Coordinate systems/Derivation of formulas

The purpose of this resource is to carefully examine the Wikipedia article Del in cylindrical and spherical coordinates for accuracy.

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Transformations between coordinates

1. w:Cartesian coordinates (x, y, z)
2. w:Cylindrical coordinates (ρ, ϕ, z)
3. w:Spherical coordinates (r, θ, ϕ)
4. w:Parabolic cylindrical coordinates (σ, τ, z)

Coordinate variable transformations*

*Asterisk indicates that the title is a link to more discussion

Cylindrical from Cartesian variable transformation

$\rho ={\sqrt {x^{2}+y^{2}}}$   ,     $\phi =\arctan(y/x)$   ,     $z=z$ verified using mathworld

Cartesian from cylindrical variable transformation

$x=\rho \cos \phi$   ,     $y=\rho \sin \phi$   ,     $z=z$  verified using mathworld

Cartesian from spherical variable transformation

$x=r\sin \theta \cos \phi$   ,     $y=r\sin \theta \sin \phi$   ,     $z=r\cos \theta$ verified using mathworld

Cartesian from parabolic cylindrical variable transformation

$x=\sigma \tau$   ,     $y={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)$   ,     $z=z$ --no reference

Spherical from Cartesian variable transformation

$r={\sqrt {x^{2}+y^{2}+z^{2}}}$   ,     $\theta =\arctan({\sqrt {x^{2}+y^{2}}}/z)$   ,     $\phi =\arctan(y/x)$ verified using mathworld

Spherical from cylindrical variable transformation

$r={\sqrt {\rho ^{2}+z^{2}}}$   ,     $\theta =\arctan {(\rho /z)}$   ,     $\phi =\phi$ no reference

Cylindrical from spherical variable transformation

$\rho =r\sin \theta$   ,     $\phi =\phi$   ,     $z=r\cos \theta$ no reference

Cylindrical from parabolic cylindrical variable transformation

$\rho \cos \phi =\sigma \tau$   ,     $\rho \sin \phi ={\tfrac {1}{2}}\left(\tau ^{2}-\sigma ^{2}\right)$   ,     $z=z$ no reference

Unit vectors

Cylindrical from Cartesian unit vectors

${\begin{aligned}{\hat {\boldsymbol {\rho }}}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$  Verified, see page linked in title

Cartesian from cylindrical unit vectors

${\begin{aligned}{\hat {\mathbf {x} }}&=\cos \phi {\hat {\boldsymbol {\rho }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \phi {\hat {\boldsymbol {\rho }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$  Verified, see page linked in title

Cartesian from spherical unit vectors

${\begin{aligned}{\hat {\mathbf {x} }}&=\sin \theta \cos \phi {\hat {\boldsymbol {r}}}+\cos \theta \cos \phi {\hat {\boldsymbol {\theta }}}-\sin \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {y} }}&=\sin \theta \sin \phi {\hat {\boldsymbol {r}}}+\cos \theta \sin \phi {\hat {\boldsymbol {\theta }}}+\cos \phi {\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\boldsymbol {r}}}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$  Verified, see page linked in title

Parabolic cylindrical from Cartesian unit vectors

${\begin{aligned}{\hat {\boldsymbol {\sigma }}}&={\frac {\tau {\hat {\mathbf {x} }}-\sigma {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\boldsymbol {\tau }}}&={\frac {\sigma {\hat {\mathbf {x} }}+\tau {\hat {\mathbf {y} }}}{\sqrt {\tau ^{2}+\sigma ^{2}}}}\\{\hat {\mathbf {z} }}&={\hat {\mathbf {z} }}\end{aligned}}$

Spherical from Cartesian unit vectors

${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {x{\hat {\mathbf {x} }}+y{\hat {\mathbf {y} }}+z{\hat {\mathbf {z} }}}{\sqrt {x^{2}+y^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {xz{\hat {\mathbf {x} }}+yz{\hat {\mathbf {y} }}-\left(x^{2}+y^{2}\right){\hat {\mathbf {z} }}}{{\sqrt {x^{2}+y^{2}}}{\sqrt {x^{2}+y^{2}+z^{2}}}}}\\{\hat {\boldsymbol {\phi }}}&={\frac {-y{\hat {\mathbf {x} }}+x{\hat {\mathbf {y} }}}{\sqrt {x^{2}+y^{2}}}}\end{aligned}}$ Verified, see page linked in title

