Operation
Cartesian coordinates (x , y , z )
Cylindrical coordinates (ρ , ϕ , z )
Spherical coordinates (r , θ , ϕ )
Parabolic cylindrical coordinates (σ , τ , z )
Definition
ρ
=
x
2
+
y
2
ϕ
=
arctan
(
y
/
x
)
z
=
z
{\displaystyle {\begin{aligned}\rho &={\sqrt {x^{2}+y^{2}}}\\\phi &=\arctan(y/x)\\z&=z\end{aligned}}}
x
=
ρ
cos
ϕ
y
=
ρ
sin
ϕ
z
=
z
{\displaystyle {\begin{aligned}x&=\rho \cos \phi \\y&=\rho \sin \phi \\z&=z\end{aligned}}}
x
=
r
sin
θ
cos
ϕ
y
=
r
sin
θ
sin
ϕ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}}}
x
=
σ
τ
y
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{aligned}x&=\sigma \tau \\y&={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}}
r
=
x
2
+
y
2
+
z
2
θ
=
arccos
(
z
/
r
)
ϕ
=
arctan
(
y
/
x
)
{\displaystyle {\begin{aligned}r&={\sqrt {x^{2}+y^{2}+z^{2}}}\\\theta &=\arccos(z/r)\\\phi &=\arctan(y/x)\end{aligned}}}
r
=
ρ
2
+
z
2
θ
=
arctan
(
ρ
/
z
)
ϕ
=
ϕ
{\displaystyle {\begin{aligned}r&={\sqrt {\rho ^{2}+z^{2}}}\\\theta &=\arctan {(\rho /z)}\\\phi &=\phi \end{aligned}}}
ρ
=
r
sin
θ
ϕ
=
ϕ
z
=
r
cos
θ
{\displaystyle {\begin{aligned}\rho &=r\sin \theta \\\phi &=\phi \\z&=r\cos \theta \end{aligned}}}
ρ
cos
ϕ
=
σ
τ
ρ
sin
ϕ
=
1
2
(
τ
2
−
σ
2
)
z
=
z
{\displaystyle {\begin{aligned}\rho \cos \phi &=\sigma \tau \\\rho \sin \phi &={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)\\z&=z\end{aligned}}}
Definition
ρ
^
=
x
x
^
+
y
y
^
x
2
+
y
2
ϕ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\boldsymbol {\hat {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}}
x
^
=
cos
ϕ
ρ
^
−
sin
ϕ
ϕ
^
y
^
=
sin
ϕ
ρ
^
+
cos
ϕ
ϕ
^
z
^
=
z
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\boldsymbol {\hat {\rho }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\boldsymbol {\hat {\rho }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}}
x
^
=
sin
θ
cos
ϕ
r
^
+
cos
θ
cos
ϕ
θ
^
−
sin
ϕ
ϕ
^
y
^
=
sin
θ
sin
ϕ
r
^
+
cos
θ
sin
ϕ
θ
^
+
cos
ϕ
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\boldsymbol {\hat {r}}}+\cos \theta \cos \phi {\boldsymbol {\hat {\theta }}}-\sin \phi {\boldsymbol {\hat {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\boldsymbol {\hat {r}}}+\cos \theta \sin \phi {\boldsymbol {\hat {\theta }}}+\cos \phi {\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta {\boldsymbol {\hat {r}}}-\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}}
σ
^
=
τ
x
^
−
σ
y
^
τ
2
+
σ
2
τ
^
=
σ
x
^
+
τ
y
^
τ
2
+
σ
2
z
^
=
z
^
{\displaystyle {\begin{aligned}{\boldsymbol {\hat {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\boldsymbol {\hat {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\\mathbf {\hat {z}} &=\mathbf {\hat {z}} \end{aligned}}}
r
^
=
x
x
^
+
y
y
^
+
z
z
^
x
2
+
y
2
+
z
2
θ
^
=
x
z
x
^
+
y
z
y
^
−
(
x
2
+
y
2
)
z
^
x
2
+
y
2
x
2
+
y
2
+
z
2
ϕ
^
=
−
y
x
^
+
x
y
^
x
2
+
y
2
{\displaystyle {\begin{aligned}\mathbf {\hat {r}} &={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z\mathbf {\hat {z}} }{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right)\mathbf {\hat {z}} }{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\boldsymbol {\hat {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}}
