Lecture Notes on Vectors and Tensors

Overview

• Introduction to mathematical concepts required for upcoming lectures.
• Focus on vectors and tensors, with a detailed discussion of their properties and operations.

Vectors

• Definition: A vector has magnitude and direction.
  • Represented visually as an arrow, where:
    • Length = Magnitude
    • Direction = Arrow pointing
• Independence of Coordinate System:
  • A vector remains unchanged regardless of the coordinate system used to describe it.
  • Example: In Cartesian coordinates (e1, e2, e3), a vector V in the e1-e2 plane at a 45-degree angle has components:
    • v1 = 1/√2, v2 = 1/√2, v3 = 0 (assuming unit magnitude).
  • In a different coordinate system aligned with vector V, components become:
    • v1' = 1, v2' = 0, v3' = 0.
• Notation: Vectors are denoted with a tilde below (e.g., ( ilde{V} )).

Operations with Vectors

Dot Product

• Defined as:
  • ( A \cdot B = \sum_{i=1}^{3} A_i B_i )
  • Can also be represented as the multiplication of a row by a column matrix.
• Result: A scalar quantity._

Cross Product

• Defined as:
  • ( A \times B = (A_2B_3 - A_3B_2, A_3B_1 - A_1B_3, A_1B_2 - A_2B_1) )
• Can be represented as a matrix product involving a skew-symmetric matrix derived from vector A.
• Result: A vector quantity.

Tensor Product

• Denoted by ( A imes B ) (with a circle).
• Result: A second-order tensor with components denoted by ( C_{ij} = A_i B_j )._

Tensors

• Definition: A tensor is a mathematical object that can be represented in multiple coordinate systems.
  • First-order tensor: Represents vectors (one tilde).
  • Second-order tensor: Represents matrices (two tildes).
• Basis Tensors: For a second-order tensor, there are 9 basis tensors defined by the indices (i, j).
• Important property: Tensor representation changes with the coordinate system, but the tensor itself does not.

Matrix Representation of Tensors

• When expressed in a coordinate system, a second-order tensor appears as a 3x3 matrix.
• Example: Given a second-order tensor C, its matrix form is represented as:
  • ( C_{ij} ) where rows and columns represent different indices._

Multiplication of Tensors

Tensor-Vector Multiplication

• For a second-order tensor C multiplied by a vector B:
  • Resulting vector A = C imes B can be derived using matrix multiplication.
  • Components of A are given by: ( A_i = \sum_{j} C_{ij} B_j ).

Tensor-Tensor Multiplication

• The product of two second-order tensors results in another second-order tensor:
  • ( C = A imes B ) leads to components given by: ( C_{il} = \sum_{k} A_{ik} B_{kl} ).

Rotation Tensor

• Represents the physical rotation of vectors in a coordinate system.
• The rotation tensor R has orthogonal properties:
  • ( R^T R = I ), where I is the identity matrix.
• Example of a rotation matrix for angle ( \theta ):
  • [ R = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \ \sin \theta & \cos \theta & 0 \ 0 & 0 & 1 \end{pmatrix} ]

Conclusion

• The properties and operations of vectors and tensors are foundational for understanding complex mathematical modeling in upcoming lectures.
• Next lecture will cover traction vectors.