- Source: Del in cylindrical and spherical coordinates
This is a list of some vector calculus formulae for working with common curvilinear coordinate systems.
Notes
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ):
The polar angle is denoted by
θ
∈
[
0
,
π
]
{\displaystyle \theta \in [0,\pi ]}
: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
The azimuthal angle is denoted by
φ
∈
[
0
,
2
π
]
{\displaystyle \varphi \in [0,2\pi ]}
: it is the angle between the x-axis and the projection of the radial vector onto the xy-plane.
The function atan2(y, x) can be used instead of the mathematical function arctan(y/x) owing to its domain and image. The classical arctan function has an image of (−π/2, +π/2), whereas atan2 is defined to have an image of (−π, π].
Coordinate conversions
Note that the operation
arctan
(
A
B
)
{\displaystyle \arctan \left({\frac {A}{B}}\right)}
must be interpreted as the two-argument inverse tangent, atan2.
Unit vector conversions
Del formula
^α This page uses
θ
{\displaystyle \theta }
for the polar angle and
φ
{\displaystyle \varphi }
for the azimuthal angle, which is common notation in physics. The source that is used for these formulae uses
θ
{\displaystyle \theta }
for the azimuthal angle and
φ
{\displaystyle \varphi }
for the polar angle, which is common mathematical notation. In order to get the mathematics formulae, switch
θ
{\displaystyle \theta }
and
φ
{\displaystyle \varphi }
in the formulae shown in the table above.
^β Defined in Cartesian coordinates as
∂
i
A
⊗
e
i
{\displaystyle \partial _{i}\mathbf {A} \otimes \mathbf {e} _{i}}
. An alternative definition is
e
i
⊗
∂
i
A
{\displaystyle \mathbf {e} _{i}\otimes \partial _{i}\mathbf {A} }
.
^γ Defined in Cartesian coordinates as
e
i
⋅
∂
i
T
{\displaystyle \mathbf {e} _{i}\cdot \partial _{i}\mathbf {T} }
. An alternative definition is
∂
i
T
⋅
e
i
{\displaystyle \partial _{i}\mathbf {T} \cdot \mathbf {e} _{i}}
.
= Calculation rules
=div
grad
f
≡
∇
⋅
∇
f
≡
∇
2
f
{\displaystyle \operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f\equiv \nabla ^{2}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)
∇
2
(
f
g
)
=
f
∇
2
g
+
2
∇
f
⋅
∇
g
+
g
∇
2
f
{\displaystyle \nabla ^{2}(fg)=f\nabla ^{2}g+2\nabla f\cdot \nabla g+g\nabla ^{2}f}
∇
2
(
P
⋅
Q
)
=
Q
⋅
∇
2
P
−
P
⋅
∇
2
Q
+
2
∇
⋅
[
(
P
⋅
∇
)
Q
+
P
×
∇
×
Q
]
{\displaystyle \nabla ^{2}\left(\mathbf {P} \cdot \mathbf {Q} \right)=\mathbf {Q} \cdot \nabla ^{2}\mathbf {P} -\mathbf {P} \cdot \nabla ^{2}\mathbf {Q} +2\nabla \cdot \left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \nabla \times \mathbf {Q} \right]\quad }
(From )
Cartesian derivation
div
A
=
lim
V
→
0
∬
∂
V
A
⋅
d
S
∭
V
d
V
=
A
x
(
x
+
d
x
)
d
y
d
z
−
A
x
(
x
)
d
y
d
z
+
A
y
(
y
+
d
y
)
d
x
d
z
−
A
y
(
y
)
d
x
d
z
+
A
z
(
z
+
d
z
)
d
x
d
y
−
A
z
(
z
)
d
x
d
y
d
x
d
y
d
z
=
∂
A
x
∂
x
+
∂
A
y
∂
y
+
∂
A
z
∂
z
{\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}}}
(
curl
A
)
x
=
lim
S
⊥
x
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
z
(
y
+
d
y
)
d
z
−
A
z
(
y
)
d
z
+
A
y
(
z
)
d
y
−
A
y
(
z
+
d
z
)
d
y
d
y
d
z
=
∂
A
z
∂
y
−
∂
A
y
∂
z
{\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}}}
The expressions for
(
curl
A
)
y
{\displaystyle (\operatorname {curl} \mathbf {A} )_{y}}
and
(
curl
A
)
z
{\displaystyle (\operatorname {curl} \mathbf {A} )_{z}}
are found in the same way.
