Vector Calculus Lecture Notes, 2016–17 1 Fields and Vector Differential Operators

Total Page:16

File Type:pdf, Size:1020Kb

Vector Calculus Lecture Notes, 2016–17 1 Fields and Vector Differential Operators Vector Calculus Andrea Moiola University of Reading, 19th September 2016 These notes are meant to be a support for the vector calculus module (MA2VC/MA3VC) taking place at the University of Reading in the Autumn term 2016. The present document does not substitute the notes taken in class, where more examples and proofs are provided and where the content is discussed in greater detail. These notes are self-contained and cover the material needed for the exam. The suggested textbook is [1] by R.A. Adams and C. Essex, which you may already have from the first year; several copies are available in the University library. We will cover only a part of the content of Chapters 10–16, more precise references to single sections will be given in the text and in the table in Appendix G. This book also contains numerous exercises (useful to prepare the exam) and applications of the theory to physical and engineering problems. Note that there exists another book by the same authors containing the worked solutions of the exercises in the textbook (however, you should better try hard to solve the exercises and, if unsuccessful, discuss them with your classmates). Several other good books on vector calculus and vector analysis are available and you are encouraged to find the book that suits you best. In particular, Serge Lang’s book [5] is extremely clear. [7] is a collection of exercises, many of which with a worked solution (and it costs less than 10£). You can find other useful books on shelf 515.63 (and those nearby) in the University library. The lecture notes [2], the book [3] and the “Vector Calculus Primer” [6] are available online; on the web page [4] of O. Knill you can find plenty of exercises, lecture notes and graphs. Note that different books inevitably use different notation and conventions. In this notes we will take for granted what you learned in the previous classes, so the first year notes might be useful from time to time (in particular those for calculus, linear algebra and analysis). Some of the figures in the text have been made with Matlab. The scripts used for generating these plots are available on Blackboard and on the web page1 of the course; most of them can be run in Octave as well. You can use and modify them: playing with the different graphical representations of scalar and vector fields is a great way to familiarise with these concepts. On the same page you can find the file VCplotter.m, which you can use to visualise fields, curves and changes of variables in Matlab or Octave. Warning 1: The paragraphs and the proofs marked with a star “ ⋆” are addressed to students willing to deepen the subject and learn about some closely related topics. Some of these remarks try to relate the topic presented here to the content of other courses (e.g., analysis or physics); some others try to explain some important details that were glossed over in the text. These parts are not requested for the exam and can safely be skipped by students with modest ambitions. Warning 2: These notes are not entirely mathematically rigorous, for example we usually assume (sometimes tacitly) that fields and domains are “smooth enough” without specifying in detail what we mean with this assumption; multidimensional analysis is treated in a rigorous fashion in other modules, e.g. “analysis in several variables”. On the other hand, the (formal) proofs of vector identities and of some theorems are a fundamental part of the lectures, and at the exam you will be asked to prove some simple results. The purpose of this course is not only to learn how to compute integrals and divergences! Suggestion: The content of this module can be seen as the extension to multiple variables and vector quantities of the calculus you learned in the first year. Hence, many results you will encounter in this course correspond to simpler similar facts, holding for real functions, you already know well from previous classes. Comparing the vector results and formulas you will learn here with the scalar ones you already know will greatly simplify the study and the understanding of the content of this course. (This also means that you need to know and remember well what you learned in the first year.) If you find typos or errors of any kind in these notes, please let me know at [email protected] or in person during the office hours, before or after the lectures. 