r
^
=
ρ
ρ
^
+
z
z
^
ρ
2
+
z
2
θ
^
=
z
ρ
^
−
ρ
z
^
ρ
2
+
z
2
ϕ
^
=
ϕ
^
{\displaystyle {\begin{aligned}\mathbf {\hat {r}} &={\frac {\rho {\boldsymbol {\hat {\rho }}}+z\mathbf {\hat {z}} }{\sqrt {\rho ^{2}+z^{2}}}}\\{\boldsymbol {\hat {\theta }}}&={\frac {z{\boldsymbol {\hat {\rho }}}-\rho \mathbf {\hat {z}} }{\sqrt {\rho ^{2}+z^{2}}}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\end{aligned}}}
ρ
^
=
sin
θ
r
^
+
cos
θ
θ
^
ϕ
^
=
ϕ
^
z
^
=
cos
θ
r
^
−
sin
θ
θ
^
{\displaystyle {\begin{aligned}{\boldsymbol {\hat {\rho }}}&=\sin \theta \mathbf {\hat {r}} +\cos \theta {\boldsymbol {\hat {\theta }}}\\{\boldsymbol {\hat {\phi }}}&={\boldsymbol {\hat {\phi }}}\\\mathbf {\hat {z}} &=\cos \theta \mathbf {\hat {r}} -\sin \theta {\boldsymbol {\hat {\theta }}}\end{aligned}}}
{\displaystyle {\begin{matrix}\end{matrix}}}
A vector field
A
{\displaystyle \mathbf {A} }
A
x
x
^
+
A
y
y
^
+
A
z
z
^
{\displaystyle A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}\mathbf {\hat {z}} }
A
ρ
ρ
^
+
A
ϕ
ϕ
^
+
A
z
z
^
{\displaystyle A_{\rho }{\boldsymbol {\hat {\rho }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}+A_{z}\mathbf {\hat {z}} }
A
r
r
^
+
A
θ
θ
^
+
A
ϕ
ϕ
^
{\displaystyle A_{r}{\boldsymbol {\hat {r}}}+A_{\theta }{\boldsymbol {\hat {\theta }}}+A_{\phi }{\boldsymbol {\hat {\phi }}}}
A
σ
σ
^
+
A
τ
τ
^
+
A
ϕ
z
^
{\displaystyle A_{\sigma }{\boldsymbol {\hat {\sigma }}}+A_{\tau }{\boldsymbol {\hat {\tau }}}+A_{\phi }\mathbf {\hat {z}} }
f
{\displaystyle f}
scalar field Gradient
∇
f
{\displaystyle \nabla f}
∂
f
∂
x
x
^
+
∂
f
∂
y
y
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}\mathbf {\hat {z}} }
∂
f
∂
ρ
ρ
^
+
1
ρ
∂
f
∂
ϕ
ϕ
^
+
∂
f
∂
z
z
^
{\displaystyle {\partial f \over \partial \rho }{\boldsymbol {\hat {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}+{\partial f \over \partial z}\mathbf {\hat {z}} }
∂
f
∂
r
r
^
+
1
r
∂
f
∂
θ
θ
^
+
1
r
sin
θ
∂
f
∂
ϕ
ϕ
^
{\displaystyle {\partial f \over \partial r}{\boldsymbol {\hat {r}}}+{1 \over r}{\partial f \over \partial \theta }{\boldsymbol {\hat {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\boldsymbol {\hat {\phi }}}}
1
σ
2
+
τ
2
∂
f
∂
σ
σ
^
+
1
σ
2
+
τ
2
∂
f
∂
τ
τ
^
+
∂
f
∂
z
z
^
{\displaystyle {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\boldsymbol {\hat {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\boldsymbol {\hat {\tau }}}+{\partial f \over \partial z}\mathbf {\hat {z}} }
Divergence
∇
⋅
A
{\displaystyle \nabla \cdot \mathbf {A} }
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\displaystyle {\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}}
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
ϕ
∂
ϕ
+
∂
A
z
∂
z
{\displaystyle {1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}}
1
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
∂
θ
(
A
θ
sin
θ
)
+
1
r
sin
θ
∂
A
ϕ
∂
ϕ
{\displaystyle {1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }}
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
σ
