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Tensor Calculus

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Tensor Calculus APPENDIX C Tensor calculus Continuum mechanics, and in particular fluid dynamics, is a theory of (classical) fields. The latter may be scalars, vectors or more generally tensors—mainly of degree at most 2—, whose dynamical behavior is governed by partial differential equations, which obviously involve various derivatives of tensorial quantities. When describing vector or tensor fields by their respective components on appropriate (local) bases, the basis vectors or tensors may actually vary from point to point. Accordingly, care must be taken when differentiating with respect to the space coordinates: instead of the usual partial derivatives, the quantities that behave in the expected manner are rather covariant derivatives (Sec. C.1), which are the main topic of this Appendix. To provide the reader with some elementary background on the proper mathematical framework to discuss vector and tensor fields and their differentiation, some basic ideas of differential geometry are gathered in Sec. C.2, C.1 Covariant differentiation of tensor fields The purpose of this Section is to introduce the covariant derivative, which is the appropriate math- ematical quantity measuring the spatial rate of change of a field on a space, irrespective of the choice of coordinates on that space. The notion is first introduced for vector fields (Sec. C.1.1) and illustrated on the example of vector fields on a plane (Sec. C.1.2). The covariant derivative of tensors of arbitrary type, in particular of one-forms, is then given in Sec. C.1.3. Eventually, the usual differential operators of vector analysis are discussed in Sec. C.1.4. Throughout this Section, we mostly list recipes, without providing proofs or the given results, nor specifying for example in which space the vector or tensor fields “live”. These more formal issues will be shortly introduced in Sec. C.2. C.1.1 Covariant differentiation of vector fields Consider a set M of points generically denoted by P , possessing the necessary properties so that the following features are realized: (a) In a neighborhood of every point P 2 M, one can find a system of local coordinates fxi(P )g. (b) It is possible to define functions on M with sufficient smoothness properties, as e.g. differen- tiable functions. (c) At each point P 2 M, one can attach vectors—and more generally tensors. Let f~ei(P )g denote a basis of the vectors at P . From the physicist’s point of view, the above requirements mean that we want to be able to define scalar, vector or tensor fields at each point [property (c)], that depend smoothly on the position [property (b)], where the latter can be labeled by local coordinates [property (a)]. Mathematically, it will be seen in Sec. C.2 that the proper framework is to look at a differentiable manifold and its tangent bundle. 62 Tensor calculus Before we go any further, let us emphasize that the results we state hereafter are independent of the dimension n of the vectors, from 1 to which the indices i, j, k, l. run. In addition, we use Einstein’s summation convention throughout. Assuming the above requirements are fulfilled, which we now do without further comment, we in addition assume that the local basis f~ei(P )g at every point is that which is “naturally induced” i (27) by the coordinates fx (P )g, and that for every possible i the mapping P 7! ~ei(P ) defines a (28) continuous, and even differentiable vector field on M. The derivative of ~ei at P with respect to any of the (local) coordinate direction xk is then itself a vector “at P ”, which may thus be expanded l on the basis f~el(P )g: denoting by Γik(P ) its coordinates @~e (P ) i = Γl (P )~e (P ): (C.1) @xk ik l l l The numbers Γik, which are also alternatively denoted as i k , are called Christoffel symbols (of the second type) or connection coefficients. Remark: The reader should remember that the local coordinates also depend on P , i.e. a better notation for the left hand side of Eq. (C.1)—and for every similar derivative in the following—could k be @~ei(P )=@x (P ). Let now ~c(P ) be a differentiable vector field defined on M, whose local coordinates at each point will be denoted by ci(P ) [cf. Eq. (B.1)]: i ~c(P ) = c (P )~ei(P ): (C.2) The spatial rate of change