is the gradient of scalar field f. A vector does not have a gradient in this sense. A vector can have curl and divergence, but not gradient. The gradient of a scalar field is a vector field.

Nabla, is defined as :

If you have a vector

, then you can have only two multiplicative operations with this vector and the nabla vector. Those are the dot product (

)and the cross product (

). And these represent divergence and curl respectively. It does not make sense to have scalar multiplcation between two vectors, which is what you are proposing with

.

You may have scalar multiplication between two scalars, and scalar multiplcation between a scalar and a vector (a lá,

, but you may not have scalar multiplication between two vectors.