d V = d x d y d z {\displaystyle \mathrm {d} V=\mathrm {d} x\mathrm {d} y\mathrm {d} z}
∇ → ψ = ( ∂ ψ ∂ x , ∂ ψ ∂ y , ∂ ψ ∂ z ) {\displaystyle {\vec {\nabla }}\psi =\left({\frac {\partial \psi }{\partial x}},{\frac {\partial \psi }{\partial y}},{\frac {\partial \psi }{\partial z}}\right)}
∇ → ⋅ v → = ∂ v x ∂ x + ∂ v y ∂ y + ∂ v z ∂ z {\displaystyle {\vec {\nabla }}\cdot {\vec {v}}={\frac {\partial v_{x}}{\partial x}}+{\frac {\partial v_{y}}{\partial y}}+{\frac {\partial v_{z}}{\partial z}}} , 有時記作 ∇ v {\displaystyle {\nabla }\mathrm {v} }
∇ → ⋅ v → = ( ∂ v z ∂ y − ∂ v y ∂ z , ∂ v x ∂ z − ∂ v z ∂ x , ∂ v y ∂ x − ∂ v x ∂ y ) {\displaystyle {\vec {\nabla }}\cdot {\vec {v}}=\left({\frac {\partial v_{z}}{\partial y}}-{\frac {\partial v_{y}}{\partial z}},{\frac {\partial v_{x}}{\partial z}}-{\frac {\partial v_{z}}{\partial x}},{\frac {\partial v_{y}}{\partial x}}-{\frac {\partial v_{x}}{\partial y}}\right)}
Δ ψ = ∂ 2 ψ x ∂ x 2 + ∂ 2 ψ y ∂ y 2 + ∂ 2 ψ z ∂ z 2 {\displaystyle \Delta \psi ={\frac {\partial ^{2}\psi _{x}}{\partial x^{2}}}+{\frac {\partial ^{2}\psi _{y}}{\partial y^{2}}}+{\frac {\partial ^{2}\psi _{z}}{\partial z^{2}}}}
轉換為新座標系 ( u 1 , u 2 , u 3 ) {\displaystyle (u_{1},u_{2},u_{3})} 的一般變換關係為：
d V = d u 1 U 1 d u 2 U 2 d u 3 U 3 {\displaystyle \mathrm {d} V={\frac {\mathrm {d} u_{1}}{U_{1}}}{\frac {\mathrm {d} u_{2}}{U_{2}}}{\frac {\mathrm {d} u_{3}}{U_{3}}}}
∇ → ψ = ( U 1 ∂ ψ ∂ u 1 , U 2 ∂ ψ ∂ u 2 , U 3 ∂ ψ ∂ u 3 ) {\displaystyle {\vec {\nabla }}\psi =\left(U_{1}{\frac {\partial \psi }{\partial u_{1}}},U_{2}{\frac {\partial \psi }{\partial u_{2}}},U_{3}{\frac {\partial \psi }{\partial u_{3}}}\right)}
∇ → ⋅ v → = U 1 U 2 U 3 ( ∂ ∂ u 1 v u 1 U 2 U 3 + ∂ ∂ u 2 v u 2 U 1 U 3 + ∂ ∂ u 3 v u 3 U 1 U 2 ) {\displaystyle {\vec {\nabla }}\cdot {\vec {v}}=U_{1}U_{2}U_{3}\left({\frac {\partial }{\partial u_{1}}}{\frac {v_{u_{1}}}{U_{2}U_{3}}}+{\frac {\partial }{\partial u_{2}}}{\frac {v_{u_{2}}}{U_{1}U_{3}}}+{\frac {\partial }{\partial u_{3}}}{\frac {v_{u_{3}}}{U_{1}U_{2}}}\right)} ∇ → × v → = ( U 2 U 3 ( ∂ ∂ u 2 v u 3 U 3 − ∂ ∂ u 3 v u 2 U 2 ) , U 1 U 3 ( ∂ ∂ u 3 v u 1 U 1 − ∂ ∂ u 1 v u 3 U 3 ) , U 1 U 2 ( ∂ ∂ u 1 v u 2 U 2 − ∂ ∂ u 2 v u 1 U 1 ) ) {\displaystyle {\vec {\nabla }}\times {\vec {v}}=\left(U_{2}U_{3}({\frac {\partial }{\partial u_{2}}}{\frac {v_{u_{3}}}{U_{3}}}-{\frac {\partial }{\partial u_{3}}}{\frac {v_{u_{2}}}{U_{2}}}),U_{1}U_{3}({\frac {\partial }{\partial u_{3}}}{\frac {v_{u_{1}}}{U_{1}}}-{\frac {\partial }{\partial u_{1}}}{\frac {v_{u_{3}}}{U_{3}}}),U_{1}U_{2}({\frac {\partial }{\partial u_{1}}}{\frac {v_{u_{2}}}{U_{2}}}-{\frac {\partial }{\partial u_{2}}}{\frac {v_{u_{1}}}{U_{1}}})\right)} Δ ψ = U 1 U 2 U 3 [ ∂ ∂ u 1 ( U 1 U 2 U 3 ∂ ψ ∂ u 1 ) + ∂ ∂ u 2 ( U 2 U 1 U 3 ∂ ψ ∂ u 2 ) + ∂ ∂ u 3 ( U 3 U 1 U 2 ∂ ψ ∂ u 3 ) ] {\displaystyle \Delta \psi =U_{1}U_{2}U_{3}\left[{\frac {\partial }{\partial u_{1}}}({\frac {U_{1}}{U_{2}U_{3}}}{\frac {\partial \psi }{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {U_{2}}{U_{1}U_{3}}}{\frac {\partial \psi }{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {U_{3}}{U_{1}U_{2}}}{\frac {\partial \psi }{\partial u_{3}}})\right]} ,
其中
U 1 − 1 = ( ∂ x ∂ u 1 ) 2 + ( ∂ y ∂ u 1 ) 2 + ( ∂ z ∂ u 1 ) 2 , {\displaystyle U_{1}^{-1}={\sqrt {\left({\frac {\partial x}{\partial u_{1}}}\right)^{2}+\left({\frac {\partial y}{\partial u_{1}}}\right)^{2}+\left({\frac {\partial z}{\partial u_{1}}}\right)^{2}}},}
U 2 − 1 = ( ∂ x ∂ u 2 ) 2 + ( ∂ y ∂ u 2 ) 2 + ( ∂ z ∂ u 2 ) 2 , {\displaystyle U_{2}^{-1}={\sqrt {\left({\frac {\partial x}{\partial u_{2}}}\right)^{2}+\left({\frac {\partial y}{\partial u_{2}}}\right)^{2}+\left({\frac {\partial z}{\partial u_{2}}}\right)^{2}}},}
U 3 − 1 = ( ∂ x ∂ u 3 ) 2 + ( ∂ y ∂ u 3 ) 2 + ( ∂ z ∂ u 3 ) 2 , {\displaystyle U_{3}^{-1}={\sqrt {\left({\frac {\partial x}{\partial u_{3}}}\right)^{2}+\left({\frac {\partial y}{\partial u_{3}}}\right)^{2}+\left({\frac {\partial z}{\partial u_{3}}}\right)^{2}}},}
v u 1 = v x U 1 ∂ x ∂ u 1 + v y U 1 ∂ y ∂ u 1 + v z U 1 ∂ z ∂ u 1 {\displaystyle v_{u_{1}}=v_{x}U_{1}{\frac {\partial x}{\partial u_{1}}}+v_{y}U_{1}{\frac {\partial y}{\partial u_{1}}}+v_{z}U_{1}{\frac {\partial z}{\partial u_{1}}}}
v u 2 = v x U 2 ∂ x ∂ u 2 + v y U 2 ∂ y ∂ u 2 + v z U 2 ∂ z ∂ u 2 {\displaystyle v_{u_{2}}=v_{x}U_{2}{\frac {\partial x}{\partial u_{2}}}+v_{y}U_{2}{\frac {\partial y}{\partial u_{2}}}+v_{z}U_{2}{\frac {\partial z}{\partial u_{2}}}}