Cylindrical derivation
div
A
=
lim
V
→
0
∬
∂
V
A
⋅
d
S
∭
V
d
V
=
A
ρ
(
ρ
+
d
ρ
)
(
ρ
+
d
ρ
)
d
ϕ
d
z
−
A
ρ
(
ρ
)
ρ
d
ϕ
d
z
+
A
ϕ
(
ϕ
+
d
ϕ
)
d
ρ
d
z
−
A
ϕ
(
ϕ
)
d
ρ
d
z
+
A
z
(
z
+
d
z
)
d
ρ
(
ρ
+
d
ρ
/
2
)
d
ϕ
−
A
z
(
z
)
d
ρ
(
ρ
+
d
ρ
/
2
)
d
ϕ
ρ
d
ϕ
d
ρ
d
z
=
1
ρ
∂
(
ρ
A
ρ
)
∂
ρ
+
1
ρ
∂
A
ϕ
∂
ϕ
+
∂
A
z
∂
z
{\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}}}
(
curl
A
)
ρ
=
lim
S
⊥
ρ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
ϕ
(
z
)
(
ρ
+
d
ρ
)
d
ϕ
−
A
ϕ
(
z
+
d
z
)
(
ρ
+
d
ρ
)
d
ϕ
+
A
z
(
ϕ
+
d
ϕ
)
d
z
−
A
z
(
ϕ
)
d
z
(
ρ
+
d
ρ
)
d
ϕ
d
z
=
−
∂
A
ϕ
∂
z
+
1
ρ
∂
A
z
∂
ϕ
{\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}}}
(
curl
A
)
ϕ
=
lim
S
⊥
ϕ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
z
(
ρ
)
d
z
−
A
z
(
ρ
+
d
ρ
)
d
z
+
A
ρ
(
z
+
d
z
)
d
ρ
−
A
ρ
(
z
)
d
ρ
d
ρ
d
z
=
−
∂
A
z
∂
ρ
+
∂
A
ρ
∂
z
{\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}}}
(
curl
A
)
z
=
lim
S
⊥
z
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
ρ
(
ϕ
)
d
ρ
−
A
ρ
(
ϕ
+
d
ϕ
)
d
ρ
+
A
ϕ
(
ρ
+
d
ρ
)
(
ρ
+
d
ρ
)
d
ϕ
−
A
ϕ
(
ρ
)
ρ
d
ϕ
ρ
d
ρ
d
ϕ
=
−
1
ρ
∂
A
ρ
∂
ϕ
+
1
ρ
∂
(
ρ
A
ϕ
)
∂
ρ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{z}&=\lim _{S^{\perp {\hat {\boldsymbol {z}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}\\[1ex]&={\frac {A_{\rho }(\phi )\,d\rho -A_{\rho }(\phi +d\phi )\,d\rho +A_{\phi }(\rho +d\rho )(\rho +d\rho )\,d\phi -A_{\phi }(\rho )\rho \,d\phi }{\rho \,d\rho \,d\phi }}\\[1ex]&=-{\frac {1}{\rho }}{\frac {\partial A_{\rho }}{\partial \phi }}+{\frac {1}{\rho }}{\frac {\partial (\rho A_{\phi })}{\partial \rho }}\end{aligned}}}
curl
A
=
(
curl
A
)
ρ
ρ
^
+
(
curl
A
)
ϕ
ϕ
^
+
(
curl
A
)
z
z
^
=
(
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}\operatorname {curl} \mathbf {A} &=(\operatorname {curl} \mathbf {A} )_{\rho }{\hat {\boldsymbol {\rho }}}+(\operatorname {curl} \mathbf {A} )_{\phi }{\hat {\boldsymbol {\phi }}}+(\operatorname {curl} \mathbf {A} )_{z}{\hat {\boldsymbol {z}}}\\[1ex]&=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\hat {\boldsymbol {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\hat {\boldsymbol {\phi }}}+{\frac {1}{\rho }}\left({\frac {\partial (\rho A_{\phi })}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\boldsymbol {z}}}\end{aligned}}}
Spherical derivation
div
A
=
lim
V
→
0
∬
∂
V
A
⋅
d
S
∭
V
d
V
=
A
r
(
r
+
d
r
)
(
r
+
d
r
)
d
θ
(
r
+
d
r
)
sin
θ
d
ϕ
−
A
r
(
r
)
r
d
θ
r
sin
θ
d
ϕ
+
A
θ
(
θ
+
d
θ
)
sin
(
θ
+
d
θ
)
r
d
r
d
ϕ
−
A
θ
(
θ
)
sin
(
θ
)
r
d
r
d
ϕ
+
A
ϕ
(
ϕ
+
d
ϕ
)
r
d
r
d
θ
−
A
ϕ
(
ϕ
)
r
d
r
d
θ
d
r
r
d
θ
r
sin
θ
d
ϕ
=
1
r
2
∂
(
r
2
A
r
)
∂
r
+
1
r
sin
θ
∂
(
A
θ
sin
θ
)
∂
θ
+
1
r
sin
θ
∂
A
ϕ
∂
ϕ
