Index Notation

Usually, rectangular Cartesian coordinates of a point are denoted by $(x,y,z)$ and the unit vectors along $x,\ y$ and $z$-axes by $\mathbf{i},\ \mathbf{j}$, and $\mathbf{k}$ respectively. In this coordinate system, the components of a vector $\mathbf{u}$ along $x,\ y$, and $z$ axes are denoted by $u_x,\ u_y$, and $u_z$. The vector $\mathbf{u}$ has the representation

$$\mathbf{u} = u_x \mathbf{i} + u_y\mathbf{j} + u_z\mathbf{k}.$$

This notation does not lend itself to any abbreviation. Therefore, instead of denoting the coordinate axes by $x,y,z$ we will denote them by $x_1,x_2,x_3$. Also we will denote unit vectors

along $x_1,x_2$ and $x_3$ axes by $\mathbf{e}_1,\ \mathbf{e}_2$, and $\mathbf{e}_3$ respectively. Naturally then components of a vector $\mathbf{u}$ along $x_1,x_2$ and $x_3$ axes will be indicated by $u_1,u_2$, and $u_3$ respectively. Hence we can write

$$\begin{split}\mathbf{u} =\ & u_1\mathbf{e}_1 + u_2\mathbf{e}_2 + u_3\mathbf{e}_3,\\ =\ & u_j\mathbf{e}_j.\end{split}$$ (2.4.1)

Similarly,

$$\begin{split}\mathbf{v} =\ & v_1\mathbf{e}_1 + v_2\mathbf{e}_2 + v_3\mathbf{e}_3,\\ =\ & v_j \mathbf{e}_j.\end{split}$$

The dot product $\mathbf{u}\cdot \mathbf{v}$ can simply be written as

$$\mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 + u_3v_3 = u_iv_i.$$ (2.4.2)

Since $\mathbf{e}_1,\ \mathbf{e}_2,\ \mathbf{e}_3$ are mutually orthogonal unit vectors, therefore,

$$\begin{split}&\mathbf{e}_1 \cdot \mathbf{e}_1 = \mathbf{e}_2 ... ... \mathbf{e}_3 = \mathbf{e}_1 \cdot \mathbf{e}_3 = 0.\end{split}$$

These equations can be summarized as

$$\mathbf{e}_i\cdot \mathbf{e}_j = \delta_{ij}.$$ (2.4.3)

Exercise. Using the index notation, write expressions for

(1)

the magnitude of a vector $\mathbf{u}$,

(2)

$\cos\theta;\ \theta$ being the angle between vectors $\mathbf{u}$ and $\mathbf{v}$.

As another illustration of the use of the index notation, consider a line element with components $dx_1,\ dx_2,\ dx_3$. The square of the length, $ds$, of the line element is given by

$$ds^2 =\ dx^2_1 + dx^2_2 + dx^2_3,$$

$$=\ dx_i dx_i$$

$$=\ \delta_{ij}dx_idx_j.$$

Finally, we note that the differential of a function $f(x_1,x_2,x_3)$ can be written as

$$df =\ \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2 + \frac{\partial f}{\partial x_3} dx_3,$$

$$=\ \frac{\partial f}{\partial x_i} dx_i.$$