What do gradient, curl, and div input and output? What do gradient, curl, and div input and output? (e.g. vector field or scalar function of several variables)
 A: I'll borrow from JohnD's answer since only a part is relevant here
\begin{align}
f\longrightarrow &\ \color{blue}{{\LARGE\boxed{\text{grad}}}} \longrightarrow \text{vector}\\
\text{vector}\longrightarrow &\ \color{blue}{{\LARGE\boxed{\text{div}}}} \longrightarrow \text{scalar}\\
\text{vector}\longrightarrow &\ \color{blue}{{\LARGE\boxed{\text{curl}}}} \longrightarrow \text{vector} 
\end{align}
A: Think about those operators as defined with the del ($\nabla$) operator: If $\nabla$ is a vector, $f$ is a scalar function of several variables and $g$ is a vector field, then $\text{grad} f = \nabla f$ is a scalar product of the scalar $f$ with the vector $\nabla$, $\text{div} g = \nabla \cdot g$ is the dot product of the vectors $\nabla$ and $g$, yielding a scalar, and $\text{curl} g = \nabla \times g$ is the cross product of the vectors $\nabla$ and $g$ resulting in a vector. So grad turns a scalar field into a vector field, div turns a vector field into a scalar field, and curl turns a vector field into another vector field. 
