Defining Del in Index Notation: Which Approach is Appropriate?

[member 428835]

Hi PF!

Which way is appropriate for defining del in index notation: $\nabla \equiv \partial_i()\vec{e_i}$ or $\nabla \equiv \vec{e_i}\partial_i()$. The two cannot be generally equivalent. Quick example.

Let $\vec{v}$ and $\vec{w}$ be vectors. Then $$\nabla \vec{v} \cdot \vec{w} = \partial_i(v_j \vec{e_j})\vec{e_i} \cdot u_k \vec{e_k}\\ = \partial_i(v_j \vec{e_j}) u_i$$ yet using the other definition for del implies $$\nabla \vec{v} \cdot \vec{w} = \vec{e_i} \partial_i(v_j \vec{e_j}) \cdot u_k \vec{e_k}\\=\vec{e_i} v_ju_k (\partial_i(\vec{e_j}) \cdot \vec{e_k}) + \vec{e_i} u_j \partial_i(v_j)$$

 

[pasmith]

Hi PF!

Which way is appropriate for defining del in index notation: $\nabla \equiv \partial_i()\vec{e_i}$ or $\nabla \equiv \vec{e_i}\partial_i()$.

The second. In orthogonal coordinates \nabla = \sum_i \mathbf{e}_i h_i \partial _i and in non-cartesian coordinates h_i is generally a non-constant function of position.

 

[member 428835]

The second. In orthogonal coordinates \nabla = \sum_i \mathbf{e}_i h_i \partial _i and in non-cartesian coordinates h_i is generally a non-constant function of position.

So is $h_i = |\partial_i \vec{r}|$ where $\vec{r}$ is the position vector, expressed in cartesian coordinates as $\vec{r} = x \hat{i} + y \hat{j} +z \hat{k}$?