Exercise 1

Revisiting Vector Calculus

Einstein Summation

In a right-handed, orthonormal system of fixed basis vectors \(\mathbf{e}_1\), \(\mathbf{e}_2\) and \(\mathbf{e}_3\) of unit length, an arbitrary vector \(\mathbf{u}\) has the following co-ordinate (component) expression

\[ \mathbf{u} = u_1 \mathbf{e}_1 + u_2 \mathbf{e}_2 + u_3 \mathbf{e}_3 = \sum_{i=1}^{3} u_i \mathbf{e}_i = \begin{pmatrix} u_1 \\ u_2 \\ u_3 \end{pmatrix} \mathbf{e}_i \]

By dropping the summation symbol, we get the vector \(\mathbf{u}\) in Einstein summation convention as follows \[ \mathbf{u} = u_i \mathbf{e}_i \] where a summation is implied in an expression, whenever indices occur twice. The repeated indices are called free or dummy indices.

Note

In this course, we use the convention that index notation signals the use of Einstein summation.

Common Symbols

In order to evaluate and simplify expressions in vector and tensor algebra we make use of

  1. The Kronecker delta

\[ \delta_{ij} = \begin{cases} 1,& \text{if } i=j\\ 0,& \text{if } i\not=j \end{cases} \]

  1. The Levi-Cevita symbol \[ \epsilon_{ijk} = \begin{cases} +1,& \text{if } (i, j, k) \text{ has even permutation of} (1, 2, 3)\\ -1,& \text{if } (i, j, k) \text{ has odd permutations of} (1, 2, 3)\\ 0 & \text{otherwise} \end{cases} \]

Useful Identities

The dot product of two orthonormal basis vectors gives \[ \mathbf{e}_i \cdot \mathbf{e}_j = \delta_{ij} \]

and the cross product of the two orthonormal basis vectors gives

\[ \mathbf{e}_i \times \mathbf{e}_j = \epsilon_{ijk} \mathbf{e}_k \]

Interchanging Notations

During the course, we often require you to change the notation from one form to another. For example, given two vectors \(\mathbf{a}\) and \(\mathbf{b}\), we have the following notations

  • vector notation:

    \[ \mathbf{a} + \mathbf{b} \]

  • index notation:

    \[ a_i + b_i \]

  • component notation

    \[ \begin{pmatrix} a_1 + b_1 \\ a_2 + b_2 \\ a_3 + b_3 \\ \end{pmatrix} \]

    or

    \[ \begin{pmatrix} a_x + b_x \\ a_y + b_y \\ a_z + b_z \\ \end{pmatrix} \]

Tasks

  1. Write out the dot-product of two arbitrary vectors \(\mathbf{u} \cdot \mathbf{v}\) in component notation and index notation.
  1. Write the cross-product of two arbitrary vectors \(\mathbf{u} \times \mathbf{v}\) in component notation and in index notation.
  1. Write the expression \(\left(\mathbf{u} \cdot \nabla \right) \mathbf{u}\) in component notation and in index notation.