# Unit Basis Components of a Vector in Tensorial Expressions?

• yucheng
In summary, the divergence formula is: $\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$
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 $$\vec{e}^{i} \cdot \vec{e}_{i} = \hat{e}^{i} \cdot \hat{e}_{i} \sqrt{g^{ii}g_{ii}} = \sqrt{g^{ii}g_{ii}} = 1$$

Last edited:
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
yucheng said:
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
ergospherical said:
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...

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

yucheng
ergospherical

## 1. What are unit basis components in tensorial expressions?

Unit basis components refer to the coefficients that represent the magnitude and direction of a vector in a specific coordinate system. In tensorial expressions, these components are used to describe the vector in terms of a basis set of unit vectors.

## 2. How are unit basis components calculated?

The unit basis components of a vector can be calculated by taking the dot product of the vector with each of the unit basis vectors in the coordinate system. This will result in a set of scalar values that represent the magnitude of the vector in each direction.

## 3. What is the significance of unit basis components in tensorial expressions?

Unit basis components are important in tensorial expressions because they allow for the representation of a vector in a specific coordinate system. This makes it easier to perform calculations and transformations on the vector.

## 4. How do unit basis components relate to tensors?

In tensorial expressions, unit basis components are used to define the elements of a tensor. The tensor is then used to represent the vector in a specific coordinate system, making it easier to manipulate and work with.

## 5. Can unit basis components be used to represent any type of vector?

Yes, unit basis components can be used to represent any type of vector in a specific coordinate system. This includes both geometric vectors, such as displacement or velocity, as well as abstract vectors, such as forces or electric fields.

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