The gradient \(\nabla \) is vector operator. In Cartessian\begin {align*} \nabla & =\hat {e}_{x}\frac {\partial }{\partial x}+\hat {e}_{y}\frac {\partial }{\partial y}+\hat {e}_{z}\frac {\partial }{\partial z}\\ \nabla f & =\begin {pmatrix} \frac {\partial f}{\partial x}\\ \frac {\partial f}{\partial y}\\ \frac {\partial f}{\partial z}\end {pmatrix} \end {align*}
In Cylindrical\begin {align*} \nabla & =\hat {e}_{\rho }\frac {\partial }{\partial \rho }+\hat {e}_{\phi }\rho \frac {\partial }{\partial \phi }+\hat {e}_{z}\frac {\partial }{\partial z}\\ \nabla f & =\begin {pmatrix} \frac {\partial f}{\partial \rho }\\ \rho \frac {\partial f}{\partial \phi }\\ \frac {\partial f}{\partial z}\end {pmatrix} \end {align*}
In spherical\begin {align*} \nabla & =\hat {e}_{\rho }\frac {\partial }{\partial \rho }+\hat {e}_{\theta }\frac {1}{\rho }\frac {\partial }{\partial \theta }+\hat {e}_{\phi }\frac {1}{\rho \sin \theta }\frac {\partial }{\partial \phi }\\ \nabla f & =\begin {pmatrix} \frac {\partial f}{\partial \rho }\\ \frac {1}{\rho }\frac {\partial f}{\partial \theta }\\ \frac {1}{\rho \sin \theta }\frac {\partial f}{\partial \phi }\end {pmatrix} \end {align*}
For conservative force\[ F=-\nabla V \] Notice that \(-\int \bar {F}\cdot d\bar {r}=\int \nabla V\cdot d\bar {r}=\int _{from}^{to}dV=V\left (to\right ) -V\left (from\right ) \) also \({\displaystyle \oint } \bar {F}\cdot d\bar {r}=0\) for conservative force.
The curl in Cartessian\[ \nabla \times \bar {F}=\begin {vmatrix} \hat {e}_{x} & \hat {e}_{y} & \hat {e}_{z}\\ \frac {\partial }{\partial x} & \frac {\partial }{\partial y} & \frac {\partial }{\partial z}\\ F_{x} & F_{y} & F_{z}\end {vmatrix} \]
In Cylinderical
\[ \nabla \times \bar {F}=\begin {vmatrix} \hat {e}_{\rho } & \hat {e}_{\phi } & \hat {e}_{z}\\ \frac {\partial }{\partial \rho } & \frac {1}{\rho }\frac {\partial }{\partial \phi } & \frac {\partial }{\partial z}\\ F_{\rho } & F_{\phi } & F_{z}\end {vmatrix} \]
In Spherical
\[ \nabla \times \bar {F}=\begin {vmatrix} \hat {e}_{\rho } & \hat {e}_{\phi } & \hat {e}_{\theta }\\ \frac {\partial }{\partial \rho } & \frac {1}{\rho \sin \theta }\frac {\partial }{\partial \phi } & \frac {1}{\rho }\frac {\partial }{\partial \theta }\\ F_{\rho } & F_{\phi } & F_{\theta }\end {vmatrix} \]
Divergence This is scalar. see cha7b.pdf\[ \nabla \cdot \bar {F}\] Gauss law
From Wiki
It states that the flux of the electric field out of an arbitrary closed surface is proportional to the electric charge enclosed by the surface.
Gauss’s law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.\[ \overset {\text {surface integral}}{\overbrace {\int \int \bar {F}\cdot d\bar {s}}}=\int _{V}\left (\nabla \cdot \bar {F}\right ) dV \] Stoke’s theorem\[ \overset {\text {line integral}}{\overbrace {{\displaystyle \oint } \bar {F}\cdot d\bar {r}}}=\int _{S}\left (\nabla \times \bar {F}\right ) \cdot d\bar {s}\] Also divergence of the curl is zero.\[ \nabla \cdot \left (\nabla \times \bar {F}\right ) =0 \] From the net
The characteristic of a conservative field is that the line integral around every simple closed contour is zero. Since the curl is defined as a particular closed contour line integral, it follows that curl(gradF) equals zero.
And curl of a gradient is the zero vector.\[ \nabla \times \left (\nabla \bar {F}\right ) =\bar {0}\]