1INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS********************************************************************************I. Basic Principles ...................................................................... 1II. Three Dimensional Spaces ............................................................. 4III. Physical Vectors ...................................................................... 8IV. Examples: Cylindrical and Spherical Coordinates .................................. 9V. Application: Special Relativity, including Electromagnetism ......................... 10VI. Covariant Differentiation ............................................................. 17VII. Geodesics and Lagrangians ............................................................. 21********************************************************************************I. Basic PrinciplesWe shall treat only the basic ideas, which will suffice for much of physics. The objective is toanalyze problems in any coordinate system, the variables of which are expressed asqj(xi) or q'j(qi) where xi : Cartesian coordinates, i = 1,2,3, ....Nfor any dimension N. Often N=3, but in special relativity, N=4, and the results apply in anydimension. Any well-defined set of qj will do. Some explicit requirements will be specified later.An invariant is the same in any system of coordinates. A vector, however, has componentswhich depend upon the system chosen. To determine how the components change (transform) withsystem, we choose a prototypical vector, a small displacement dxi. (Of course, a vector is ageometrical object which is, in some sense, independent of coordinate system, but since it can beprescribed or quantified only as components in each particular coordinate system, the approach here isthe most straightforward.) By the chain rule, dqi = ( ∂qi / ∂xj ) dxj , where we use the famoussummation convention of tensor calculus: each repeated index in an expression, here j, is to besummed from 1 to N. The relation above gives a prescription for transforming the (contravariant)vector dxi to another system. This establishes the rule for transforming any contravariant vector fromone system to another.Ai (q) = ( ∂qi∂xj ) Aj (x)Ai(q') = ( ∂q'i∂qj ) Aj(q) = ( ∂q'i∂qj ) ( ∂qj∂xk ) Ak(x) ≡ ( ∂q'i ∂xk ) Ak(x)Λij (q,x) ≡ ∂qi∂xj Contravariant vector transformThe (contravariant) vector is a mathematical object whose representation in terms of componentstransforms according to this rule. The conventional notation represents only the object, Ak, withoutindicating the coordinate system. To clarify this discussion of transformations, the coordinate systemwill be indicated by Ak(x), but this should not be misunderstood as implying that the components inthe "x" system are actually expressed as functions of the xi. (The choice of variables to be used toexpress the results is totally independent of the choice coordinate system in which to express thecomponents Ak. The Ak(q) might still be expressed in terms of the xi, or Ak(x) might be moreconveniently expressed in terms of some qi.)Distance is the prototypical invariant. In Cartesian coordinates, ds2 = δij dxi dxj , whereδij is the Kroneker delta: unity if i=j, 0 otherwise. Using the chain rule,INTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS2dxi = ( ∂xi∂qj ) dqj ds2 = δij ( ∂xi∂qk ) ( ∂xj∂ql ) dqk dql = gkl (q) dqk dqlgkl (q) ≡ ( ∂xi∂qk ) ( ∂xj∂ql ) δij (definition of the metric tensor)One is thus led to a new object, the metric tensor, a (covariant) tensor, and by analogy, the covarianttransform coefficients:Λji(q,x) ≡ ( ∂xj∂qi ) Covariant vector transform{More generally, one can introduce an arbitrary measure (a generalized notion of 'distance') ina chosen reference coordinate system by ds2 = gkl (0) dqk dql , and that measure will be invariant ifgkl transforms as a covariant tensor. A space having a measure is a metric space.}Unfortunately, the preservation of an invariant has required two different transformation rules,and thus two types of vectors, covariant and contravariant, which transform by definition according tothe rules above. (The root of the problem is that our naive notion of 'vector' is simple and well-defined only in simple coordinate systems. The appropriate generalizations will all be developed indue course here.) Further, we define tensors as objects with arbitrary covariant and contravariantindices which transform in the manner of vectors with each index. For example,Tijk(q) ≡ Λim (q,x) Λjn(q,x) Λlk(q,x) Tmnl (x)The metric tensor is a special tensor. First, note that distance is indeed invariant:ds2(q') = gkl (q') dq'k dq'l= ( ∂qi ∂q'k ) ( ∂qj ∂q'l ) gij (q) ( ∂q'k∂qs ) dqs ( ∂q'l∂qt ) dqt= gij (q) ( ∂qi ∂q'k )( ∂q'k∂qs ) ( ∂qj ∂q'l )( ∂q'l∂qt ) dqs dqt ⇓ ⇓∂qi∂qs = δis δjt= gij (q) dqi dqj ≡ ds2(q)There is also a consistent and unique relation between the covariant and contravariantcomponents of a vector. (There is indeed a single 'object' with two representations in each coordinatesystem.)dqj ≡ gji dqiINTRODUCTION TO THE ESSENTIALS OF TENSOR CALCULUS3dq'j ≡ gji (q') dq'i = gkl (q) ( ∂qk ∂q'j ) ( ∂ql ∂q'i ) (∂q'i∂qp ) dqp ⇓ δlp = ( ∂qk ∂q'j ) gkl (q) dql = ( ∂qk ∂q'j ) dqkThus it transforms properly as a covariant vector.These results are quite general; summing on an index (contraction) produces a new objectwhich is a tensor of lower rank (fewer indices).Tijk Gkl = RijlThe use of the metric tensor to convert contravariant to covariant indices can be generalized to'raise' and 'lower' indices in all cases. Since gij = δij in Cartesian coordinates, dxi =dxi ; there isno difference between co- and contra-variant. Hence gij = δij , too, and one can thus define gij inother coordinates. {More generally, if an arbitrary measure and metric have been defined, thecomponents of the contravariant metric tensor may be found by inverting the [N(N+1)/2] equations(symmetric g) of gij (0) gik(0) gnj(0) = gkn(0). The matrices are inverses.}Ai(q) ≡ gij (q) Aj(q)gij = gik gkj = ( ∂qi ∂xm ) ( ∂qk∂xn ) δmn ( ∂xr∂qk ) ( ∂xs∂qj ) δrs | | ⇓ δrn= ( ∂qi∂xs ) ( ∂xs∂qj ) = δij = δijThus gij is a unique tensor which is the same