Coordinate Systems, Vectors, TensorsAstr/Phys 469/569Fall 2008The fundamental requirement for the mathematical expression of each physicallaw is that it is written in a way that is independent of the particular coordinatesystem that is being used. For example, in Newtonian dynamics, the second law isexpressed using vector notation asmd2!rdt=!F, (1)where m and !r are the mass and position vector of a particle that is moving underthe influence of a force!F . If, moreover, the force is potential, i.e., it can be writtenas the gradient of a scalar potential function Φ, then Newton’s second law takes theformmd2!rd2t= −!∇Φ . (2)This is a symbolic representation of the vectorial for m of Newton’s second law, Manifestlycovariant formsbut it is not particular ly useful for computations. What we would like to have isa form of Newton’ s second law in terms of the components of the various vectors,but written in a way that is invariant under coordinate transformations. We wouldcall such an expression the manifestly covariant form of a physical law.In an orthonormal Cartesian coordinate sys tem with unit vectors ˆxi(i=1,2,3),we can write the position vector in component form as !r = xiˆxi, and the gradi-ent operator as!∇ =(∂/∂xi)ˆxi. In this case, Newton’s second law in Cartesiancoordinates takes the formmd2xidt2= −∂Φ∂xi. (3)This is, however, not a manifestly covar iant form of the physical law, as we caneasily prove. Consider a transformatio n from the Cartesian coordinates xito a setof general co ordinates ξithat may be ne ither orthogo nal nor normal . We understandthis transformation to imply the existence of one-to- one functions of the formxi= xi(ξ1,ξ2,ξ3) (4)as well as of their inverseξi= ξi(x1,x2,x3) (5)that are well defined in all but a small number of spatial points in space that wewill call the po les. Then, we can use the chain rule of differentiation to writed2xidt2=3!j=1∂xi∂ξjd2ξjdt2(6)and∂Φ∂xi=3!j=1∂Φ∂ξj∂ξj∂xi. (7)Inserting these two expressions in equation (3) we obtain3!j=1"∂xi∂ξj#md2ξjdt2= −3!j=1"∂ξj∂xi#∂Φ∂ξj. (8)1Figure 1: Coordinate lines and coordinate basis vectors for two different coordinate s ystems. Theleft panel depicts a Cartesian orthonormal coordinate system. In both cases, a third coordinate isassumed to extend from each point, perpendicular to the plane of the pap er.This last expression can be put in the form of equation (3) if and only if∂xi∂ξj= δij, (9)whereδij≡$1, if i = j0, if i $= j(10)is Kronecker’s delta. As a result, Newton’s second law in the coordinate form ofequation (3) is not manifestly covariant.What went wrong in this derivation? As we will see below, in writing thecomp o n e nts of the two vectors as !r = xiˆxiand!∇ =(∂/∂xi)ˆxi, we actually used twodifferent basis vectors, even though we denoted them both by ˆxi. We, therefor e,need to start the discussion fro m the beginning by defining pro perly coordinatesystems and basis vectors.1 Coordinate and Dual Basis VectorsWe consider a fla t space with N dimensions and define a coordinate system as anone-to-one map between an ordered set of N rea l numbers ξ1, ξ2, ..., ξNand eachindividual point in spa ce. The position of each point in space can, therefore, b ewritten in the form!r = !r(ξ1,ξ2, ..., ξN) . (11)We define coordinate lines as the curves along which only one of the coordinates Coordinate Linesand Surfaceschanges, whereas the other remain constant. This is illustrated in Figur e 1 for twosample coordinate systems. At the same time, we also define coordinate surfaces asthe surfaces on which only one of the coordinates remains constant. It follows fromtheir definition that, e.g., in a three dimensional spa ce, the intersection betweentwo c oordinate surfaces is a coordinate line.Using this definition of coordinate lines and surfaces, we have an infinite num-ber of options of defining basis vectors, three of which are pa rticularly useful indescribing physical laws.2Figure 2: Coordinates lines and dual basis vectors for the coordinate systems shown in Figure 1.We define the coordinate basis vectors at each point in space as the ordered setof vectors ˆei(i =1, 2, ..., N ), with the property that each of them is tangent to thecorresponding coordinate line. Formally, we define them by relations of the formˆei≡∂!r∂ξi(12)CoordinateBasis Vectorsand use a subscript notation to denote the coordinate along which this vector istangent. Figure 1 shows the coordina te basis vectors for two sample coordinatesystems.We can also define the dual basis vectors at each point in space in terms o f thecoordinate surfaces. Given tha n each coordinate surface can be represented by anequation of the form ξi(!r ) =constant, we define the dual basis vectors asˆei≡!∇ξi. (13)Dual BasisVectorsNote that we use a subscript notation for the dual basis vectors, in order to dis-tinguish them from the coordinate basis vectors. Figure 2 shows the dual basisvectors for two sample coordinate sy stems. Note that for an orthonormal Cartesiancoordinate systemˆe∗i=ˆe∗i, (14)where we have used the star to denote a Cartesian system.In general, the coordinate a nd dual basis vectors depend on position in space.However, they always satisfyˆeiˆej= δji. (15)In order to prove this property, we will use an orthonormal Cartesian coordinatesystem (x1,x2, ..., xN) and make a coordinate transformation to some unspecifiedcoordinate system (ξ1,ξ2, ..., ξN). The position vector of any point in space can bewritten in te rms of the Cartesia n basis vectors as!r =N!k=1xkˆe∗k. (16)3Figure 3: A example set of coordinate basis vectors (ˆe1, ˆe2) and of the corresponding dual basisvectors (ˆe1, ˆe2). Equation (15) requires that ˆe1⊥ ˆe2and ˆe2⊥ ˆe1but not necessarily that ˆe1↑↑ ˆe1or ˆe2↑↑ ˆe2.We use this together with the definition of the coordinate basis vectors to obtainˆei=∂!r∂ξi=N!k=1∂xk∂ξiˆe∗k(17)We also write explicitly the definition of the dual basis vectors asˆej=!∇ξj=N!k=1∂ξj∂xkˆe∗k. (18)(Don’t worry for the moment about the apparent asymmetry in the k−index in thelast expression; this is a Cartesian system for which ˆe∗k=ˆe∗k.) Taking the productof the two vectors we finally obtainˆeiˆej=N!k=1∂ξj∂xk∂xk∂ξi=∂ξj∂ξi= δji. (19)Relation (15) leads to a number of important results regarding the two sets ofbasis vectors. In gener al, it