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.