Vector Calculus Question in Lagrangian Mechanics

Hi guys. I hope this isn't a bad place to post my question, which is:

I'm reading some lecture notes on Lagrangian mechanics, and we've just derived the Euler-Lagrange equations of motion for a particle in an electromagnetic field. It reads:

[tex] m \ddot{\vec{r}} = -\frac{e}{c} \frac{\partial \vec{A}}{\partial t}-e\nabla \phi(\vec{r})+ \frac{e}{c} \nabla (\dot{\vec{r}}\cdot \vec{A}) [/tex]

(A and phi are the vector and scalar potentials, respectively). Now the author switches to index notation, and I get lost in the process. The author gives:

[tex]
m \ddot{r}^a = - \frac{e}{c} \frac{\partial A^a}{\partial t} - e \frac{\partial \phi(\vec{r})}{\partial r^a} + \frac{e}{c} \left( \frac{\partial A^b}{\partial r^a} - \frac{\partial A^a}{\partial r^b}\right) \dot{r}^b
[/tex]

My problem is with the third term above. Is the summation over b implied? I thought that it is bad form to repeat a summation index three or more times. Also, if we are to look at the a-th component of the gradient of the inner product, it should be:

[tex]
\left(\nabla (\dot{\vec{r}}.\vec{A})\right)_a = \left(\nabla (\dot{r}_i A^i)\right)_a=\frac{\partial}{\partial r^a} \left ( \dot{r}_i A^i \right) = \dot{r_i}\frac{\partial A^i}{\partial r^a}
[/tex]

which is clearly not equal to the text. I have a feeling I'm doing something stupid, and it would be great if someone can point out my mistake(s).

Thanks.