Spirituality, Quantum Physics & Life: T1. Covarient Derivative:

a) Co-variant Derivative of First Rank Tensor: $\bar{A}_{\alpha }$ .

>> Solution:

We know that:
                        $\bar{A}_{\alpha } = \frac{\partial x^i}{\partial\bar{x}^\alpha }A_{i}$                                                                       -------------------- (1)
Differentiating with respect to $\bar{x}^\beta$ . We get,

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\frac{\partial A_{i}}{\partial {x}^j} + \frac{\partial^2 x^{i}}{\partial \bar{x}^{\alpha }{\partial \bar{x}^\beta }}A_{i}$                                                              -------------------- (2)

Now,
$\frac{\partial^2 x^{i}}{\partial \bar{x}^{\alpha }{\partial \bar{x}^\beta }} = \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\frac{\partial x^i}{\partial \bar{x^\eta }} - \frac{\partial x^j}{\partial \bar{x^\alpha}}\frac{\partial x^k}{\partial \bar{x^\beta}}\begin{Bmatrix} i \\ j &k \end{Bmatrix}$                                      -------------------- (3)

Put this value of equation (3) in equation (2). We get,

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\frac{\partial A_{i}}{\partial {x}^j} + \left [ \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\frac{\partial x^i}{\partial \bar{x^\eta }} - \frac{\partial x^j}{\partial \bar{x^\alpha}}\frac{\partial x^k}{\partial \bar{x^\beta}}\begin{Bmatrix} i \\ j &k \end{Bmatrix} \right ]A_i$

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\frac{\partial A_{i}}{\partial {x}^j} + \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\frac{\partial x^i}{\partial \bar{x^\eta }}A_i - \frac{\partial x^j}{\partial \bar{x^\alpha}}\frac{\partial x^k}{\partial \bar{x^\beta}}\begin{Bmatrix} i \\ j &k \end{Bmatrix}A_i$

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} - \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\frac{\partial x^i}{\partial \bar{x^\eta }}A_i = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\frac{\partial A_{i}}{\partial {x}^j}- \frac{\partial x^j}{\partial \bar{x^\alpha}}\frac{\partial x^k}{\partial \bar{x^\beta}}\begin{Bmatrix} i \\ j &k \end{Bmatrix}A_i$

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} - \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\frac{\partial x^i}{\partial \bar{x^\eta }}A_i = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\frac{\partial A_{i}}{\partial {x}^j}- \frac{\partial x^k}{\partial \bar{x^\alpha }}\frac{\partial x^j}{\partial \bar{x^\beta}}\begin{Bmatrix} i \\ k &j \end{Bmatrix}A_i$

In the last term: $k \leftrightharpoons i$ . Then we get,

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} - \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\bar{A_\eta } = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\frac{\partial A_{i}}{\partial {x}^j}- \frac{\partial x^i}{\partial \bar{x^\alpha }}\frac{\partial x^j}{\partial \bar{x^\beta}}\begin{Bmatrix} k \\ i &j \end{Bmatrix}A_k$

$\frac{\partial \bar{A}_{\alpha }}{\partial \bar{x^{\beta }}} - \bar{\begin{Bmatrix} \eta \\ \alpha &\beta \end{Bmatrix}}\bar{A_\eta } = \frac{\partial x^{i}}{\partial \bar{x}^{\alpha }} \frac{\partial x^{j}}{\partial \bar{x}^\beta}\left [ \frac{\partial A_{i}}{\partial {x}^j}- \begin{Bmatrix} k \\ i &j \end{Bmatrix}A_k \right ]$                     -------------------- (4)

The term: $\left [ \frac{\partial A_{i}}{\partial {x}^j}- \begin{Bmatrix} k \\ i &j \end{Bmatrix}A_k \right ]$ is a co-variant tensor of first rank, called the Co-variant derivative of $A_i$ with respect to $A^j$ .

It is denoted as:

$A_{i,j} = \left [ \frac{\partial A_{i}}{\partial {x}^j}- \begin{Bmatrix} k \\ i &j \end{Bmatrix}A_k \right ]$    called the Co-variant derivative.

T1. Covarient Derivative:

a) Co-variant Derivative of First Rank Tensor: .

a) Co-variant Derivative of First Rank Tensor: $\bar{A}_{\alpha }$ .