1http://www.personal.reading.ac.uk/~st904897/VC2016/VC2016.html A. Moiola, University of Reading 2 Vector calculus lecture notes, 2016–17 1 Fields and vector differential operators For simplicity, in these notes we only consider the 3-dimensional Euclidean space R3, and, from time to time, the plane R2. However, all the results not involving neither the vector product nor the curl operator can be generalised to Euclidean spaces RN of any dimensions N N. ∈ 1.1 Review of vectors in 3-dimensional Euclidean space We quickly recall some notions about vectors and vector operations known from previous modules; see Sections 10.2–10.3 of [1] and the first year calculus and linear algebra notes. We use the word “scalar” simply to denote any real number x R. We denote by R3 the three-dimensional∈ Euclidean vector space. A vector is an element of R3. Notation. In order to distinguish scalar from vector quantities, we denote vectors with boldface and a little arrow: ~u R3. Note that several books use underlined (u) symbols. We use the hat symbol (ˆ) to denote unit vectors∈ , i.e. vectors of length 1. You are probably used to write a vector ~u R3 in “matrix notation” as a “column vector” ∈ u1 ~u = u , 2 u3 where the scalars u1, u2 and u3 R are the components, or coordinates, of u~ . We will always use the equivalent notation ∈ ~u = u1ˆı + u2ˆ+ u3kˆ, where ˆı, ˆand kˆ are three fixed vectors that constitute the canonical basis of R3. When we draw vectors, we always assume that the canonical basis has a right-handed orientation, i.e. it is ordered according to the right-hand rule: closing the fingers of the right hand from the ˆı direction to the ˆdirection, the thumb 1 0 0 points towards the kˆ direction. You can think at ˆı, ˆ and kˆ as the vectors ˆı = 0 , ˆ= 1 and kˆ = 0 , 0 0 1 ˆ 1 0 0 u1 so that the two notations above are consistent: ~u = u1ˆı + u2ˆ+ u3k = u1 0 + u2 1 +u3 0 = u2 . 0 0 1 u3 u3 ~u kˆ ˆı ˆ u2 u1 Figure 1: The basis vectors ˆı, ˆ, kˆ and the components of the vector ~u. Definition 1.1. The magnitude, or length, or norm, of the vector ~u is the scalar defined as 2 u := ~u := u2 + u2 + u2. | | 1 2 3 q The direction of a (non-zero) vector u~ is the unit vector defined as u~ uˆ := . ~u | | Note that often the magnitude of a vector ~u is written as ~u (e.g. in your Linear Algebra lecture notes). We use the same notation ~u for the magnitude of a vectork k and x for the absolute value of a | | | | 2The notation “A := B” means “the object A is defined to be equal to the object B”. A and B may be scalars, vectors, matrices, sets. A. Moiola, University of Reading 3 Vector calculus lecture notes, 2016–17 scalar, which can be thought as the magnitude of a one-dimensional vector; the meaning of the symbol should always be clear from the argument. Every vector satisfies ~u = ~u uˆ. Therefore length and direction uniquely identify a vector. The vector of length 0 (i.e. ~0 := 0|ˆı +0| ˆ+0kˆ) does not have a specified direction. Note that, if we want to represent vectors with arrows, the point of application (“where” we draw the arrow) is not relevant: two arrows with the same direction and length represent the same vector; we can imagine that all the arrows have the application point in the origin ~0. Physics provides many examples of vectors: e.g. velocity, acceleration, displacement, force, momentum. Example 1.2 (The position vector). The position vector ~r = xˆı + yˆ+ zkˆ represents the position of a point in the three-dimensional Euclidean space relative to the origin. This is the only vector for which we do not use the notation ~r = r1ˆı + r2ˆ+ r3kˆ. We will use the position vector mainly for two purposes: (i) to describe subsets of R3 (for example ~r R3, ~r < 1 is the ball of radius 1 centred at the origin), and (ii) as argument of fields, which will be defined{ ∈ in Section| | 1.2} . ⋆ Remark 1.3 (Are vectors arrows, points, triples of numbers, or elements of a vector space?). There are several different definitions of vectors, this fact may lead to some confusion. Vectors defined as geometric entities fully described by magnitude and direction are sometimes called “Euclidean vectors” or “geometric vectors”. Note that, even though a vector is represented as an arrow, in order to be able to sum any two vectors the position of the arrow (the application point) has no importance. One can use vectors to describe points, or positions, in the three-dimensional space, after an origin has been fixed. These are sometimes called “bound vectors”. In this interpretation, the sum of two vectors does not make sense, while their difference is an Euclidean vector as described above.
