# Einstein notation non-repeating indices

I forget the rule for Einstein notation. If I have something like the gradient:

$$\vec\nabla f = \frac{\partial f}{\partial x_i} = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \rangle$$

$$\vec\nabla \vec F = \frac{\partial F_i}{\partial x_j} = \begin{bmatrix} \frac{\partial F_x}{\partial x} & \frac{\partial F_x}{\partial y} & \frac{\partial F_x}{\partial z} \\ \frac{\partial F_y}{\partial x} & \frac{\partial F_y}{\partial y} & \frac{\partial F_y}{\partial z} \\ \frac{\partial F_z}{\partial x} & \frac{\partial F_z}{\partial y} & \frac{\partial F_z}{\partial z} \end{bmatrix}$$

First off, is this correct? For $\frac{\partial F_i}{\partial x_j}$, I forget which are the columns and which are the rows.

And secondly, am I remembering this rule correctly — "the dimension/rank of the resulting tensor is the number of non-repeated indices"? For instance, $\alpha_{ij}\beta_{j}\gamma_{k}\delta_{l}$ would be dimension 3, since it has three unique indices, right?

Thanks for the clarification.

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What operation is denoted by $\vec\nabla \vec F$? Tensor product? –  Ruslan Apr 15 '13 at 21:32