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The first method involves multiplying the gradient vector (del) by the vector in question to form a matrix. I believe the resulting matrix is in the form of a 3 by 3 Jacobian matrix. With this method I am unsure what to do if this was then involved in a dot product with another vector, or even a cross product with another vector for that matter.

The second method (and the one I am more familiar with) simply results in a vector of the 3 partial derivatives with respect to each dimension x, y and z. This method was the one I learnt and all you had to do was solve the partial derivatives to get the resulting vector.

Are these both equivalent with the exception that the first method is more detailed? or are these methods each meant for a different situation?