What is the expanded form of (v ∙ grad)w?
Since we have a dot product involving v, I assume that v is a vector but then (v- grad) evaluates to a scalar operator and w could be either a vector or a scalar.
Let and
If w is a vector, then this is a scalar product- the scalar multiplying each part of the vector:
If w is a scalar function then this is a product of two scalars: