent variable. Often associate variables to coordinates (Cartesian, polar, etc.) of a physical system, e.g. u(x; y; z; t) or w(r; θ) The domain of the functions involved (usually denoted Ω) is an essential detail. Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable. Often associate variables to coordinates (Cartesian, polar, etc.) of a physical system, e.g. u(x; y; z; t) or w(r; θ) The domain of the functions involved (usually denoted Ω) is an essential detail. Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable. In contrast, ordinary differential equations have only one independent variable. The domain of the functions involved (usually denoted Ω) is an essential detail. Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable. In contrast, ordinary differential equations have only one independent variable. Often associate variables to coordinates (Cartesian, polar, etc.) of a physical system, e.g. u(x; y; z; t) or w(r; θ) Often other conditions placed on domain boundary, denoted @Ω. Partial differential equations A partial differential equation is a relationship among partial derivatives of a function (or functions) of more than one variable.
    [Show full text]
  • Notes for Math 136: Review of Calculus
    Notes for Math 136: Review of Calculus Gyu Eun Lee These notes were written for my personal use for teaching purposes for the course Math 136 running in the Spring of 2016 at UCLA. They are also intended for use as a general calculus reference for this course. In these notes we will briefly review the results of the calculus of several variables most frequently used in partial differential equations. The selection of topics is inspired by the list of topics given by Strauss at the end of section 1.1, but we will not cover everything. Certain topics are covered in the Appendix to the textbook, and we will not deal with them here. The results of single-variable calculus are assumed to be familiar. Practicality, not rigor, is the aim. This document will be updated throughout the quarter as we encounter material involving additional topics. We will focus on functions of two real variables, u = u(x;t). Most definitions and results listed will have generalizations to higher numbers of variables, and we hope the generalizations will be reasonably straightforward to the reader if they become necessary. Occasionally (notably for the chain rule in its many forms) it will be convenient to work with functions of an arbitrary number of variables. We will be rather loose about the issues of differentiability and integrability: unless otherwise stated, we will assume all derivatives and integrals exist, and if we require it we will assume derivatives are continuous up to the required order. (This is also generally the assumption in Strauss.) 1 Differential calculus of several variables 1.1 Partial derivatives Given a scalar function u = u(x;t) of two variables, we may hold one of the variables constant and regard it as a function of one variable only: vx(t) = u(x;t) = wt(x): Then the partial derivatives of u with respect of x and with respect to t are defined as the familiar derivatives from single variable calculus: ¶u dv ¶u dw = x ; = t : ¶t dt ¶x dx c 2016, Gyu Eun Lee.
    [Show full text]
  • Multivariable and Vector Calculus
    Multivariable and Vector Calculus Lecture Notes for MATH 0200 (Spring 2015) Frederick Tsz-Ho Fong Department of Mathematics Brown University Contents 1 Three-Dimensional Space ....................................5 1.1 Rectangular Coordinates in R3 5 1.2 Dot Product7 1.3 Cross Product9 1.4 Lines and Planes 11 1.5 Parametric Curves 13 2 Partial Differentiations ....................................... 19 2.1 Functions of Several Variables 19 2.2 Partial Derivatives 22 2.3 Chain Rule 26 2.4 Directional Derivatives 30 2.5 Tangent Planes 34 2.6 Local Extrema 36 2.7 Lagrange’s Multiplier 41 2.8 Optimizations 46 3 Multiple Integrations ........................................ 49 3.1 Double Integrals in Rectangular Coordinates 49 3.2 Fubini’s Theorem for General Regions 53 3.3 Double Integrals in Polar Coordinates 57 3.4 Triple Integrals in Rectangular Coordinates 62 3.5 Triple Integrals in Cylindrical Coordinates 67 3.6 Triple Integrals in Spherical Coordinates 70 4 Vector Calculus ............................................ 75 4.1 Vector Fields on R2 and R3 75 4.2 Line Integrals of Vector Fields 83 4.3 Conservative Vector Fields 88 4.4 Green’s Theorem 98 4.5 Parametric Surfaces 105 4.6 Stokes’ Theorem 120 4.7 Divergence Theorem 127 5 Topics in Physics and Engineering .......................... 133 5.1 Coulomb’s Law 133 5.2 Introduction to Maxwell’s Equations 137 5.3 Heat Diffusion 141 5.4 Dirac Delta Functions 144 1 — Three-Dimensional Space 1.1 Rectangular Coordinates in R3 Throughout the course, we will use an ordered triple (x, y, z) to represent a point in the three dimensional space.
    [Show full text]
  • Lie Time Derivative £V(F) of a Spatial Field F
    Lie time derivative $v(f) of a spatial field f • A way to obtain an objective rate of a spatial tensor field • Can be used to derive objective Constitutive Equations on rate form D −1 Definition: $v(f) = χ? Dt (χ? (f)) Procedure in 3 steps: 1. Pull-back of the spatial tensor field,f, to the Reference configuration to obtain the corresponding material tensor field, F. 2. Take the material time derivative on the corresponding material tensor field, F, to obtain F_ . _ 3. Push-forward of F to the Current configuration to obtain $v(f). D −1 Important!|Note that the material time derivative, i.e. Dt (χ? (f)) is executed in the Reference configuration (rotation neutralized). Recall that D D χ−1 (f) = (F) = F_ = D F Dt ?(2) Dt v d D F = F(X + v) v d and hence, $v(f) = χ? (DvF) Thus, the Lie time derivative of a spatial tensor field is the push-forward of the directional derivative of the corresponding material tensor field in the direction of v (velocity vector). More comments on the Lie time derivative $v() • Rate constitutive equations must be formulated based on objective rates of stresses and strains to ensure material frame-indifference. • Rates of material tensor fields are by definition objective, since they are associated with a frame in a fixed linear space. • A spatial tensor field is said to transform objectively under superposed rigid body motions if it transforms according to standard rules of tensor analysis, e.g. A+ = QAQT (preserves distances under rigid body rotations).
    [Show full text]