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Cartesian Coordinates B Cartesian coordinates The space in which we live is nearly flat everywhere. Its geometry is Euclidean, meaning that Euclid’s axioms and the theorems deduced from them are valid everywhere. After Einstein published his General Relativity in 1916 we know, however, that space is not perfectly flat. In the field of gravity from a massive body, space necessarily curves, but only very little unless the body is a black hole. The kind of physics that is the subject of this book may always be assumed to take place in flat Euclidean space. Rene´ Descartes (1596–1650). One of the consequences of the Euclidean geometry is Pythagoras’ theorem that in the French scientist and philosopher, well-known way relates the lengths of the sides of any right-angled triangle. The simplicity father of analytic geometry. De- of Pythagoras’ theorem favors the use of right-angled Cartesian coordinate systems in which veloped a theory of mechani- the distance between two points in space is the squareroot of the sum of the squares of their cal philosophy, later to be super- coordinate differences. In Cartesian coordinates, vector algebra also finds its simplest form. seded by Newton’s work. Con- This chapter serves in most respects to define the mathematical notation and present the fronted with doubts about reality, efficient modern methods of Cartesian vector and tensor algebra. In appendix C vector no- he saw thought as the only ar- tation is extended to include differentiation and integration, and in appendix D a couple of gument for existence: “I think, useful non-Cartesian coordinate systems are introduced and related to Cartesian coordinates. therefore I am”. One should, however, be aware that the emphasis on strictly algebraic treatment of geometric concepts differs from what is usually seen in texts at this level. B.1 Cartesian vectors z In a Cartesian coordinate system the distance between any two points, x .x; y; z/ and 6 x .x ; y ; z /, is given by the expression1 D 0 D 0 0 0 z r x p 2 2 2 d.x0; x/ .x x/ .y y/ .z z/ : (B.1) D 0 C 0 C 0 s - y y This distance function implies that space is Euclidean, and therefore has all the properties r r one learns about in elementary geometry. Although we could prove this claim right here, it ©x becomes nearly trivial after vector algebra has been established. x r 1In this book we have chosen to follow the physics tradition in which the Cartesian coordinates are labeled x, A Cartesian coordinate system y and z. Mathematicians would prefer instead to label the coordinates x1, x2, and x3, which is definitely a more with axes labeled x, y, and z, systematic notation. Boldface symbols, x .x; y; z/, are used to denote Cartesian coordinate triplets and vectors looks just like the one you got to (see below). For calculations with pencil onD paper several different notations can be used to distinguish a triplet from know (and love) in high school. other symbols, for example a bar (x), an arrow (x) or underlining (x). E Copyright c 1998–2010 Benny Lautrup 598 PHYSICS OF CONTINUOUS MATTER Definition of a vector The Cartesian distance function (1.14) depends only on the coordinate differences between two points, not on their individual values. Since all geometry is contained in the distance func- tion, coordinate differences will be of such importance in the analytic description of space in Cartesian coordinate systems that a special notation and a special type of algebra is necessary to deal efficiently with them. The mathematical concept of a vector is usually attributed to R. W. Hamilton (1845). A vector is a triplet of Cartesian coordinate differences between two points, say x and x0, u0 a .ax; ay ; az/ .x0 x; y0 y; z0 z/; (B.2) x0 a r D D a r u This particular vector is naturally visualized by a straight arrow pointing from x to x0. Since r a vector only depends on the coordinate differences of its endpoints, the same symbol a can x be used for the vector connecting any other pair of points, say u .u; v; w/ and u r D 0 D .u0; v0; w0/, as long as they have the same coordinate differences. A vector has like a real The vector a connects the point x arrow both direction and length, but no fixed origin. In accordance with the etymology of the with the point x . The same vec- 0 word2, the same vector a will carry you from x to x , and from u to u . tor also connects the points u and 0 0 u . Notice how the three components of the vector, ax, ay and az, are labeled by the coor- 0 dinate symbols