Spherical from cylindrical unit vectors

${\begin{aligned}{\hat {\mathbf {r} }}&={\frac {\rho {\hat {\boldsymbol {\rho }}}+z{\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\theta }}}&={\frac {z{\hat {\boldsymbol {\rho }}}-\rho {\hat {\mathbf {z} }}}{\sqrt {\rho ^{2}+z^{2}}}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\end{aligned}}$

Cylindrical from spherical unit vectors

${\begin{aligned}{\hat {\boldsymbol {\rho }}}&=\sin \theta {\hat {\mathbf {r} }}+\cos \theta {\hat {\boldsymbol {\theta }}}\\{\hat {\boldsymbol {\phi }}}&={\hat {\boldsymbol {\phi }}}\\{\hat {\mathbf {z} }}&=\cos \theta {\hat {\mathbf {r} }}-\sin \theta {\hat {\boldsymbol {\theta }}}\end{aligned}}$

Vector and scalar fields

$\mathbf {A}$ is vector field and f is a scalar field. The vector field can be expressed as:

1. $A_{x}{\hat {\mathbf {x} }}+A_{y}{\hat {\mathbf {y} }}+A_{z}{\hat {\mathbf {z} }}$
2. $A_{\rho }{\hat {\boldsymbol {\rho }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}+A_{z}{\hat {\mathbf {z} }}$
3. $A_{r}{\hat {\boldsymbol {r}}}+A_{\theta }{\hat {\boldsymbol {\theta }}}+A_{\phi }{\hat {\boldsymbol {\phi }}}$
4. $A_{\sigma }{\hat {\boldsymbol {\sigma }}}+A_{\tau }{\hat {\boldsymbol {\tau }}}+A_{\phi }{\hat {\mathbf {z} }}$

Gradient of a scalar field

$\nabla f$ is the w:gradient of a scalar field.

1. ${\partial f \over \partial x}{\hat {\mathbf {x} }}+{\partial f \over \partial y}{\hat {\mathbf {y} }}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$
2. ${\partial f \over \partial \rho }{\hat {\boldsymbol {\rho }}}+{1 \over \rho }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$
3. ${\partial f \over \partial r}{\hat {\boldsymbol {r}}}+{1 \over r}{\partial f \over \partial \theta }{\hat {\boldsymbol {\theta }}}+{1 \over r\sin \theta }{\partial f \over \partial \phi }{\hat {\boldsymbol {\phi }}}$
4. ${\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \sigma }{\hat {\boldsymbol {\sigma }}}+{\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\partial f \over \partial \tau }{\hat {\boldsymbol {\tau }}}+{\partial f \over \partial z}{\hat {\mathbf {z} }}$

Divergence of a vector field*

$\nabla \cdot \mathbf {A}$ is the w:divergence of a vector field

1. ${\partial A_{x} \over \partial x}+{\partial A_{y} \over \partial y}+{\partial A_{z} \over \partial z}$
2. ${1 \over \rho }{\partial \left(\rho A_{\rho }\right) \over \partial \rho }+{1 \over \rho }{\partial A_{\phi } \over \partial \phi }+{\partial A_{z} \over \partial z}$
3. ${1 \over r^{2}}{\partial \left(r^{2}A_{r}\right) \over \partial r}+{1 \over r\sin \theta }{\partial \over \partial \theta }\left(A_{\theta }\sin \theta \right)+{1 \over r\sin \theta }{\partial A_{\phi } \over \partial \phi }$
4. ${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\sigma }) \over \partial \sigma }+{\partial ({\sqrt {\sigma ^{2}+\tau ^{2}}}A_{\tau }) \over \partial \tau }\right)+{\partial A_{z} \over \partial z}$

