This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
Note that the operation
must be interpreted as the two-argument inverse tangent, atan2.
2 cos θ
2 cos θ
cot θ
cot θ
{\displaystyle {\begin{aligned}\left({\frac {\partial T_{xx}}{\partial x}}+{\frac {\partial T_{yx}}{\partial y}}+{\frac {\partial T_{zx}}{\partial z}}\right)&{\hat {\mathbf {x} }}\\+\left({\frac {\partial T_{xy}}{\partial x}}+{\frac {\partial T_{yy}}{\partial y}}+{\frac {\partial T_{zy}}{\partial z}}\right)&{\hat {\mathbf {y} }}\\+\left({\frac {\partial T_{xz}}{\partial x}}+{\frac {\partial T_{yz}}{\partial y}}+{\frac {\partial T_{zz}}{\partial z}}\right)&{\hat {\mathbf {z} }}\end{aligned}}}
φ ρ
ρ φ
ρ φ
φ ρ
cot θ
cot θ
φ θ
θ φ
θ φ
φ θ
{\displaystyle {\begin{aligned}\left[{\frac {\partial T_{rr}}{\partial r}}+2{\frac {T_{rr}}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta r}}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta r}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi r}}{\partial \varphi }}-{\frac {1}{r}}(T_{\theta \theta }+T_{\varphi \varphi })\right]&{\hat {\mathbf {r} }}\\+\left[{\frac {\partial T_{r\theta }}{\partial r}}+2{\frac {T_{r\theta }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \theta }}{\partial \theta }}+{\frac {\cot \theta }{r}}T_{\theta \theta }+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \theta }}{\partial \varphi }}+{\frac {T_{\theta r}}{r}}-{\frac {\cot \theta }{r}}T_{\varphi \varphi }\right]&{\hat {\boldsymbol {\theta }}}\\+\left[{\frac {\partial T_{r\varphi }}{\partial r}}+2{\frac {T_{r\varphi }}{r}}+{\frac {1}{r}}{\frac {\partial T_{\theta \varphi }}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial T_{\varphi \varphi }}{\partial \varphi }}+{\frac {T_{\varphi r}}{r}}+{\frac {\cot \theta }{r}}(T_{\theta \varphi }+T_{\varphi \theta })\right]&{\hat {\boldsymbol {\varphi }}}\end{aligned}}}
{\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 {A_{x}(x+dx)\,dy\,dz-A_{x}(x)\,dy\,dz+A_{y}(y+dy)\,dx\,dz-A_{y}(y)\,dx\,dz+A_{z}(z+dz)\,dx\,dy-A_{z}(z)\,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}}&={\frac {A_{z}(y+dy)\,dz-A_{z}(y)\,dz+A_{y}(z)\,dy-A_{y}(z+dz)\,dy}{dy\,dz}}\\&={\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\end{aligned}}}
{\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 {A_{\rho }(\rho +d\rho )(\rho +d\rho )\,d\phi \,dz-A_{\rho }(\rho )\rho \,d\phi \,dz+A_{\phi }(\phi +d\phi )\,d\rho \,dz-A_{\phi }(\phi )\,d\rho \,dz+A_{z}(z+dz)\,d\rho \,(\rho +d\rho /2)\,d\phi -A_{z}(z)\,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}}}
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\theta }=\lim _{S^{\perp {\boldsymbol {\hat {\theta }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\phi }(r)r\sin \theta \,d\phi +A_{r}(\phi +d\phi )\,dr-A_{\phi }(r+dr)(r+dr)\sin \theta \,d\phi -A_{r}(\phi )\,dr}{dr\,r\sin \theta \,d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {1}{r}}{\frac {\partial (rA_{\phi })}{\partial r}}\end{aligned}}}
The unit vector of a coordinate parameter u is defined in such a way that a small positive change in u causes the position vector
where s is the arc length parameter.
For two sets of coordinate systems
, according to chain rule,
Then divide on both sides by