Recommended publications
  • Vector Calculus and Multiple Integrals Rob Fender, HT 2018
    Vector Calculus and Multiple Integrals Rob Fender, HT 2018 COURSE SYNOPSIS, RECOMMENDED BOOKS Course syllabus (on which exams are based): Double integrals and their evaluation by repeated integration in Cartesian, plane polar and other specified coordinate systems. Jacobians. Line, surface and volume integrals, evaluation by change of variables (Cartesian, plane polar, spherical polar coordinates and cylindrical coordinates only unless the transformation to be used is specified). Integrals around closed curves and exact differentials. Scalar and vector fields. The operations of grad, div and curl and understanding and use of identities involving these. The statements of the theorems of Gauss and Stokes with simple applications. Conservative fields. Recommended Books: Mathematical Methods for Physics and Engineering (Riley, Hobson and Bence) This book is lazily referred to as “Riley” throughout these notes (sorry, Drs H and B) You will all have this book, and it covers all of the maths of this course. However it is rather terse at times and you will benefit from looking at one or both of these: Introduction to Electrodynamics (Griffiths) You will buy this next year if you haven’t already, and the chapter on vector calculus is very clear Div grad curl and all that (Schey) A nice discussion of the subject, although topics are ordered differently to most courses NB: the latest version of this book uses the opposite convention to polar coordinates to this course (and indeed most of physics), but older versions can often be found in libraries 1 Week One A review of vectors, rotation of coordinate systems, vector vs scalar fields, integrals in more than one variable, first steps in vector differentiation, the Frenet-Serret coordinate system Lecture 1 Vectors A vector has direction and magnitude and is written in these notes in bold e.g.
    [Show full text]
  • Lecture 9: Partial Derivatives
    Math S21a: Multivariable calculus Oliver Knill, Summer 2016 Lecture 9: Partial derivatives ∂ If f(x,y) is a function of two variables, then ∂x f(x,y) is defined as the derivative of the function g(x) = f(x,y) with respect to x, where y is considered a constant. It is called the partial derivative of f with respect to x. The partial derivative with respect to y is defined in the same way. ∂ We use the short hand notation fx(x,y) = ∂x f(x,y). For iterated derivatives, the notation is ∂ ∂ similar: for example fxy = ∂x ∂y f. The meaning of fx(x0,y0) is the slope of the graph sliced at (x0,y0) in the x direction. The second derivative fxx is a measure of concavity in that direction. The meaning of fxy is the rate of change of the slope if you change the slicing. The notation for partial derivatives ∂xf,∂yf was introduced by Carl Gustav Jacobi. Before, Josef Lagrange had used the term ”partial differences”. Partial derivatives fx and fy measure the rate of change of the function in the x or y directions. For functions of more variables, the partial derivatives are defined in a similar way. 4 2 2 4 3 2 2 2 1 For f(x,y)= x 6x y + y , we have fx(x,y)=4x 12xy ,fxx = 12x 12y ,fy(x,y)= − − − 12x2y +4y3,f = 12x2 +12y2 and see that f + f = 0. A function which satisfies this − yy − xx yy equation is also called harmonic. The equation fxx + fyy = 0 is an example of a partial differential equation: it is an equation for an unknown function f(x,y) which involves partial derivatives with respect to more than one variables.
    [Show full text]
  • A Brief Tour of Vector Calculus
    A BRIEF TOUR OF VECTOR CALCULUS A. HAVENS Contents 0 Prelude ii 1 Directional Derivatives, the Gradient and the Del Operator 1 1.1 Conceptual Review: Directional Derivatives and the Gradient........... 1 1.2 The Gradient as a Vector Field............................ 5 1.3 The Gradient Flow and Critical Points ....................... 10 1.4 The Del Operator and the Gradient in Other Coordinates*............ 17 1.5 Problems........................................ 21 2 Vector Fields in Low Dimensions 26 2 3 2.1 General Vector Fields in Domains of R and R . 26 2.2 Flows and Integral Curves .............................. 31 2.3 Conservative Vector Fields and Potentials...................... 32 2.4 Vector Fields from Frames*.............................. 37 2.5 Divergence, Curl, Jacobians, and the Laplacian................... 41 2.6 Parametrized Surfaces and Coordinate Vector Fields*............... 48 2.7 Tangent Vectors, Normal Vectors, and Orientations*................ 52 2.8 Problems........................................ 58 3 Line Integrals 66 3.1 Defining Scalar Line Integrals............................. 66 3.2 Line Integrals in Vector Fields ............................ 75 3.3 Work in a Force Field................................. 78 3.4 The Fundamental Theorem of Line Integrals .................... 79 3.5 Motion in Conservative Force Fields Conserves Energy .............. 81 3.6 Path Independence and Corollaries of the Fundamental Theorem......... 82 3.7 Green's Theorem.................................... 84 3.8 Problems........................................ 89 4 Surface Integrals, Flux, and Fundamental Theorems 93 4.1 Surface Integrals of Scalar Fields........................... 93 4.2 Flux........................................... 96 4.3 The Gradient, Divergence, and Curl Operators Via Limits* . 103 4.4 The Stokes-Kelvin Theorem..............................108 4.5 The Divergence Theorem ...............................112 4.6 Problems........................................114 List of Figures 117 i 11/14/19 Multivariate Calculus: Vector Calculus Havens 0.
