Del or Nabla

Slightly ambiguous vector differential operator $\nabla$, defined as the vector operator $(\frac{\partial }{\partial x_1},\cdots ,\frac{\partial }{\partial x_n})$, which can represent a few different operations:

• Gradient of a scalar field $f:{\mathbb{R}}^n\to \mathbb{R}$, where $\nabla f$ = $(\frac{\partial }{\partial x_1}f,\cdots ,\frac{\partial }{\partial x_n}f)$
• Jacobian of a Vector field $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$, where $\nabla f$ = $J_f$. The Jacobian equals the Gradient when $m=1$.
• Divergence of a Vector field $\overrightarrow{F}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ , where $\nabla \cdot \overrightarrow{v}=(\frac{\partial }{\partial x_1}{F}_1,\cdots ,\frac{\partial }{\partial x_n}{F}_n)$,
• Curl of a Vector field $\overrightarrow{F}:{\mathbb{R}}^3\to {\mathbb{R}}^3$ , where $\nabla \times \overrightarrow{v}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right)e_x+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right)e_y+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right)e_z$,

In the above $e_x,e_y,e_z$ are the 3 vectors of length $1$ along the $x,y,z$ axis, respectively