Curl of a vector field

$\nabla \times \mathbf {A}$ is the w:curl (mathematics) of A

1. $\left({\frac {\partial A_{z}}{\partial y}}-{\frac {\partial A_{y}}{\partial z}}\right){\hat {\mathbf {x} }}+\left({\frac {\partial A_{x}}{\partial z}}-{\frac {\partial A_{z}}{\partial x}}\right){\hat {\mathbf {y} }}+\left({\frac {\partial A_{y}}{\partial x}}-{\frac {\partial A_{x}}{\partial y}}\right){\hat {\mathbf {z} }}$
2. $\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \phi }}-{\frac {\partial A_{\phi }}{\partial z}}\right){\hat {\boldsymbol {\rho }}}+\left({\frac {\partial A_{\rho }}{\partial z}}-{\frac {\partial A_{z}}{\partial \rho }}\right){\hat {\boldsymbol {\phi }}}+{\frac {1}{\rho }}\left({\frac {\partial \left(\rho A_{\phi }\right)}{\partial \rho }}-{\frac {\partial A_{\rho }}{\partial \phi }}\right){\hat {\mathbf {z} }}$
3. ${\frac {1}{r\sin \theta }}\left({\frac {\partial }{\partial \theta }}\left(A_{\phi }\sin \theta \right)-{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}+{\frac {1}{r}}\left({\frac {1}{\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}-{\frac {\partial }{\partial r}}\left(rA_{\phi }\right)\right){\hat {\boldsymbol {\theta }}}+{\frac {1}{r}}\left({\frac {\partial }{\partial r}}\left(rA_{\theta }\right)-{\frac {\partial A_{r}}{\partial \theta }}\right){\hat {\boldsymbol {\phi }}}$
4. $\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \tau }}-{\frac {\partial A_{\tau }}{\partial z}}\right){\hat {\boldsymbol {\sigma }}}-\left({\frac {1}{\sqrt {\sigma ^{2}+\tau ^{2}}}}{\frac {\partial A_{z}}{\partial \sigma }}-{\frac {\partial A_{\sigma }}{\partial z}}\right){\hat {\boldsymbol {\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){\hat {\mathbf {z} }}$

Laplacian of a scalar field

$\Delta f\equiv \nabla ^{2}f$ is the w:Laplace operator on a scalar field

1. ${\partial ^{2}f \over \partial x^{2}}+{\partial ^{2}f \over \partial y^{2}}+{\partial ^{2}f \over \partial z^{2}}$
2. ${1 \over \rho }{\partial \over \partial \rho }\left(\rho {\partial f \over \partial \rho }\right)+{1 \over \rho ^{2}}{\partial ^{2}f \over \partial \phi ^{2}}+{\partial ^{2}f \over \partial z^{2}}$
3. ${1 \over r^{2}}{\partial \over \partial r}\!\left(r^{2}{\partial f \over \partial r}\right)\!+\!{1 \over r^{2}\!\sin \theta }{\partial \over \partial \theta }\!\left(\sin \theta {\partial f \over \partial \theta }\right)\!+\!{1 \over r^{2}\!\sin ^{2}\theta }{\partial ^{2}f \over \partial \phi ^{2}}$
4. ${\frac {1}{\sigma ^{2}+\tau ^{2}}}\left({\frac {\partial ^{2}f}{\partial \sigma ^{2}}}+{\frac {\partial ^{2}f}{\partial \tau ^{2}}}\right)+{\frac {\partial ^{2}f}{\partial z^{2}}}$

Laplacian of a vector field

$\Delta \mathbf {A} \equiv \nabla ^{2}\mathbf {A}$ is the w:Vector Laplacian of $\mathbf {A}$

