This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.

Note that the operation ${\displaystyle \arctan \left({\frac {A}{B}}\right)}$ must be interpreted as the two-argument inverse tangent, atan2.

$${\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\[1ex]&={\frac {\left[A_{x}(x{+}dx)-A_{x}(x)\right]dy\,dz+\left[A_{y}(y{+}dy)-A_{y}(y)\right]dx\,dz+\left[A_{z}(z{+}dz)-A_{z}(z)\right]dx\,dy}{dx\,dy\,dz}}\\&={\frac {\partial A_{x}}{\partial x}}+{\frac {\partial A_{y}}{\partial y}}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}$$

$${\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{x}&=\lim _{S^{\perp \mathbf {\hat {x}} }\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\[1ex]&={\frac {\left[A_{z}(y{+}dy)-A_{z}(y)\right]dz-\left[A_{y}(z{+}dz)-A_{y}(z)\right]dy}{dy\,dz}}\\&={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\end{aligned}}}$$

The expressions for ${\displaystyle (\operatorname {curl} \mathbf {A} )_{y}}$ and ${\displaystyle (\operatorname {curl} \mathbf {A} )_{z}}$ are found in the same way.

$${\displaystyle {\begin{aligned}\operatorname {div} \mathbf {A} &=\lim _{V\to 0}{\frac {\iint _{\partial V}\mathbf {A} \cdot d\mathbf {S} }{\iiint _{V}dV}}\\&={\frac {\left[A_{\rho }(\rho {+}d\rho )(\rho {+}d\rho )-A_{\rho }(\rho )\rho \right]d\phi \,dz+\left[A_{\phi }(\phi {+}d\phi )-A_{\phi }(\phi )\right]d\rho \,dz+\left[A_{z}(z{+}dz)-A_{z}(z)\right]d\rho (\rho {+}d\rho /2)\,d\phi }{\rho \,d\phi \,d\rho \,dz}}\\&={\frac {1}{\rho }}{\frac {\partial (\rho A_{\rho })}{\partial \rho }}+{\frac {1}{\rho }}{\frac {\partial A_{\phi }}{\partial \phi }}+{\frac {\partial A_{z}}{\partial z}}\end{aligned}}}$$

$${\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\rho }&=\lim _{S^{\perp {\hat {\boldsymbol {\rho }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}}{\iint _{S}dS}}\\[1ex]&={\frac {A_{\phi }(z)\left(\rho +d\rho \right)\,d\phi -A_{\phi }(z+dz)\left(\rho +d\rho \right)\,d\phi +A_{z}(\phi +d\phi )\,dz-A_{z}(\phi )\,dz}{\left(\rho +d\rho \right)\,d\phi \,dz}}\\[1ex]&=-{\frac {\partial A_{\phi }}{\partial z}}+{\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}\end{aligned}}}$$

$${\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }&=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d{\boldsymbol {\ell }}}{\iint _{S}dS}}\\&={\frac {A_{z}(\rho )\,dz-A_{z}(\rho +d\rho )\,dz+A_{\rho }(z+dz)\,d\rho -A_{\rho }(z)\,d\rho }{d\rho \,dz}}\\&=-{\frac {\partial A_{z}}{\partial \rho }}+{\frac {\partial A_{\rho }}{\partial z}}\end{aligned}}}$$