I'm currently teaching in a course on vector calculus, and in this course we use a strange method for computing vector calculus expressions. We also teach standard suffix notation in parallel, but this other method is supposed to simplify the computations for the weaker students that don't want to pursue more involved theoretical courses.
Now, what I want to know is if anyone of you recognizes this method and perhaps knows what it is called in English? Maybe you haves some proof as to why it works? The methods is as follows.
Assume that we want to prove the identity
$$\nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A}) + (\mathbf{B}\cdot\nabla)\mathbf{A} - (\mathbf{A}\cdot\nabla)\mathbf{B}.$$
Then the idea is that we copy the expression as many times as we have variables that the del operator is acting on, marking a different variable with a dot in each copy. After doing this, the idea is that we can treat nabla as an ordinary vector. Therefore, we have
$$\nabla \times (\mathbf{A} \times \mathbf{B}) = \nabla \times (\underset{\cdot}{\mathbf{A}} \times \mathbf{B}) + \nabla \times (\mathbf{A} \times \underset{\cdot}{\mathbf{B}}). $$
Now, treating nabla as an ordinary vector, we want to use vector identities to rearrange this expression so that only the variable marked with a dot is to the right of the del operator. Therefore, we use the bac-cab rule to obtain
$$ \nabla \times (\underset{\cdot}{\mathbf{A}} \times \mathbf{B}) + \nabla \times (\mathbf{A} \times \underset{\cdot}{\mathbf{B}}) = \underset{\cdot}{\mathbf{A}}(\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \underset{\cdot}{\mathbf{A}}) + \mathbf{A}(\nabla \cdot \underset{\cdot}{\mathbf{B}}) - \underset{\cdot}{\mathbf{B}} (\nabla \cdot \mathbf{A}). $$
Rearranging to have only the dotted variable to the right of nabla, we get
$$ \underset{\cdot}{\mathbf{A}}(\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \underset{\cdot}{\mathbf{A}}) + \mathbf{A}(\nabla \cdot \underset{\cdot}{\mathbf{B}}) - \underset{\cdot}{\mathbf{B}} (\nabla \cdot \mathbf{A}) = (\mathbf{B} \cdot \nabla)\underset{\cdot}{\mathbf{A}} - \mathbf{B} (\nabla \cdot \underset{\cdot}{\mathbf{A}}) + \mathbf{A}(\nabla \cdot \underset{\cdot}{\mathbf{B}}) - (\mathbf{A} \cdot \nabla ) \underset{\cdot}{\mathbf{B}}. $$
We can now remove the dots, and the identity is shown:
$$ \nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla)\mathbf{A} - \mathbf{B} (\nabla \cdot \mathbf{A}) + \mathbf{A}(\nabla \cdot \mathbf{B}) - (\mathbf{A} \cdot \nabla ) \mathbf{B}. $$
Personally, I don't like this method at all, and prefer the suffix notation, but I am still curious to where is comes from and whether you have come across it before, as I have only seen it be used at my own university.