    [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]
  • Policy Gradient
    Lecture 7: Policy Gradient Lecture 7: Policy Gradient David Silver Lecture 7: Policy Gradient Outline 1 Introduction 2 Finite Difference Policy Gradient 3 Monte-Carlo Policy Gradient 4 Actor-Critic Policy Gradient Lecture 7: Policy Gradient Introduction Policy-Based Reinforcement Learning In the last lecture we approximated the value or action-value function using parameters θ, V (s) V π(s) θ ≈ Q (s; a) Qπ(s; a) θ ≈ A policy was generated directly from the value function e.g. using -greedy In this lecture we will directly parametrise the policy πθ(s; a) = P [a s; θ] j We will focus again on model-free reinforcement learning Lecture 7: Policy Gradient Introduction Value-Based and Policy-Based RL Value Based Learnt Value Function Implicit policy Value Function Policy (e.g. -greedy) Policy Based Value-Based Actor Policy-Based No Value Function Critic Learnt Policy Actor-Critic Learnt Value Function Learnt Policy Lecture 7: Policy Gradient Introduction Advantages of Policy-Based RL Advantages: Better convergence properties Effective in high-dimensional or continuous action spaces Can learn stochastic policies Disadvantages: Typically converge to a local rather than global optimum Evaluating a policy is typically inefficient and high variance Lecture 7: Policy Gradient Introduction Rock-Paper-Scissors Example Example: Rock-Paper-Scissors Two-player game of rock-paper-scissors Scissors beats paper Rock beats scissors Paper beats rock Consider policies for iterated rock-paper-scissors A deterministic policy is easily exploited A uniform random policy
    [Show full text]
  • Introduction to Shape Optimization
    Introduction to Shape optimization Noureddine Igbida1 1Institut de recherche XLIM, UMR-CNRS 6172, Facult´edes Sciences et Techniques, Universit´ede Limoges 123, Avenue Albert Thomas 87060 Limoges, France. Email : [email protected] Preliminaries on PDE 1 Contents 1. PDE ........................................ 2 2. Some differentiation and differential operators . 3 2.1 Gradient . 3 2.2 Divergence . 4 2.3 Curl . 5 2 2.4 Remarks . 6 2.5 Laplacian . 7 2.6 Coordinate expressions of the Laplacian . 10 3. Boundary value problem . 12 4. Notion of solution . 16 1. PDE A mathematical model is a description of a system using mathematical language. The process of developing a mathematical model is termed mathematical modelling (also written model- ing). Mathematical models are used in many area, as in the natural sciences (such as physics, biology, earth science, meteorology), engineering disciplines (e.g. computer science, artificial in- telligence), in the social sciences (such as economics, psychology, sociology and political science); Introduction to Shape optimization N. Igbida physicists, engineers, statisticians, operations research analysts and economists. Among this mathematical language we have PDE. These are a type of differential equation, i.e., a relation involving an unknown function (or functions) of several independent variables and their partial derivatives with respect to those variables. Partial differential equations are used to formulate, and thus aid the solution of, problems involving functions of several variables; such as the propagation of sound or heat, electrostatics, electrodynamics, fluid flow, and elasticity. Seemingly distinct physical phenomena may have identical mathematical formulations, and thus be governed by the same underlying dynamic. In this section, we give some basic example of elliptic partial differential equation (PDE) of second order : standard Laplacian and Laplacian with variable coefficients.
