1. An introduction to Cartesian Vector and Tensors Dr Karl Travis Immobilisation Science Laboratory, Department of Engineering Materials, University of Sheffield, UK k.travis@sheffield.ac.uk

2. Analytic definition of vectors and tensors Let a be a vector expanded in an orthogonal basis: where  i are unit vectors along the 3 Cartesian axes. scalar product of 2 vectors where the third line follows from the orthogonality relations between the unit vectors

3. where  ijk is defined by i.e. {  123,  312,  231 } = +1, {  213,  132,  321 } = -1 Cross (vector) product of 2 vectors

4. Other useful relations when dealing with vector cross products are Example: Proove that

6. Differential operators The  (‘del’) operator is also a vector, and is defined as Forming the scalar product of  with another vector is called the divergence, or simply ‘div’. The divergence of a vector a, say, is which is a scalar quantity

7. The Laplacian operator or ‘del squared’ is just the scalar product of ‘del’ with itself, and is a scalar. The curl of a vector is formed from the vector cross product of ‘del’ with the vector, and is itself a vector:

8. Cartesian tensors. Some physical quantities require both a magnitude and at least two directions to define them: eg Inertia tensor, Pressure tensor. The number of directions required defines the rank of the tensor. Pressure and inertia are examples of 2nd rank tensors (vectors and scalars are tensors of rank 1 and 0 respectively). Let A be a 2nd rank tensor expanded in an orthogonal basis: where  i  j is called the unit dyad. It is sometimes easier to think of the components of a tensor as a rectangular array of numbers i.e. a matrix. So for A,

9. Operations for the unit dyads The transpose of a tensor

10. The magnitude of a tensor Invariants of a tensor 3 independent scalars can be formed from a tensor by taking the trace of A, A 2 and A 3. These scalars are invariants since they do do change value upon a change of the coordinate system.

11. Operations between vectors and tensors Dyadic product of two vectors, a and b, is written as ab and is a 2nd rank tensor which is defined as Scalar product between a vector and a 2nd rank tensor forms a new vector defined by where we have used the orthogonality relation

12. Double contraction of two 2nd rank tensors gives a scalar Einstein notation for tensors Repeated index means sum over that index. The double contraction above becomes in Einstein notation. Where we have made use of

13. Other products: The non-repeated indices give the tensor character. The rank of the product is the sum of the ranks of the two quantities less 2 for each dot appearing in the operator. common tensors 2nd rank isotropic tensor, 1 =  ij  i  j. 3rd rank Levi-Cevita tensor,  =  ijk  i  j  k which is also referred to as the alternating tensor.

14. Parity: polar and pseudo vectors Vectors which change sign under a mirror inversion of the coordinate axes are called pseudovectors. Vectors which are invariant to a mirror inversion of the coordinate system are called polar vectors. The spin angular momentum is an example of a pseudovector since it is defined by a vector cross product: There are also polar and pseudo scalars and tensors. Decomposition of tensors A cartesian tensor can be decomposed into a symmetric and antisymmetric part:

15. Where a superscript ‘T’ denotes the transpose. A 2nd rank antisymmetric tensor has the form: And hence has only 3 independent components. These components transform like a vector, so antisymmetric 2nd rank tensors are often represented as a pseudovector dual. If A a is an antisymmetric tensor, its pseudovector dual, a d is given by which involves the alternating tensor.

16. Symmetric tensors can be further split into an isotropic component and a traceless symmetric component: where Tr(A) = A ii in Einstein notation. Note that when a tensor is formed from a dyadic product of 2 vectors, say C (2) = ab, the trace is given as the scalar product of the two vectors: Tr(C) = ab Pressure and strain rate tensors  = P - p1 The viscous pressure tensor, , is defined as Where p is the equilibrium scalar pressure.

17. Now we decompose  to give The strain rate tensor can be decomposed into

18. Fourier transforms of quantities involving vectors and tensors. Define the Fourier transform pair by: (i) Fourier transform of the divergence of a tensor quantity. Let then

19. Now we can integrate by parts: Writing out the multiple integrals explicitly, Where we have used the fact that the boundary term is zero.