How would you write $\frac{\partial m_j}{\partial x_i} = \frac{\partial H_j}{\partial x_i}m_j$ in vector notation?

Question

The original paper uses Einstein summation notation as in the question title and I'm trying to write it using vector product notation. $\mathbf{m}$, $\mathbf{x}$ and $\mathbf{H}$ are all 3-component vectors.

My attempt is:

$$\frac{\partial\mathbf{m}}{\partial\mathbf{x}} = \mathbf{m} \cdot \nabla\mathbf{H}$$

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Is this correct? If it is, what does a dot product with the gradient of a vector look like, given that is a tensor?