Larger assignment on Vector Fields

[MarkBrezina]

TL;DR Summary

I have a series of assignment questions on Vectorfields and I'd love some feedback on whether I'm doing it properly

Dear everyone.

I'm doing an assignment on vectorfields and for most of the assignment I have to deal with tensors and tensornotation.

The first assignment asks me to express the following vector and matrixproducts in tensornotation.
$$\overline c = \overline a + \overline b \\ d=(\overline a + \overline b) \cdot \overline c \\ \overline c = \bar{\bar a}\bar b $$ Which I've written as $$c_i=a_i+b_i \\ d = (a_i+b_i)c_i=c_i^2 \\ c_i=a_{ij} b_i$$

Those are my current results in tensornotation, please do enlighten me on any wrongs. As I was also for a short period confused on whether $\bar{\bar a}$ was actually meant as a matrix or a vector.Furthermore I have to calculate $\delta_{ii}$, which I've gotten to 3
How do I show this best?

I also have to express $\nabla \phi$ and $\nabla \cdot \mathbf{v}$ in tensornotation.
Which I've gotten to be $\nabla \phi = \frac{\partial \phi}{\partial x_i} e_i=\phi_i e_i$ and $\nabla \cdot \mathbf{v} = \nabla_i \mathbf{v}_i = \frac{\partial v_i}{\partial x_i}$

And at the end I have to express the following in tensornotation.
$\frac{\partial x_1}{\partial x_1} =1 , \frac{\partial x_1}{\partial x_2} =0 , \frac{\partial x_1}{\partial x_3} =0$ thereby producing the kronecker delta as a series of differentials in the indices i for $x_i$ and j for $d_j$ I've tried writing the following, but I'm near 100% certain it might be wrong

$\delta_{ij}=\partial_j x_i$

Please do share comments, feedback, critic and much more. I'm eager to learn

 

[fresh_42]

What do the bars mean? If you want to write tensor products, then you have to start with tensors, neither with vectors nor with matrices. This means a vector is written $v=\sum_k v_k \vec{e}_k$ and a matrix $M=\sum_{p,q} M_{pq} \vec{e}_p \otimes \vec{e}_q$ with some basis $\{\,\vec{e}_i\,\}$ of $V$.

 

[MarkBrezina]

I agree Fresh. I'm guessing it's because it's common in Denmark to use bars over ordinary letters $\bar a$ to represent vectors. This is also why it confuses me not to see a boldface $\mathbf{A}$ matrix but I instead see a $\bar{\bar a}$ which might be a matrix.

Assuming the above were vectors and matrices I'd end up with the same results

 

[MarkBrezina]

I've changed all my assignment replies from usage of $\bar v$ to $\mathbf{v}=\sum_k v_k \mathbf{e}_k$