Unit Basis Components of a Vector in Tensorial Expressions?

[yucheng]

Homework Statement

Vanderlinde, Classical Electromagnetic Theory

Relevant Equations

N/A

Divergence formula

$$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$$

If we express it in terms of the components of $\vec{A}$ in unit basis using

$$A^{*j} = \sqrt{g^{jj}} A^{j}$$

, we get $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{*j} \sqrt{g^{jj} G})$$

However, did the author intend
$$A^{j} = \sqrt{g^{jj}} A^{*j}$$

(quick check: with this one can directly substitute to get the correct divergence formula)? Because
$$\vec{A} = A^{j} \vec{e}_j = A^{*j} \hat{e}_j = A^{*j} \frac{1}{\sqrt{g_{jj}}} \sqrt{g_{jj}} \hat{e}_j = A^{*j} \sqrt{g^{jj}} \, \vec{e}_j$$

, since $$\vec{e}^{i} \cdot \vec{e}_{i} = \hat{e}^{i} \cdot \hat{e}_{i} \sqrt{g^{ii}g_{ii}} = \sqrt{g^{ii}g_{ii}} = 1$$

 

[ergospherical]

Let $q^i$ be orthogonal co-ordinates and let the scale factors in the holonomic basis be $h_i \equiv \sqrt{\boldsymbol{e}_i \cdot \boldsymbol{e}_i} = \sqrt{g_{ii}}$ (no sum). Then $\boldsymbol{e}_i = h_i \hat{\boldsymbol{e}}_i$ (no sum) and
\begin{align*}
\boldsymbol{A} = \sum_i A^i \boldsymbol{e}_i = \sum_i A^i h_i \hat{\boldsymbol{e}}_i \equiv \sum_i {A^*}^i \hat{\boldsymbol{e}}_i
\end{align*}from which one identifies the physical components in the normalised basis as ${A^*}^i = h_i A^i$ (no sum), which is as you wrote. The divergence formula:\begin{align*}
\nabla \cdot \boldsymbol{A} = \sum_i \nabla_i A^i &= \sum_i \frac{1}{\sqrt{|g|}} \frac{\partial}{\partial q^{i}} \left(A^{i} \sqrt{|g|} \right) \\
&= \sum_i \frac{1}{\sqrt{|g|}} \frac{\partial}{\partial q^{i}} \left( \frac{1}{h_i} {A^*}^i \sqrt{|g|} \right)
\end{align*}And of course $h_i = \sqrt{g_{ii}} = 1/\sqrt{g^{ii}}$ (no sum).

 

[yucheng]

${A^*}^i = h_i A^i = \sqrt{g_{ii}} A^{i}$ So would you agree that the formula the author gave, $A^{*j} = \sqrt{g^{jj}} A^{j}$ is... at best... confusing?

 

[ergospherical]

So would you agree that the formula the author gave, $A^{*j} = \sqrt{g^{jj}} A^{j}$ is... at best... confusing?

I agree with you that it is wrong! It should, of course, be $A^{*j} = \sqrt{g_{jj}} A^{j} = \frac{1}{\sqrt{g^{jj}}} A^j$ (no sum).

 

[yucheng]

And of course hi=gii=1/gii (no sum), but I'm not sure why you'd want to start re-writing it in this form. Writing it in terms of the scale factors is simplest, no?

Because I am trying to compute the divergence of functions, and since most functions we have are given in spherical/cylindrical coordinates, they are usually expressed in terms of the unit vectors... So the original formula does not work...

 

[ergospherical]

The quantities $h_i = \sqrt{g_{ii}}$ are usually called the scale factors of the co-ordinates.

 

[yucheng]

@ergospherical thanks!