1. 1 Part B Tensors Or, inverting The graph above represents a transformation of coordinates when the system is rotated at an angle  CCW

2. 2 Let be two vectors in a Cartesian coordinate system. If is a transformation, which transforms any vector into some other vector, we can write 2B1 Tensors whereandare two different vectors. If For any arbitrary vectorandand scalar is called a LINEAR TRANSFORMATION and a Second Order Tensor. (1.1a) and (1.2a) can be written as,

3. 3 2B1 Tensors Questions: Yes

4. 4 2B1 Tensors Questions: Yes

5. 5 2B2 Components of a Tensor Scalar has 1 component Vector has 3 component Tensor has ? component?

6. 6 2B2 Components of a Tensor In general The components of T ij can be written as

7. 7 2B2 Components of a Tensor (cont.) Example 2B2.3 Thus Q: Let R correspond to a righ-hand rotation of a rigid body about the x 3 axis by an angle . Find the matrix of R Q: Let R correspond to a 90° right-hand rigid body rotation about the x 3 axis. Find the matrix of R

8. 8 2B3 Components of a transformed vector The transformationis linear if is then called a LINEAR TRANSFORMATION and is a second-order tensor.

9. 9 2B3 Components of a transformed vector Now what is b? what are the components of b?

10. 10 2B2 Components of a Tensor (cont.)

11. 11 2B4 Sum of Tensors Definition:

12. 12 2B5 Product of Two Tensors Components: or In general, the product of two tensors is not commutative:

13. 13 Q: Let R correspond to a 90° right-hand rigid body rotation about the x 3 axis. Find the matrix of R 2B5 Product of Two Tensors Q: Let R correspond to a 90° right-hand rigid body rotation about the x 1 axis. Find the matrix of S Q: Let W correspond to a 90° right-hand rigid body rotation about the x 3 axis, then a 90° right-hand rigid body rotation about the x 1 axis. Find the matrix of W

14. 14 2B6 Transpose of a Tensor Transpose of ~ T is denoted by ~ T T. If a  and then Note

15. 15 2B7 Dyadic Product Definition, dyadic product of two vectors is also a tensor. It transforms a vector into a vector parallel to

16. 16 2B8 Trace of a Tensor

17. 17 2B9 Identity Tensor A linear transformation which transforms every vector into itself is an identify tensor

18. 18 2B9 Inverse Tensor Note that

19. 19 2B10 Orthogonal Tensor Thus and

20. 20 2B10 Orthogonal Tensor (cont.) Thus ~~~ QQI T  Also, we have ~~ QQI   1, So Thus for an orthogonal matrix, the transpose is also its inverse. Example: A rigid body rotation is an ORTHOGONAL tensor,

21. 21 2B11 Transformation Matrix Between Two Coordinate Systems We see that,

22. 22 2B11 Transformation Matrix Between Two Coordinate Systems (cont.) Thus Q: Let be obtained by rotating the basis about the x 3 axis by an angle . Find the transformation matrix of R from to

23. 23 2B12 Transformation laws for Vectors.

24. 24 2B12 Transformation laws for Vectors. (cont.) Answer

25. 25 2B13 Transformation law for tensor Let and Recall that the components of a tensor are:

26. 26 2B13 Transformation law for tensor (cont.)

27. 27 2B13 Transformation law for tensor (cont.) Find the matrix of T with respect to the basis

28. 28 2B14 Tensors by Transformation laws

29. 29 2B14 Tensors by Transformation laws (cont.) The components transform as follows:

30. 30 2B14 Tensors by Transformation laws (cont.)