+
∂
(
σ
2
+
τ
2
A
τ
)
∂
τ
)
+
∂
A
z
∂
z
{\displaystyle {\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}}
Curl
∇
×
A
{\displaystyle \nabla \times \mathbf {A} }
(
∂
A
z
∂
y
−
∂
A
y
∂
z
)
x
^
+
+
(
∂
A
x
∂
z
−
∂
A
z
∂
x
)
y
^
+
+
(
∂
A
y
∂
x
−
∂
A
x
∂
y
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right)&{\hat {\mathbf {x} }}+\\+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right)&{\hat {\mathbf {y} }}+\\+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right)&\mathbf {\hat {z}} \end{aligned}}}
(
1
ρ
∂
A
z
∂
ϕ
−
∂
A
ϕ
∂
z
)
ρ
^
+
(
∂
A
ρ
∂
z
−
∂
A
z
∂
ρ
)
ϕ
^
+
1
ρ
(
∂
(
ρ
A
ϕ
)
∂
ρ
−
∂
A
ρ
∂
ϕ
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right)&{\boldsymbol {\hat {\rho }}}\\+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right)&{\boldsymbol {\hat {\phi }}}\\+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right)&\mathbf {\hat {z}} \end{aligned}}}
1
r
sin
θ
(
∂
∂
θ
(
A
ϕ
sin
θ
)
−
∂
A
θ
∂
ϕ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
ϕ
−
∂
∂
r
(
r
A
ϕ
)
)
θ
^
+
1
r
(
∂
∂
r
(
r
A
θ
)
−
∂
A
r
∂
θ
)
ϕ
^
{\displaystyle {\begin{aligned}{\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right)&{\boldsymbol {\hat {r}}}\\+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right)&{\boldsymbol {\hat {\theta }}}\\+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right)&{\boldsymbol {\hat {\phi }}}\end{aligned}}}
(
1
σ
2
+
τ
2
∂
A
z
∂
τ
−
∂
A
τ
∂
z
)
σ
^
−
(
1
σ
2
+
τ
2
∂
A
z
∂
σ
−
∂
A
σ
∂
z
)
τ
^
+
1
σ
2
+
τ
2
(
∂
(
σ
2
+
τ
2
A
σ
)
∂
τ
−
∂
(
σ
2
+
τ
2
A
τ
)
∂
σ
)
z
^
{\displaystyle {\begin{aligned}\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right)&{\boldsymbol {\hat {\sigma }}}\\-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right)&{\boldsymbol {\hat {\tau }}}\\+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}\left({\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }\right)}{\partial \tau }}-{\frac {\partial \left({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }\right)}{\partial \sigma }}\right)&\mathbf {\hat {z}} \end{aligned}}}
Laplace operator
Δ
f
≡
∇
2
f
{\displaystyle \Delta f\equiv \nabla ^{2}f}
∂
2
f
∂
x
2
+
∂
2
f
∂
y
2
+
∂
2
f
∂
z
2
{\displaystyle {\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
ρ
∂
∂
ρ
(
ρ
∂
f
∂
ρ
)
+
1
ρ
2
∂
2
f
∂
ϕ
2
+
∂
2
f
∂
z
2
{\displaystyle {1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}}
1
r
2
∂
∂
r
(
r
2
∂
f
∂
r
)
+
1
r
2
sin
θ
∂
∂
θ
(
sin
θ
∂
f
∂
θ
)
+
1
r
2
sin
2
θ
∂
2
f
∂
ϕ
2
{\displaystyle {1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}}
1
σ
2
+
τ
2
(
∂
2
f
∂
σ
2
+
∂
2
f
∂
τ
2
)
+
∂
2
f
∂
z
2
{\displaystyle {\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}}
Vector Laplacian
Δ
A
≡
∇
2
A
{\displaystyle \Delta \mathbf {A} \equiv \nabla ^{2}\mathbf {A} }
Δ
A
x
x
^
+
Δ
A
y
y
^
+
Δ
A
z
z
^
{\displaystyle \Delta A_{x}{\hat {\mathbf {x} }}+\Delta A_{y}{\hat {\mathbf {y} }}+\Delta A_{z}\mathbf {\hat {z}} }
Modèle:Collapsible section
Modèle:Collapsible section
Material derivative [ 1]
(
A
⋅
∇
)
B
{\displaystyle (\mathbf {A} \cdot \nabla )\mathbf {B} }