    [Show full text]
  • Calculus Terminology
    AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential
    [Show full text]
  • Vector Calculus and Differential Forms with Applications To
    Vector Calculus and Differential Forms with Applications to Electromagnetism Sean Roberson May 7, 2015 PREFACE This paper is written as a final project for a course in vector analysis, taught at Texas A&M University - San Antonio in the spring of 2015 as an independent study course. Students in mathematics, physics, engineering, and the sciences usually go through a sequence of three calculus courses before go- ing on to differential equations, real analysis, and linear algebra. In the third course, traditionally reserved for multivariable calculus, stu- dents usually learn how to differentiate functions of several variable and integrate over general domains in space. Very rarely, as was my case, will professors have time to cover the important integral theo- rems using vector functions: Green’s Theorem, Stokes’ Theorem, etc. In some universities, such as UCSD and Cornell, honors students are able to take an accelerated calculus sequence using the text Vector Cal- culus, Linear Algebra, and Differential Forms by John Hamal Hubbard and Barbara Burke Hubbard. Here, students learn multivariable cal- culus using linear algebra and real analysis, and then they generalize familiar integral theorems using the language of differential forms. This paper was written over the course of one semester, where the majority of the book was covered. Some details, such as orientation of manifolds, topology, and the foundation of the integral were skipped to save length. The paper should still be readable by a student with at least three semesters of calculus, one course in linear algebra, and one course in real analysis - all at the undergraduate level.
    [Show full text]
  • Course Material
    SREENIVASA INSTITUTE of TECHNOLOGY and MANAGEMENT STUDIES (autonomous) (ENGINEERING MATHEMATICS-III) Course Material II- B.TECH / I - SEMESTER regulation: r18 Course Code: 18SAH211 Compiled by Department OF MATHEMATICS Unit-I: Numerical Integration Source:https://www.intmath.com/integration/integration-intro.php Numerical Integration: Simpson’s 1/3- Rule Note: While applying the Simpson’s 1/3 rule, the number of sub-intervals (n) should be taken as multiple of 2. Simpson’s 3/8- Rule Note: While applying the Simpson’s 3/8 rule, the number of sub-intervals (n) should be taken as multiple of 3. Numerical solution of ordinary differential equations Taylor’s Series Method Picard’s Method Euler’s Method Runge-Kutta Formula UNIT-II Multiple Integrals 1. Double Integration Evaluation of Double Integration Triple Integration UNIT-III Partial Differential Equations Partial differential equations are those equations which contain partial differential coefficients, independent variables and dependent variables. The independent variables are denoted by x and y and dependent variable by z. the partial differential coefficients are denoted as follows The order of the partial differential equation is the same as that of the order of the highest differential coefficient in it. UNIT-IV Vector Differentiation Elementary Vector Analysis Definition (Scalar and vector): Scalar is a quantity that has magnitude but not direction. For instance mass, volume, distance Vector is a directed quantity, one with both magnitude and direction. For instance acceleration, velocity, force Basic Vector System Magnitude of vectors: Let P = (x, y, z). Vector OP P is defined by OP p x i + y j + z k []x, y, z with magnitude (length) OP p x2 + y 2 + z2 Calculation of Vectors Vector Equation Two vectors are equal if and only if the corresponding components are equals Let a a1i + a2 j + a3 k and b b1i + b2 j + b3 k.
    [Show full text]
  • Part IA — Vector Calculus
    Part IA | Vector Calculus Based on lectures by B. Allanach Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. 3 Curves in R 3 Parameterised curves and arc length, tangents and normals to curves in R , the radius of curvature. [1] 2 3 Integration in R and R Line integrals. Surface and volume integrals: definitions, examples using Cartesian, cylindrical and spherical coordinates; change of variables. [4] Vector operators Directional derivatives. The gradient of a real-valued function: definition; interpretation as normal to level surfaces; examples including the use of cylindrical, spherical *and general orthogonal curvilinear* coordinates. Divergence, curl and r2 in Cartesian coordinates, examples; formulae for these oper- ators (statement only) in cylindrical, spherical *and general orthogonal curvilinear* coordinates. Solenoidal fields, irrotational fields and conservative fields; scalar potentials. Vector derivative identities. [5] Integration theorems Divergence theorem, Green's theorem, Stokes's theorem, Green's second theorem: statements; informal proofs; examples; application to fluid dynamics, and to electro- magnetism including statement of Maxwell's equations. [5] Laplace's equation 2 3 Laplace's equation in R and R : uniqueness theorem and maximum principle. Solution of Poisson's equation by Gauss's method (for spherical and cylindrical symmetry) and as an integral. [4] 3 Cartesian tensors in R Tensor transformation laws, addition, multiplication, contraction, with emphasis on tensors of second rank. Isotropic second and third rank tensors.