v u 3 = v x U 3 ∂ x ∂ u 3 + v y U 3 ∂ y ∂ u 3 + v z U 3 ∂ z ∂ u 3 {\displaystyle v_{u_{3}}=v_{x}U_{3}{\frac {\partial x}{\partial u_{3}}}+v_{y}U_{3}{\frac {\partial y}{\partial u_{3}}}+v_{z}U_{3}{\frac {\partial z}{\partial u_{3}}}}
柱座標 ( u 1 = r , u 2 = φ , u 3 = z ) {\displaystyle (u_{1}=r,u_{2}=\varphi ,u_{3}=z)} 與直角座標的關係為
( x , y , z ) = ( r cos φ , r sin φ , z ) {\displaystyle (x,y,z)=(r\cos \varphi ,r\sin \varphi ,z)}
d V = r d r d φ d z {\displaystyle \mathrm {d} V=r\ \mathrm {d} r\mathrm {d} \varphi \mathrm {d} z}
∇ → ψ = ( ∂ ψ ∂ r , 1 r ∂ ψ ∂ φ , ∂ ψ ∂ z ) {\displaystyle {\vec {\nabla }}\psi =\left({\frac {\partial \psi }{\partial r}},{\frac {1}{r}}{\frac {\partial \psi }{\partial \varphi }},{\frac {\partial \psi }{\partial z}}\right)}
∇ → ⋅ v → = 1 r ∂ ∂ r ( r v r ) + 1 r ∂ v φ ∂ φ + ∂ v z ∂ z {\displaystyle {\vec {\nabla }}\cdot {\vec {v}}={\frac {1}{r}}{\frac {\partial }{\partial r}}(rv_{r})+{\frac {1}{r}}{\frac {\partial v_{\varphi }}{\partial \varphi }}+{\frac {\partial v_{z}}{\partial z}}}
∇ → × v → = ( 1 r ∂ v z ∂ φ − ∂ v ϕ v z , ∂ v r ∂ z − ∂ v z ∂ r , 1 r ∂ ∂ r ( r v ϕ ) − 1 r ∂ v r ∂ ϕ ) {\displaystyle {\vec {\nabla }}\times {\vec {v}}=\left({\frac {1}{r}}{\frac {\partial v_{z}}{\partial \varphi }}-{\frac {\partial v_{\phi }}{v_{z}}},{\frac {\partial v_{r}}{\partial z}}-{\frac {\partial v_{z}}{\partial r}},{\frac {1}{r}}{\frac {\partial }{\partial r}}(rv_{\phi })-{\frac {1}{r}}{\frac {\partial v_{r}}{\partial \phi }}\right)}
Δ ψ = ∂ 2 ψ ∂ r 2 + 1 r ∂ ψ ∂ r + 1 r 2 ∂ 2 ψ ∂ φ 2 + ∂ 2 ψ ∂ z 2 {\displaystyle \Delta \psi ={\frac {\partial ^{2}\psi }{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial \psi }{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\psi }{\partial \varphi ^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}}
球面極座標 ( u 1 = r , u 2 = θ , u 3 = φ ) {\displaystyle (u_{1}=r,u_{2}=\theta ,u_{3}=\varphi )} ，或簡稱為球座標，在 R 3 {\displaystyle \mathrm {R} ^{3}} 中具有球對稱性的系統（例如，在中心力影響下粒子的運動）特別有用。
( x , y , z ) = ( r sin θ cos φ , r sin θ sin φ , r cos θ ) {\displaystyle (x,y,z)=(r\sin \theta \cos \varphi ,r\sin \theta \sin \varphi ,r\cos \theta )}
U 1 − 1 = 1 , U 2 − 1 = r , U 3 − 1 = r sin θ {\displaystyle U_{1}^{-1}=1,\ U_{2}^{-1}=r,\ U_{3}^{-1}=r\sin \theta }
d V = r 2 sin θ d r d φ d θ {\displaystyle \mathrm {d} V=r^{2}\sin \theta \ \mathrm {d} r\mathrm {d} \varphi \mathrm {d} \theta }