A
⋅
∇
B
x
x
^
+
A
⋅
∇
B
y
y
^
+
A
⋅
∇
B
z
z
^
{\displaystyle \mathbf {A} \cdot \nabla B_{x}{\hat {\mathbf {x} }}+\mathbf {A} \cdot \nabla B_{y}{\hat {\mathbf {y} }}+\mathbf {A} \cdot \nabla B_{z}{\hat {\mathbf {z} }}}
Modèle:Collapsible section
Modèle:Collapsible section
Differential displacement
d
l
=
d
x
x
^
+
d
y
y
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,\mathbf {\hat {z}} }
d
l
=
d
ρ
ρ
^
+
ρ
d
ϕ
ϕ
^
+
d
z
z
^
{\displaystyle d\mathbf {l} =d\rho \,{\boldsymbol {\hat {\rho }}}+\rho \,d\phi \,{\boldsymbol {\hat {\phi }}}+dz\,\mathbf {\hat {z}} }
d
l
=
d
r
r
^
+
r
d
θ
θ
^
+
r
sin
θ
d
ϕ
ϕ
^
{\displaystyle d\mathbf {l} =dr\,\mathbf {\hat {r}} +r\,d\theta \,{\boldsymbol {\hat {\theta }}}+r\,\sin \theta \,d\phi \,{\boldsymbol {\hat {\phi }}}}
d
l
=
σ
2
+
τ
2
d
σ
σ
^
+
σ
2
+
τ
2
d
τ
τ
^
+
d
z
z
^
{\displaystyle d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\boldsymbol {\hat {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\boldsymbol {\hat {\tau }}}+dz\,\mathbf {\hat {z}} }
Differential normal area
d
S
{\displaystyle d\mathbf {S} }
d
y
d
z
x
^
+
d
x
d
z
y
^
+
d
x
d
y
z
^
{\displaystyle {\begin{aligned}dy\,dz&{\hat {\mathbf {x} }}\\+dx\,dz&{\hat {\mathbf {y} }}\\+dx\,dy&\mathbf {\hat {z}} \end{aligned}}}
ρ
d
ϕ
d
z
ρ
^
+
d
ρ
d
z
ϕ
^
+
ρ
d
ρ
d
ϕ
z
^
{\displaystyle {\begin{aligned}\rho \,d\phi \,dz&{\boldsymbol {\hat {\rho }}}\\+d\rho \,dz&{\boldsymbol {\hat {\phi }}}\\+\rho \,d\rho \,d\phi &\mathbf {\hat {z}} \end{aligned}}}
r
2
sin
θ
d
θ
d
ϕ
r
^
+
r
sin
θ
d
r
d
ϕ
θ
^
+
r
d
r
d
θ
ϕ
^
{\displaystyle {\begin{aligned}r^{2}\sin \theta \,d\theta \,d\phi &\mathbf {\hat {r}} \\+r\sin \theta \,dr\,d\phi &{\boldsymbol {\hat {\theta }}}\\+r\,dr\,d\theta &{\boldsymbol {\hat {\phi }}}\end{aligned}}}
σ
2
+
τ
2
d
τ
d
z
σ
^
+
σ
2
+
τ
2
d
σ
d
z
τ
^
+
(
σ
2
+
τ
2
)
d
σ
d
τ
z
^
{\displaystyle {\begin{aligned}{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz&{\boldsymbol {\hat {\sigma }}}\\+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz&{\boldsymbol {\hat {\tau }}}\\+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau &\mathbf {\hat {z}} \end{aligned}}}
Differential volume
d
V
{\displaystyle dV}
d
x
d
y
d
z
{\displaystyle dx\,dy\,dz}
ρ
d
ρ
d
ϕ
d
z
{\displaystyle \rho \,d\rho \,d\phi \,dz}
r
2
sin
θ
d
r
d
θ
d
ϕ
{\displaystyle r^{2}\sin \theta \,dr\,d\theta \,d\phi }
(
σ
2
+
τ
2
)
d
σ
d
τ
d
z
{\displaystyle \left(\sigma ^{2}+\tau ^{2}\right)d\sigma \,d\tau \,dz}
Non-trivial calculation rules:
div
grad
f
≡
∇
⋅
∇
f
=
∇
2
f
≡
Δ
f
{\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f}
curl
grad
f
≡
∇
×
∇
f
=
0
{\displaystyle \operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0} }
div
curl
A
≡
∇
⋅
(
∇
×
A
)
=
0
{\displaystyle \operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0}
curl
curl
A
≡
∇
×
(
∇
×
A
)
=
∇
(
∇
⋅
A
)
−
∇
2
A
{\displaystyle \operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A} }
Lagrange's formula for del)
Δ
(
f
g
)
=
f
Δ
g
+
2
∇
f
⋅
∇
g
+
g
Δ
f
{\displaystyle \Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f}