x, y, and z. This is a general rule which will also be used for curvilinear coordinates (see appendix D). Position vectors Conceptually there is a great difference between the triplet of real numbers making up the x coordinates of a point, also called its position, and the triplet of coordinate differences making r up a vector. Whereas it a priori makes no geometric sense to add the coordinates of two points, x the sum of the components of two vectors is just another vector. The distinction is, however, not so clear in Cartesian coordinate systems, because the three real numbers in the triplet x can be formally viewed as the difference between the true coordinates of a point and the 0 t coordinates, 0 .0; 0; 0/, of the origin of the coordinate system. We shall for this reason The position vector x connects permit ourselvesD the ambiguity of also calling x the position vector, thereby allowing the the origin 0 with the point x. rules of vector algebra defined below also to be applied to coordinate triplets. This notational ambiguity does not cause any problems. It must be stressed that the identification of positions and vectors is absolutely not possi- ble in curvilinear coordinates, for example cylindrical or spherical, and even less so in non- Euclidean spaces. In those cases, true vectors can only be defined from infinitesimal coordi- nate differences in the local Cartesian coordinate systems that always exist in the neighbor- hood of any point. B.2 Vector algebra In keeping with the algebraization of geometry initiated four hundred years ago by Descartes, we shall focus on the algebraic rather than the geometric properties of vectors. As discussed in section 1.4 on page 11 the coordinate system is viewed as the physical reference frame for the determination of the coordinates of any point in space according to well-defined operational procedures. In this way we avoid all undefined geometrical primitives, such as the points, lines and circles of Euclidean geometry. The following definitions endow vectors with the usual properties of the familiar geomet- ric vectors. Geometric visualization is of course as useful as ever, and we shall whenever possible use simple sketches to illustrate what is meant. 2The word “vector” is Latin for “one who carries”, derived from the verb “vehere”, meaning to carry, and also known from “vehicle”. In epidemiology a “vector” denotes the carrier of disease. Copyright c 1998–2010 Benny Lautrup B. CARTESIAN COORDINATES 599 Linear operations Linear operations lie at the core of vector algebra, ka k a .kax; kay ; kaz/ (scaling by a factor); (B.3) a D a b .ax bx; ay by ; az bz/ (addition); (B.4) C D C C C r r a b .ax bx; ay by ; az bz/ (subtraction): (B.5) Geometric scaling of a vector by D a factor k. Mathematically, these rules show that the set of all vectors constitute a three-dimensional £AK vector space. b £ Ab a AK A set of vectors a1; a2;:::; aN are said to be linearly dependent if there exists a vanishing a b £ A A linear combination with non-zero coefficients, k1a1 k2a2 kN aN 0. More than C £ -A three vectors are always linearly dependent because spaceC is three-dimensional.C C D £ a a b A straight line with origin a and direction vector b 0 is described by the linear vector £ function x.s/ a b s with < s < . Note that the¤ origin a x.0/ is a position vector Geometric addition and subtrac- whereas the directionD C vector b1x.1/ x1.0/ is the difference betweenD the coordinates of two tion of vectors. points, and thus a true vector.D The straight line is in fact the shortest path between any two b points, and its length is equal to the distance between them, as it must in Euclidean geometry ¨¨* (see problem B.3). a r Dot product A straight line with origin a and The dot product or scalar product of two vectors is familiar from geometry, direction vector b. ¡ a b axbx ay by azbz (dot product): (B.6) D C C ¡ Two vectors are said to be orthogonal when their dot product vanishes, a b 0. The square a ¡ ......... 2 2 2 2 D ............. ¡ ...... of a vector equals its dot product with itself, a a a a a a . The length of a ..... x y z ..... .... : D D C C ¡Â ... vector is a pa2. ¡ j j D b r Cross product Geometrically, the dot product of two vectors is a b a b cos  D j j j j The cross product or vector product is also familiar from geometry, where a and b are the lengths of a andj jb, andj Âj is the angle be- a b .ay bz azby ; azbx axbz; axby ay bx/ (cross product): (B.7) D tween them.
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