How to interperet calculus thing I have $\nabla \times (f\mathbb{F})$ where $f$ is a twice continuously differentiable scalar field and $\mathbb{F}$ is a twice continuously differentiable vector field.
Is it right to interpret $f$ as $ax+by+cz$ where $a,b,c\in \mathbb{R}^3$ and $\mathbb{F}$ as $(\alpha,\beta,\gamma)\in\mathbb{R}^3$
Hence $\nabla \times (f\mathbb{F}) = \nabla \times (\alpha fi+\beta fj + \gamma f k) $?
Not sure how to treat the form of any of these three, or how the operators will apply.

No, it is not. You have to use a general interpretation as $f(x,y,z)$ twice continuously differentiable. Your interpretation assumes that $f$ is a linear functional, putting a strong unnecessary restriction.
If $$\overrightarrow{F} = P(x,y,z) \overrightarrow{i} + Q(x,y,z) \overrightarrow{j} + R(x,y,z) \overrightarrow{k}$$ then $$f \cdot \overrightarrow{F} = f(x,y,z) P(x,y,z) \overrightarrow{i} + f(x,y,z) Q(x,y,z) \overrightarrow{j} + f(x,y,z) R(x,y,z) \overrightarrow{k}.$$
I've explicitly written all dependences. In order to make sense of $\nabla \times (f \overrightarrow{F})$, think of the following:


*

*Your operation has to return a vector.

*You have to apply a differential operation to $f$ and $\overrightarrow{F}$ while combining them to result in a vector.

*What differential operator can you apply to $f$ to turn it into a vector?

*What differential operator can you apply to $\overrightarrow{F}$ while keeping it a vector?

*What kind of product can you have between $f, \overrightarrow{F}$ and their transformations and resulting in vectors?

*Notice that you are applying a differential operator in a product, therefore you should apply a product rule. What should it look like?

