gradient of vectors/tensors

Hi. My class notes define ( grad **u** ) like this :

(all the x are tensor products, and the d's are partial derivatives)

grad **u** = **del** x **u**

= (d**u**/dxq) x **e**q

= (d/dxq)(up**e**p) x **e**q

= (dup/dxq) **e**p x **e**q

where **u** is a vector function of **x**, and {**e**i} are basis vectors.

I'm confused about how the **e**q is put on the right hand side of the tensor product - seems to me it should be on the left, since it's associated with **del**, and not **u**.

That is, I think it should be:

**del** x **u**

**= e**q d/dxq x up**e**p

= (dup/dxq) **e**q x **e**p

i.e. with **e**p and **e**q the other way around to my notes. This is also how my textbook seems to define it.

So my question is, does the order of **e**p and **e**q make a difference? It seems to me it would result in a different matrix (the transpose??), but I'm not sure.

If they're different, which definition is correct?

Thanks,

~squiggles