Cartesian Tensors Reference: Jeffreys Cartesian Tensors 1 Coordinates and Vectors z x 3 e 3 y x 2 e 2 e 1 x x 1 Coordinates x i, i 123,, Unit vectors: e i, i 123,, General vector (formal definition to follow) denoted by components e.g. u u i Summation convention (Einstein) repeated index means summation: C54H -Astrophysical Fluid Dynamics 1
u i v i u i v i 2 Orthogonal Transformations of Coordinates u ii x 3 3 i 1 3 u ii i 1 x 3 x2 x 2 x 1 x 1 x i a ij x j a ij Transformation Matrix 2 C54H - Astrophysical Fluid Dynamics
Position vector r x i e i x j e j a ji x i e j x i e i i.e. the transformation of coordinates from the unprimed to the primed frame implies the reverse transformation from the primed to the unprimed frame for the unit vectors. Kronecker Delta x i ( a ji e j ) x i e i e i a ji e j δ ij 1 if i j 0 otherwise 2.1 Orthonormal Condition: Now impose the condition that the primed reference is orthonormal Use transformation i e j δ ij and e i e j δ ij e i a ki e k a lj e l e j a ki a lj e k e l a ki a lj δ kl a ki a kj NB the last operation is an example of the substitution property of the Kronecker Delta. Since e i e j δ ij, then the orthonormal condition on a ij is C54H -Astrophysical Fluid Dynamics 3
a ki a kj δ ij In matrix notation: a T a I Also have ik a jk aa T δ ij 2.2 Reverse transformations i.e. the reverse transformation is simply given by the transpose. Similarly, ( x i a ij x j ) a ik x i a ik a ij x j δ kj x j x k x k a ik x i x i a ji x j e i a ij e j 2.3 Interpretation of a ij Since e i a ij e j then the a ij are the components of e i wrt the unit vectors in the unprimed system. 4 C54H - Astrophysical Fluid Dynamics
3 Scalars, Vectors & Tensors 3.1 Scalar (f): f( x i) f( x i ) Example of a scalar is f r 2 x i x i. Examples from fluid dynamics are the density and temperature. 3.2 Vector (u): Prototype vector: x i General transformation law: 3.2.1 Gradient operator i a ij x j u i a ij u j Suppose that f is a scalar. Gradient defined by f ( grad f ) i ( f ) i x i Need to show this is a vector by its transformation properties. f x i f x j x j x i Since, x j a kj x k then C54H -Astrophysical Fluid Dynamics 5
x j a x kj δ ki a ij i and f x i a ij f x j Hence the gradient operator satisfies our definition of a vector. 3.2.2 Scalar Product is the scalar product of the vectors and. Exercise: u v u i v i u 1 v 1 + u 2 v 2 + u 3 v 3 u i v i Show that u v is a scalar. 3.3 Tensor Prototype second rank tensor x i x j General definition: Exercise: ij a ik a jl T kl Show that u i v j is a second rank tensor if u i and v j are vectors. Exercise: i, j u i x j 6 C54H - Astrophysical Fluid Dynamics
is a second rank tensor. (Introduces the comma notation for partial derivatives.) In dyadic form this is written as grad u or u. 3.3.1 Divergence Exercise: Show that the quantity v div v v i x i is a scalar. 4 Products and Contractions of Tensors It is easy to form higher order tensors by multiplication of lower rank tensors, e.g. ijk T ij u k is a third rank tensor if T ij is a second rank tensor and u k is a vector (first rank tensor). It is straightforward to show that T ijk has the relevant transformation properties. T ijk Similarly, if is a third rank tensor, then is a vector. Again the relevant tr4ansformation properties are easy to prove. 5 Differentiation following the motion This involves a common operator occurring in fluid dynamics. Suppose the coordinates of an element of fluid are given as a function of time by T ijj x i x i () t C54H -Astrophysical Fluid Dynamics 7