{\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_{r}(r+dr)(r+dr)\,d\theta \,(r+dr)\sin \theta \,d\phi -A_{r}(r)r\,d\theta \,r\sin \theta \,d\phi +A_{\theta }(\theta +d\theta )\sin(\theta +d\theta )r\,dr\,d\phi -A_{\theta }(\theta )\sin(\theta )r\,dr\,d\phi +A_{\phi }(\phi +d\phi )r\,dr\,d\theta -A_{\phi }(\phi )r\,dr\,d\theta }{dr\,r\,d\theta \,r\sin \theta \,d\phi }}\\&={\frac {1}{r^{2}}}{\frac {\partial (r^{2}A_{r})}{\partial r}}+{\frac {1}{r\sin \theta }}{\frac {\partial (A_{\theta }\sin \theta )}{\partial \theta }}+{\frac {1}{r\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\end{aligned}}}
(
curl
A
)
r
=
lim
S
⊥
r
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
θ
(
ϕ
)
r
d
θ
+
A
ϕ
(
θ
+
d
θ
)
r
sin
(
θ
+
d
θ
)
d
ϕ
−
A
θ
(
ϕ
+
d
ϕ
)
r
d
θ
−
A
ϕ
(
θ
)
r
sin
(
θ
)
d
ϕ
r
d
θ
r
sin
θ
d
ϕ
=
1
r
sin
θ
∂
(
A
ϕ
sin
θ
)
∂
θ
−
1
r
sin
θ
∂
A
θ
∂
ϕ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{r}=\lim _{S^{\perp {\boldsymbol {\hat {r}}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{\theta }(\phi )r\,d\theta +A_{\phi }(\theta +d\theta )r\sin(\theta +d\theta )\,d\phi -A_{\theta }(\phi +d\phi )r\,d\theta -A_{\phi }(\theta )r\sin(\theta )\,d\phi }{r\,d\theta \,r\sin \theta \,d\phi }}\\&={\frac {1}{r\sin \theta }}{\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {1}{r\sin \theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\end{aligned}}}
(
curl
A
)
θ
=
lim
S
⊥
θ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
ϕ
(
r
)
r
sin
θ
d
ϕ
+
A
r
(
ϕ
+
d
ϕ
)
d
r
−
A
ϕ
(
r
+
d
r
)
(
r
+
d
r
)
sin
θ
d
ϕ
−
A
r
(
ϕ
)
d
r
d
r
r
sin
θ
d
ϕ
=
1
r
sin
θ
∂
A
r
∂
ϕ
−
1
r
∂
(
r
A
ϕ
)
∂
r
{\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}}}
(
curl
A
)
ϕ
=
lim
S
⊥
ϕ
^
→
0
∫
∂
S
A
⋅
d
ℓ
∬
S
d
S
=
A
r
(
θ
)
d
r
+
A
θ
(
r
+
d
r
)
(
r
+
d
r
)
d
θ
−
A
r
(
θ
+
d
θ
)
d
r
−
A
θ
(
r
)
r
d
θ
r
d
r
d
θ
=
1
r
∂
(
r
A
θ
)
∂
r
−
1
r
∂
A
r
∂
θ
{\displaystyle {\begin{aligned}(\operatorname {curl} \mathbf {A} )_{\phi }=\lim _{S^{\perp {\boldsymbol {\hat {\phi }}}}\to 0}{\frac {\int _{\partial S}\mathbf {A} \cdot d\mathbf {\ell } }{\iint _{S}dS}}&={\frac {A_{r}(\theta )\,dr+A_{\theta }(r+dr)(r+dr)\,d\theta -A_{r}(\theta +d\theta )\,dr-A_{\theta }(r)r\,d\theta }{r\,dr\,d\theta }}\\&={\frac {1}{r}}{\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {1}{r}}{\frac {\partial A_{r}}{\partial \theta }}\end{aligned}}}
curl
A
=
(
curl
A
)
r
r
^
+
(
curl
A
)
θ
θ
^
+
(
curl
A
)
ϕ
ϕ
^
=
1
r
sin
θ
(
∂
(
A
ϕ
sin
θ
)
∂
θ
−
∂
A
θ
∂
ϕ
)
r
^
+
1
r
(
1
sin
θ
∂
A
r
∂
ϕ
−
∂
(
r
A
ϕ
)
∂
r
)
θ
^
+
1
r
(
∂
(
r
A
θ
)
∂
r
−
∂
A
r
∂
θ
)
ϕ
^