    [Show full text]
  • Using Surface Integrals for Checking the Archimedes' Law of Buoyancy
    Using surface integrals for checking the Archimedes’ law of buoyancy F M S Lima Institute of Physics, University of Brasilia, P.O. Box 04455, 70919-970, Brasilia-DF, Brazil E-mail: [email protected] Abstract. A mathematical derivation of the force exerted by an inhomogeneous (i.e., compressible) fluid on the surface of an arbitrarily-shaped body immersed in it is not found in literature, which may be attributed to our trust on Archimedes’ law of buoyancy. However, this law, also known as Archimedes’ principle (AP), does not yield the force observed when the body is in contact to the container walls, as is more evident in the case of a block immersed in a liquid and in contact to the bottom, in which a downward force that increases with depth is observed. In this work, by taking into account the surface integral of the pressure force exerted by a fluid over the surface of a body, the general validity of AP is checked. For a body fully surrounded by a fluid, homogeneous or not, a gradient version of the divergence theorem applies, yielding a volume integral that simplifies to an upward force which agrees to the force predicted by AP, as long as the fluid density is a continuous function of depth. For the bottom case, this approach yields a downward force that increases with depth, which contrasts to AP but is in agreement to experiments. It also yields a formula for this force which shows that it increases with the area of contact. PACS numbers: 01.30.mp, 01.55.+b, 47.85.Dh Submitted to: Eur.
    [Show full text]
  • Vector Calculus MA3VC 2016-17
    Vector calculus MA3VC 2016–17: Assignment 1 SOLUTIONS AND FEEDBACK (Exercise 1) Prove that if f is a (smooth) scalar field and G~ is an irrotational vector field, then (∇~ f × G~ )f is solenoidal. Hint: do not expand in coordinates and partial derivatives. Use instead the vector differential identities of §1.4 and the properties of the vector product from §1.1.2 (recall in particular Exercise 1.15). (Version 1) We compute the divergence of the vector field (∇~ f × G~ )f using the product rules (29) and (30) for the divergence: (29) ∇~ · (∇~ f × G~ )f = ∇~ f · (∇~ f × G~ )+ f∇~ · (∇~ f × G~ ) (30) = ∇~ f · (∇~ f × G~ )+ f(∇×~ ∇~ f) · G~ − f∇~ f · (∇×~ G~ ). The first term in this expression vanishes1 because of the identities u~ ·(~u×w~ )= w~ ·(~u×~u) (by Exercise 1.15) and ~u × u~ = ~0 (by the anticommutativity of the vector product (3)), which hold for all u~ , w~ ∈ R3. Applying these formulas with ~u = ∇~ f and w~ = G~ we get ∇~ f · (∇~ f × G~ )= G~ · (∇~ f × ∇~ f)= G~ · ~0 = 0. The second term vanishes because of the “curl-grad identity” (26): ∇×~ ∇~ f = ~0. The last term vanishes because G~ is irrotational: ∇×~ G~ = ~0. So we conclude that the field (∇~ f × G~ )f is solenoidal because its divergence vanishes: ∇~ · (∇~ f × G~ )f = G~ · (∇~ f × ∇~ f)+ f(∇×~ ∇~ f) · G~ − f∇~ f · (∇×~ G~ )=0. =~0 =~0 =~0 | {z } | {z } | {z } (Version 2) Alternatively, one could expand the field in coordinates. This solution is of course much more complicated and prone to errors than the previous one. We first write the product rule for products of three scalar fields, which is proved by applying twice the usual product rule for partial derivatives (8): ∂ ∂ (8) ∂(fg) ∂h (8) ∂f ∂g ∂h ∂f ∂g ∂h (fgh)= (fg)h = h + fg = g + f h + fg = gh + f h + fg .
    [Show full text]