Del is denoted by \nabla, it is a vector of derivatives of each component: \nabla = \frac{\delta}{\delta x} \hat{\imath} + \frac{\delta}{\delta y} \hat{\jmath} + \frac{\delta}{\delta z} \hat{k}
It doesn’t exist by itself, it will appear next to a vector or a multivariate function.
\Delta = \nabla \nabla = \nabla^2
Curl (sometimes called rotation) describes the rotation of a vector field.
Let \vec{F} = [F_x, F_y, F_z], then:
curl(F) = \nabla \times \vec{F} = \begin{vmatrix} \hat{\imath} & \hat{\jmath} & \hat{k} \\ \frac{\delta}{\delta x} & \frac{\delta}{\delta y} & \frac{\delta}{\delta x} \\ F_x & F_y & F_z \\ \end{vmatrix} = \begin{bmatrix} \frac{\delta F_z}{\delta y} - \frac{\delta F_y}{\delta z} \\ \frac{\delta F_x}{\delta z} - \frac{\delta F_z}{\delta x} \\ \frac{\delta F_y}{\delta x} - \frac{\delta F_x}{\delta y} \\ \end{bmatrix}
Gradient of a scalar field f shows the direction to the local maximum of f.
grad(f) = \nabla f = \frac{\delta f}{\delta x}\hat{\imath} + \frac{\delta f}{\delta y}\hat{\jmath} + \frac{\delta f}{\delta z}\hat{k}
Divergence of a vector field \vec v is a measure of its increase in the direction it points
div(\vec v) = \nabla \cdot \vec v = \frac{\delta v_x}{\delta x} + \frac{\delta v_y}{\delta y} + \frac{\delta v_z}{\delta z}
\nabla \times \vec E = -\frac{\delta \vec B}{\delta t}
\nabla \times \vec H = \vec J + \frac{\delta \vec D}{\delta t}
\nabla \cdot \vec D = \rho_v
\nabla \cdot \vec B = 0
\nabla \times \underline{\vec E} = -j\omega \underline{\vec B}
\nabla \times \underline{\vec H} = \underline{\vec J} + j\omega \underline{\vec D}
\nabla \cdot \underline{\vec D} = \rho_v
\nabla \cdot \underline{\vec B} = 0