1. Continuum Mechanics (MTH487)
Lecture 4
Instructor
Dr. Junaid Anjum

2. Recap Dot (scalar) product of two vectors Cross product of two vectors
Triple scalar product (box product)
Triple cross product

3. Recap Vector-Dyad products Dyad-Dyad products Vector-Tensor product
Tensor-Tensor product

4. Recap
Rank (Order) of a Tensor: Tensor rank (order) is the number of free indices.
Scalar (0th order tensor) 3
Vector (1st order tensor)
3 components
Dyad (2nd order tensor)
9 components
Dyadic (2nd order tensor)
Triadic (3rd order tensor)
27 components
Tetradic (4th order tensor)
81 components
Contraction: The process of identifying (that is setting equal to one another) any two indices of a tensor term.
Outer products
Contraction(s)
Inner products
Note that the rank of a given tensor is reduced by 2 for each contraction

5. Recap Symmetric and Antisymmetric tensors Symmetric tensors in i & j
Antisymmetric in i & j
Antisymmetric in all indices

6. Aims and Objectives Matrices and Determinants
Some fundamental matrices
Properties of Transpose
Determinant
Cofactor Matrix
Determinants using cofactor expansion
Inverse matrix
Transformations of Cartesian Tensors

7. Matrices and Determinants ….
M=N, square matrix
M=1, row matrix
N=1, column matrix
3X3 square matrix represents 2nd order tensor.

8. Matrices and Determinants ….
Example: Show that the square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix by the decomposition

9. Matrices and Determinants ….
Example: Use indicial notation to show that for arbitrary matrices and
(a)
(b)
(c)

10. Matrices and Determinants ….
Determinant of a square matrix :
Cofactor of a matrix:

11. Matrices and Determinants ….
Determinant of a square matrix :
Cofactor Expansion:

12. Matrices and Determinants ….
Example: Show that for arbitrary matrices and

13. Matrices and Determinants ….
Example: Show that for arbitrary matrices and

14. Matrices and Determinants ….
Inverse of a Matrix The inverse of a matrix is written as and is defined by
where is the identity matrix.
The adjoint matrix is defined as the transpose of a cofactor matrix
In terms of the adjoint matrix the inverse matrix is expressed by

15. Matrices and Determinants ….
Example: Show from the definition of the inverse, that
(a)
(b)

16. Transformations of Cartesian Tensors
Consider two sets of rectangular Cartesian axes Ox1x2x3 and Ox’1x’2x’3 having a common origin.

17. Aims and Objectives Matrices and Determinants
Some fundamental matrices
Properties of Transpose
Determinant
Cofactor Matrix
Determinants using cofactor expansion
Inverse matrix
Transformations of Cartesian Tensors

18. Quiz…
True or False
The determinant of any orthogonal matrix is +1 or -1

19. Quiz… Using the square matrices below, demonstrate
That the transpose of the square of a matrix is equal to the square of its transpose.
that