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1.18 Curvilinear Coordinates: Tensor Calculus

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1.18 Curvilinear Coordinates: Tensor Calculus Section 1.18 1.18 Curvilinear Coordinates: Tensor Calculus 1.18.1 Differentiation of the Base Vectors Differentiation in curvilinear coordinates is more involved than that in Cartesian coordinates because the base vectors are no longer constant and their derivatives need to be taken into account, for example the partial derivative of a vector with respect to the Cartesian coordinates is i v vi 1 v v i g i ei but j j g i v j x j x j The Christoffel Symbols of the Second Kind First, from Eqn. 1.16.19 – and using the inverse relation, g x m 2 x m k i e g (1.18.1) j j i m i j m k x this can be written as g i k g Partial Derivatives of Covariant Base Vectors (1.18.2) j ij k where 2 x m k k , (1.18.3) ij i j x m k and ij is called the Christoffel symbol of the second kind; it can be seen to be j equivalent to the kth contravariant component of the vector g i / . One then has {▲Problem 1} g g k k i g k j g k Christoffel Symbols of the 2nd kind (1.18.4) ij ji j i and the symmetry in the indices i and j is evident2. Looking now at the derivatives of the i k k contravariant base vectors g : differentiating the relation g i g i leads to g k g g i g k mg g k k j i j ij m ij 1 of course, one could express the g i in terms of the e i , and use only the first of these expressions 2 note that, in non-Euclidean space, this symmetry in the indices is not necessarily valid Solid Mechanics Part III 164 Kelly Section 1.18 and so g i i g k Partial Derivatives of Contravariant Base Vectors (1.18.5) j jk Transformation formulae for the Christoffel Symbols The Christoffel symbols are not the components of a (third order) tensor. This follows from the fact that these components do not transform according to the tensor transformation rules given in §1.17. In fact, p q k 2 s k k r ij i j r pq i j s The Christoffel Symbols of the First Kind The Christoffel symbols of the second kind relate derivatives of covariant (contravariant) base vectors to the covariant (contravariant) base vectors. A second set of symbols can be introduced relating the base vectors to the derivatives of the reciprocal base vectors, called the Christoffel symbols of the first kind: g g i g j g Christoffel Symbols of the 1st kind (1.18.6) ijk jik j k i k so that the partial derivatives of the covariant base vectors can be written in the alternative form g i g k , (1.18.7) j ijk and it also follows from Eqn. 1.18.2 that m k mk ijk ij g mk , ij ijm g (1.18.8) showing that the index k here can be raised or lowered using the metric coefficients as for a third order tensor (but the first two indexes, i and j, cannot and, as stated, the Christoffel symbols are not the components of a third order tensor). Example: Newton’s Second Law The position vector can be expressed in terms of curvilinear coordinates, x xi . The velocity is then dx x di di v g dt i dt dt i Solid Mechanics Part III 165 Kelly Section 1.18 and the acceleration is dv d 2i d j g d k d 2i d j d k a g j i g 2 i k 2 jk i dt dt dt dt dt dt dt Equating the contravariant components of Newton’s second law f ma then gives the general curvilinear expression i i i j k f m jk ■ Partial Differentiation of the Metric Coefficients The metric coefficients can be differentiated with the aid of the Christoffel symbols of the first kind {▲Problem 3}: g ij (1.18.9) k ikj jki Using the symmetry of the metric coefficients and the Christoffel symbols, this equation can be written in a number of different ways: gij,k kij jki , g jk,i ijk kij , g ki, j jki ijk Subtracting the first of these from the sum of the second and third then leads to the useful relations (using also 1.18.8) 1 ijk g jk,i g ki, j gij,k 2 (1.18.10) 1 k g mk g g g ij 2 jm,i mi, j ij,m which show that the Christoffel symbols depend on the metric coefficients only. Alternatively, one can write the derivatives of the metric coefficients in the form (the first of these is 1.18.9) gij,k ikj jki (1.18.11) ij im j jm i g ,k g km g km Also, directly from 1.15.7, one has the relations g ij g gg , ij gg ij (1.18.12) g ij g Solid Mechanics Part III 166 Kelly Section 1.18 and from these follow other useful relations, for example {▲Problem 4} i log g 1 g 1 J ij J (1.18.13) j g j j and e g ijk g n e n m ijk m ijk mn ijk mn (1.18.14) ijk e ijk 1/ g ijk 1 n ijk n mn e mn m m g 1.18.2 Partial Differentiation of Tensors The Partial Derivative of a Vector The derivative of a vector in curvilinear coordinates can be written as i i v v i g i v vi i g g i v g v j j j j j i j v i v g v i k g or i g i v i g k (1.18.15) j i ij k j i jk i i v g vi g j i j where i i i k v | j v , j kj v Covariant Derivative of Vector Components (1.18.16) k vi | j vi, j ij vk The first term here is the ordinary partial derivative of the vector components. The second term enters the expression due to the fact that the curvilinear base vectors are changing. The complete quantity is defined to be the covariant derivative of the vector components. The covariant derivative reduces to the ordinary partial derivative in the case of rectangular Cartesian coordinates. The vi | j is the ith component of the j – derivative of v. The vi | j are also the components of a second order covariant tensor, transforming under a change of coordinate system according to the tensor transformation rule 1.17.4 (see the gradient of a vector below). Solid Mechanics Part III 167 Kelly Section 1.18 The covariant derivative of vector components is given by 1.18.16. In the same way, the covariant derivative of a vector is defined to be the complete expression in 1.18.15, v , j , i with v , j v | j g i . The Partial Derivative of a Tensor The rules for covariant differentiation of vectors can be extended to higher order tensors. The various partial derivatives of a second-order tensor ij i j j i i j A A g i g j Aij g g Ai g g j A j g i g are indicated using the following notation: A Aij |g g A | g i g j A j | g i g A i | g g j (1.18.17) k k i j ij k i k j j k i Thus, for example, i j i j i j A ,k Aij,k g g Aij g ,k g Aij g g ,k i j i m j i j m Aij,k g g Aij mk g g Aij g kmg m m i j Aij,k ik Amj jk Aim g g and, in summary, m m Aij |k Aij,k ik Amj jk Aim ij ij i mj j im A |k A ,k mk A mk A j j j m m j (1.18.18) A i |k Ai ,k mk Ai ik Am i i i m m i A j |k A j,k mk A j jk Am Covariant Derivative of Tensor Components The covariant derivative formulas can be remembered as follows: the formula contains the usual partial derivative plus for each contravariant index a term containing a Christoffel symbol in which that index has been inserted on the upper level, multiplied by the tensor component with that index replaced by a dummy summation index which also appears in the Christoffel symbol for each covariant index a term prefixed by a minus sign and containing a Christoffel symbol in which that index has been inserted on the lower level, multiplied by the tensor with that index replaced by a dummy which also appears in the Christoffel symbol. the remaining symbol in all of the Christoffel symbols is the index of the variable with respect to which the covariant derivative is taken. For example, i i i m m i m i A jk |l A jk,l ml A jk jl Amk kl A jm Solid Mechanics Part III 168 Kelly Section 1.18 Note that the covariant derivative of a product obeys the same rules as the ordinary differentiation, e.g. jk jk jk ui A |m ui| |m A ui A |m Covariantly Constant Coefficients It can be shown that the metric coefficients are covariantly constant3 {▲Problem 5}, ij g ij |k g |k 0 , This implies that the metric (identity) tensor I is constant, I ,k 0 (see Eqn.
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