∇ → ψ = ( ∂ ψ ∂ r , 1 r ∂ ψ ∂ θ , 1 r sin θ ∂ ψ ∂ φ ) {\displaystyle {\vec {\nabla }}\psi =\left({\frac {\partial \psi }{\partial r}},{\frac {1}{r}}{\frac {\partial \psi }{\partial \theta }},{\frac {1}{r\sin \theta }}{\frac {\partial \psi }{\partial \varphi }}\right)}
∇ → ⋅ v → = 1 r 2 ∂ ∂ r ( r 2 v r ) + 1 r sin θ ∂ ∂ θ ( v θ sin θ ) + 1 r sin θ ∂ v φ ∂ φ {\displaystyle {\vec {\nabla }}\cdot {\vec {v}}={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}(r^{2}v_{r})+{\frac {1}{r\sin \theta }}{\frac {\partial }{\partial \theta }}(v_{\theta }\sin \theta )+{\frac {1}{r\sin \theta }}{\frac {\partial v_{\varphi }}{\partial \varphi }}}
∇ → × v → = ( 1 r sin θ ( ∂ ∂ θ ( sin θ v φ ) − ∂ v θ ∂ φ ) , 1 r sin θ ( ∂ v r ∂ φ − sin θ ∂ ∂ r ( r v φ ) ) , 1 r ( ∂ ∂ r ( r v θ ) − ∂ v r ∂ θ ) ) {\displaystyle {\vec {\nabla }}\times {\vec {v}}=\left({\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}(\sin \theta v_{\varphi })-{\frac {\partial v_{\theta }}{\partial \varphi }}\right),{\frac {1}{r\sin \theta }}\left({\frac {\partial v_{r}}{\partial \varphi }}-\sin \theta {\frac {\partial }{\partial r}}(rv_{\varphi })\right),{\frac {1}{r}}\left({\frac {\partial }{\partial r}}(rv_{\theta })-{\frac {\partial v_{r}}{\partial \theta }}\right)\right)}
Δ ψ = 1 r 2 ∂ ∂ r ( r 2 ∂ ψ ∂ r ) + 1 r 2 sin θ ∂ ∂ θ ( sin θ ∂ ψ ∂ θ ) + 1 r 2 sin 2 θ ∂ 2 ψ ∂ φ 2 {\displaystyle \Delta \psi ={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial \psi }{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial \psi }{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}\psi }{\partial \varphi ^{2}}}}
, 有時記作 
的一般變換關係為：
![{\displaystyle \Delta \psi =U_{1}U_{2}U_{3}\left[{\frac {\partial }{\partial u_{1}}}({\frac {U_{1}}{U_{2}U_{3}}}{\frac {\partial \psi }{\partial u_{1}}})+{\frac {\partial }{\partial u_{2}}}({\frac {U_{2}}{U_{1}U_{3}}}{\frac {\partial \psi }{\partial u_{2}}})+{\frac {\partial }{\partial u_{3}}}({\frac {U_{3}}{U_{1}U_{2}}}{\frac {\partial \psi }{\partial u_{3}}})\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee0cb28a690df3c54792d953cee9613efe8c809)
,
與直角座標的關係為
大多數物理學約定中的球面座標。請注意，這與數學約定不同，在數學約定中 
和 
與所示圖形相比是互換的。
，或簡稱為球座標，在 
中具有球對稱性的系統（例如，在中心力影響下粒子的運動）特別有用。
