Notational problem in tensor calculus

[jacobrhcp]

Using the Einstein convention, is this about right? (indexes run from 1 to 3):

$$\nabla\bullet(x_{i}a)=div(x_{i}a)=\partial_{j}a_{j}x_{i}=3x_{i}$$

 

[dextercioby]

The partial derivatives are take wrt what variable ? x or a ?

 

[jacobrhcp]

it's said x above the exercises $$\partial_{j}=\frac{\partial}{\partial x_{j}}}$$

and the question is:

calculate explicitly: $$\nabla \bullet (x_{i}\vec{a})$$, where a is a constant vector...

my attempt is way off, but I don't feel at home in these new symbols enough to get the right answer.

 

[pam]

$$x_i{\vec a}$$ is not clear notation. Does it mean the i component of a tensor
[xa], or does it mean $${\vec\hat i}\cdot{\vec r}{\vec a}$$?

 

[kdv]

Using the Einstein convention, is this about right? (indexes run from 1 to 3):

$$\nabla\bullet(x_{i}a)=div(x_{i}a)=\partial_{j}a_{j}x_{i}=3x_{i}$$

You are right that it means $$\partial_{j} (a_{j}x_{i})$$ But the components of a are constants so this is equal to $$a_j \partial_{j} x_{i}$$ Now, what does $$\partial_{j} x_{i}$$give ?