v i The velocities of elements of fluid at all spatial locations within a given region constitute a vector field, i.e. v i ( x j, t) If we follow the trajectory of an element of fluid, then on a particular trajectory x i x i () t. The acceleration of an element is then given by: v i f i dv i d v vi ( x dt dt j ()t t, ) i v i dx j + t dt x j v i v i + v t j x j Exercise: Show that is a vector. 6 The permutation tensor f i ε ijk ε ijk 0 if any of i, j, k are equal 1 if i, j, k unequal and in cyclic order 1 if i, j, k unequal and not in cyclic order e.g. ε 112 0 ε 123 1 ε 321 1 8 C54H - Astrophysical Fluid Dynamics
Is ε ijk a tensor? In order to show this we have to demonstrate that ε ijk, when defined the same way in each coordinate system has the correct transformation properties. Define ε ijk ε lmn a il a jm a kn ε 123 a i1 a j2 a k3 + ε 312 a i3 a j1 a k2 + ε 231 a i2 a j1 a k2 + ε 213 a i2 a j1 a k3 + ε 321 a i3 a j2 a k1 + ε 132 a i1 a j3 a k2 a i1 ( a j2 a k3 a j3 a k2 ) a i2 ( a j1 a k3 a j3 a k2 ) + a i3 ( a j1 a k2 a j2 a k1 ) a i1 a i2 a i3 a j1 a j2 a j3 a k1 a k2 a k3 In view of the interpretation of the a ij, the rows of this determinant represent the components of the primed unit vectors in the unprimed system. Hence: ε ijk e e e i j k This is zero if any 2 of, jk, are equal, is +1 for a cyclic permutation of unequal indices and -1 for a noncyclic permutation of unequal indices. This is just the definition of ε ijk. Thus ε ijk transforms as a tensor. C54H -Astrophysical Fluid Dynamics 9
6.1 Uses of the permutation tensor 6.1.1 Cross product Define c i ε ijk a j b k then c 1 ε 123 a 2 b 3 + ε 132 a 3 b 2 a 2 b 3 a 3 b 2 c 2 ε 231 a 3 b 1 + ε 213 a 1 b 3 a 3 b 1 a 1 b 3 c 3 ε 312 a 1 b 2 + ε 321 a 2 b 1 a 1 b 2 a 2 b 1 These are the components of c a b. 6.1.2 Triple Product In dyadic notation the triple product of three vectors is: t u v w In tensor notation this is u i ε ijk v j w k ε ijk u i v j w k 6.1.3 Curl u k ( curl u) i ε ijk x j e.g. u 3 u 2 ( curl u) 1 ε 123 + ε x 132 2 x 3 u 3 u 2 x 2 x 3 etc. 10 C54H - Astrophysical Fluid Dynamics
6.1.4 The tensor ε iks ε mps Define T ikmp ε iks ε mps Properties: If i k or m p then T ikmp 0. If i m we only get a contribution from the terms s i and k i, s. Consequently k p. Thus ε iks ± 1 and ε mps ε iks ± 1 and the product ε iks ε iks ( ± 1) 2 1. If i p, similar argument tells us that we must have s i and k m i. Hence, ε iks ± 1, ε mps + 1 ε iks ε mps 1. So, i m, k p 1 unless i i p, k m 1 unless i k 0 k 0 These are the components of the tensor δ im δ kp δ ip δ km. ε iks ε mps δ im δ kp δ ip δ km C54H -Astrophysical Fluid Dynamics 11
6.1.5 Application of ε iks ε mps ( curl ( u v) ) i ε ijk ( εklm u x l v m ) ε ijk ε klm ( ul v j x m ) j ( δ il δ jm δ im δ jl ) u l x v u v m + m l x j j u i v x u j m v i m x u v m v i + i x u j j x m j 7 The Laplacean u i v i v j u x j j x j u v j u j + i x v i x j j ( v u u v+ u v v u) i 2 φ 2 2 φ φ φ + + x 2 1 x 2 2 x 2 3 2 2 φ -------------- x i x i 12 C54H - Astrophysical Fluid Dynamics
8 Tensor Integrals 8.1 Green s Theorem n i V S In dyadic form: In tensor form: V vdv ( v n) ds S u i dv x i V S u i n i ds Flux of u through S Extend this to tensors: T ij dv T x ij n j ds j S Flux of T ij through S C54H -Astrophysical Fluid Dynamics 13
8.2 Stoke s Theorem t i C S In dyadic form: In tensor form: S ( curl u) nds u tds C u k n i S ε ijk ds u x i t i ds j C 14 C54H - Astrophysical Fluid Dynamics