in ~c between a point P and a neighboring point P 0 situated in the xk-direction with respect to P is given by @~c(P ) dci(P ) = ~e (P ); (C.3a) @xk dxk k where the component along ~ei(P ) is the so-called covariant derivative dci(P ) @ci(P ) = + Γi (P )cl(P ): (C.3b) dxk @xk lk i k i Remark: The covariant derivative dc =dx is often denoted by c ;k, with a semicolon in front of the index (or indices) related to the direction(s) along which one differentiates. In contrast, the partial i k i derivative @c =@x is then written as c ;k, with a comma. That is, Eq. (C.3b) is recast as i i i l c ;k(P ) = c ;k(P ) + Γlk(P )c (P ): (C.3c) The proof of Eqs. (C.3) is rather straightforward. Differentiating relation (C.2) with the product rule first gives @~c(P ) @ci(P ) @~e (P ) @ci(P ) = ~e (P ) + ci i = ~e (P ) + ci(P )Γl (P )~e (P ) @xk @xk i @xk @xk i ik l where we have used the derivative (C.1). In the rightmost term, the dummy indices i and l may i l l i be relabeled as l and i, respectively, yielding c Γik ~el = c Γlk ~ei, i.e. @~c(P ) @ci(P ) dci(P ) = ~e (P ) + cl(P )Γi (P )~e (P ) = ~e (P ): @xk @xk i lk i dxk i i k 1 One can show that the covariant derivatives dc (P )=dx are the components of a 1 -tensor field, the (1-form-)gradient of the vector field ~c, which may be denoted by r~c. On the other hand, neither the partial derivative on the right hand side of Eq. (C.3b) nor the Christoffele symbols are tensors. (27)This requirement will be made more precise in Sec. C.2. (28)This implicitly relies on the fact that the vectors attached to every point P 2 M all have the same dimension. C.1 Covariant differentiation of tensor fields 63 The Christoffel symbols can be expressed in terms of the (local) metric tensor g(P ), whose components are in agreement with relation (B.6) given by(29) gij(P ) = ~ei(P ) ·~ej(P ); (C.4) and of its partial derivatives. Thus 1 @g (P ) @g (P ) @g (P ) Γi (P ) = gip(P ) pl + pk − kl (C.5) lk 2 @xk @xl @xp with gip(P ) the components of the inverse metric tensor g−1(P ). i This relation shows that Γlk(P ) is symmetric under the exchange of the lower indices l and l, i i i.e. Γkl(P ) = Γlk(P ). C.1.2 Examples: differentiation in Cartesian and in polar coordinates To illustrate the results introduced in the previous Section, we calculate the derivatives of vector fields defined at each point of the real plane R2, which plays the role of the set M. :::::::C.1.2 a :::::::::::::::::::::::Cartesian coordinates As a first, trivial example, let us associate to each point P 2 R2 local coordinates x1(P ) = x, 2 x (P ) = y that coincide with the usual global Cartesian coordinates on the plane. Let ~e1(P ) = ~ex, ~e2(P ) = ~ey denote the corresponding local basis vectors—which actually happen to be the same at every point P , i.e. which represent constant vector fields. k Either by writing down the vanishing derivatives @~ei(P )=@x , i.e. using Eq. (C.1), or by invoking relation (C.5)—where the metric tensor is trivial: g11 = g22 = 1, g12 = g21 = 0 everywhere—, one finds that every Christoffel symbol vanishes. This means [Eq. (C.3b)] that covariant and partial derivative coincide. which is why one need not worry about “covariant differentiation” when working in Cartesian coordinates. :::::::C.1.2 b :::::::::::::::::::Polar coordinates It is thus more instructive to associate to each point P 2 R2, with the exception of the origin, polar coordinates x10 = r ≡ xr, x20 = θ ≡ xθ. The corresponding local basis vectors are ( ~e (r; θ) = cos θ~e + sin θ~e r x y (C.6) ~eθ(r; θ) = −r sin θ~ex + r cos θ~ey: To recover the usual inner product on R2, the metric tensor g(P ) should have components 2 grr(r; θ) = 1; gθθ(r; θ) = r ; grθ(r; θ) = gθr(r; θ) = 0: (C.7a) That is, the components of g−1(P ) are 1 grr(r; θ) = 1; gθθ(r; θ) = ; grθ(r; θ) = gθr(r; θ) = 0: (C.7b) r2 Computing the derivatives @~e (r; θ) @~e (r; θ) 1 @~e (r; θ) 1 @~e (r; θ) r = ~0; r = ~e (r; θ); θ = ~e (r; θ); θ = −r~e (r; θ) @xr @xθ r θ @xr r θ @xθ r and using Eq. (C.1), or relying on relation (C.5), one finds the Christoffel symbols 1 Γr = Γθ = 0; Γθ = Γθ = ; Γr = −r; Γr = Γr = 0; Γθ = 0 (C.8) rr rr rθ θr r θθ rθ θr θθ where for the sake of brevity the (r; θ)-dependence of the Christoffel symbols was dropped. (29)Remember that the metric tensor g actually defines the inner product.
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