Solving Chain Rule A(r,t) Problem

[Confundo]

Problem

Use the chain rule to proof
$$\dot{A}=\partial_t A+v_j\partial_jA_i$$

Attempt at Solution

$$\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i}$$

Obviously
$$v_j = \frac{dr_j}{dt}$$

I'm puzzled where the v_j and partial d_j come in

 

[fzero]

If you're talking about why the index is j and not i, the answer is simple. The problem is using the summation convention that a repeated index is understood to be summed over:

$$v_j\partial_j \equiv \sum_j v_j\partial_j.$$

When you wrote

$$\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+\frac{dr_i}{dt}\frac{\partial A_i}{\partial r_i},$$

the index in the derivatives should not have been the same index on $$A_i$$, since it is confusing to have the same index appearing three times when it is being implicitly summed over in part of the expression. You should have written

$$\dot{A}=\frac{dA_i}{dt} = \partial_t A_i+ \sum_j \frac{dr_j}{dt}\frac{\partial A_i}{\partial r_j} = \partial_t A_i+ \frac{dr_j}{dt}\frac{\partial A_i}{\partial r_j},$$

where the summation convention is being used in the last expression.