{\displaystyle {\begin{aligned}\operatorname {curl} \mathbf {A} &=(\operatorname {curl} \mathbf {A} )_{r}\,{\hat {\boldsymbol {r}}}+(\operatorname {curl} \mathbf {A} )_{\theta }\,{\hat {\boldsymbol {\theta }}}+(\operatorname {curl} \mathbf {A} )_{\phi }\,{\hat {\boldsymbol {\phi }}}\\[1ex]&={\frac {1}{r\sin \theta }}\left({\frac {\partial (A_{\phi }\sin \theta )}{\partial \theta }}-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial (rA_{\phi })}{\partial r}}\right){\hat {\boldsymbol {\theta }}}+{\frac {1}{r}}\left({\frac {\partial (rA_{\theta })}{\partial r}}-{\frac {\partial A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}\end{aligned}}}
Unit vector conversion formula
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
r
{\displaystyle \mathbf {r} }
to change in
u
{\displaystyle \mathbf {u} }
direction.
Therefore,
∂
r
∂
u
=
∂
s
∂
u
u
{\displaystyle {\frac {\partial {\mathbf {r} }}{\partial u}}={\frac {\partial {s}}{\partial u}}\mathbf {u} }
where s is the arc length parameter.
For two sets of coordinate systems
u
i
{\displaystyle u_{i}}
and
v
j
{\displaystyle v_{j}}
, according to chain rule,
d
r
=
∑
i
∂
r
∂
u
i
d
u
i
=
∑
i
∂
s
∂
u
i
u
^
i
d
u
i
=
∑
j
∂
s
∂
v
j
v
^
j
d
v
j
=
∑
j
∂
s
∂
v
j
v
^
j
∑
i
∂
v
j
∂
u
i
d
u
i
=
∑
i
∑
j
∂
s
∂
v
j
∂
v
j
∂
u
i
v
^
j
d
u
i
.
{\displaystyle d\mathbf {r} =\sum _{i}{\frac {\partial \mathbf {r} }{\partial u_{i}}}\,du_{i}=\sum _{i}{\frac {\partial s}{\partial u_{i}}}{\hat {\mathbf {u} }}_{i}du_{i}=\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\hat {\mathbf {v} }}_{j}\,dv_{j}=\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\hat {\mathbf {v} }}_{j}\sum _{i}{\frac {\partial v_{j}}{\partial u_{i}}}\,du_{i}=\sum _{i}\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\frac {\partial v_{j}}{\partial u_{i}}}{\hat {\mathbf {v} }}_{j}\,du_{i}.}
Now, we isolate the
i
{\displaystyle i}
th component. For
i
≠
k
{\displaystyle i{\neq }k}
, let
d
u
k
=
0
{\displaystyle \mathrm {d} u_{k}=0}
. Then divide on both sides by
d
u
i
{\displaystyle \mathrm {d} u_{i}}
to get:
∂
s
∂
u
i
u
^
i
=
∑
j
∂
s
∂
v
j
∂
v
j
∂
u
i
v
^
j
.
{\displaystyle {\frac {\partial s}{\partial u_{i}}}{\hat {\mathbf {u} }}_{i}=\sum _{j}{\frac {\partial s}{\partial v_{j}}}{\frac {\partial v_{j}}{\partial u_{i}}}{\hat {\mathbf {v} }}_{j}.}
See also
Del
Orthogonal coordinates
Curvilinear coordinates
Vector fields in cylindrical and spherical coordinates
References
External links
Maxima Computer Algebra system scripts to generate some of these operators in cylindrical and spherical coordinates.
Kata Kunci Pencarian:
- Del in cylindrical and spherical coordinates
- Vector fields in cylindrical and spherical coordinates
- Divergence
- Spherical coordinate system
- Cylindrical coordinate system
- Laplace operator
- Del
- Potential flow around a circular cylinder
- Curl (mathematics)
- Curvilinear coordinates