1. $\Delta A_{x}{\hat {\mathbf {x} }}+\Delta A_{y}{\hat {\mathbf {y} }}+\Delta A_{z}{\hat {\mathbf {z} }}$
2. ${\mathopen {}}\left(\Delta A_{\rho }-{\frac {A_{\rho }}{\rho ^{2}}}-{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\mathclose {}}{\hat {\boldsymbol {\rho }}}$$+{\mathopen {}}\left(\Delta A_{\phi }-{\frac {A_{\phi }}{\rho ^{2}}}+{\frac {2}{\rho ^{2}}}{\frac {\partial A_{\rho }}{\partial \phi }}\right){\mathclose {}}{\hat {\boldsymbol {\phi }}}$$+\Delta A_{z}{\hat {\mathbf {z} }}$
3. $\left(\Delta A_{r}-{\frac {2A_{r}}{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial \left(A_{\theta }\sin \theta \right)}{\partial \theta }}-{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\hat {\boldsymbol {r}}}$$+\left(\Delta A_{\theta }-{\frac {A_{\theta }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}}}{\frac {\partial A_{r}}{\partial \theta }}-{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\phi }}{\partial \phi }}\right){\hat {\boldsymbol {\theta }}}$$+\left(\Delta A_{\phi }-{\frac {A_{\phi }}{r^{2}\sin ^{2}\theta }}+{\frac {2}{r^{2}\sin \theta }}{\frac {\partial A_{r}}{\partial \phi }}+{\frac {2\cos \theta }{r^{2}\sin ^{2}\theta }}{\frac {\partial A_{\theta }}{\partial \phi }}\right){\hat {\boldsymbol {\phi }}}$

Material derivative of a vector field

$(\mathbf {A} \cdot \nabla )\mathbf {B}$ might be called the "convective derivative of B along A" (appropriate description if A' is a unit vector) 

1. $\left(A_{x}{\frac {\partial B_{x}}{\partial x}}+A_{y}{\frac {\partial B_{x}}{\partial y}}+A_{z}{\frac {\partial B_{x}}{\partial z}}\right){\hat {\mathbf {x} }}$$+\left(A_{x}{\frac {\partial B_{y}}{\partial x}}+A_{y}{\frac {\partial B_{y}}{\partial y}}+A_{z}{\frac {\partial B_{y}}{\partial z}}\right){\hat {\mathbf {y} }}$$+\left(A_{x}{\frac {\partial B_{z}}{\partial x}}+A_{y}{\frac {\partial B_{z}}{\partial y}}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right){\hat {\mathbf {z} }}$
2. $\left(A_{\rho }{\frac {\partial B_{\rho }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\rho }}{\partial \phi }}+A_{z}{\frac {\partial B_{\rho }}{\partial z}}-{\frac {A_{\phi }B_{\phi }}{\rho }}\right){\hat {\boldsymbol {\rho }}}$$+\left(A_{\rho }{\frac {\partial B_{\phi }}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{\phi }}{\partial \phi }}+A_{z}{\frac {\partial B_{\phi }}{\partial z}}+{\frac {A_{\phi }B_{\rho }}{\rho }}\right){\hat {\boldsymbol {\phi }}}$$+\left(A_{\rho }{\frac {\partial B_{z}}{\partial \rho }}+{\frac {A_{\phi }}{\rho }}{\frac {\partial B_{z}}{\partial \phi }}+A_{z}{\frac {\partial B_{z}}{\partial z}}\right){\hat {\mathbf {z} }}$
3. $\left(A_{r}{\frac {\partial B_{r}}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{r}}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{r}}{\partial \phi }}-{\frac {A_{\theta }B_{\theta }+A_{\phi }B_{\phi }}{r}}\right){\hat {\boldsymbol {r}}}$$+\left(A_{r}{\frac {\partial B_{\theta }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\theta }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\theta }}{\partial \phi }}+{\frac {A_{\theta }B_{r}}{r}}-{\frac {A_{\phi }B_{\phi }\cot \theta }{r}}\right){\hat {\boldsymbol {\theta }}}$$+\left(A_{r}{\frac {\partial B_{\phi }}{\partial r}}+{\frac {A_{\theta }}{r}}{\frac {\partial B_{\phi }}{\partial \theta }}+{\frac {A_{\phi }}{r\sin \theta }}{\frac {\partial B_{\phi }}{\partial \phi }}+{\frac {A_{\phi }B_{r}}{r}}+{\frac {A_{\phi }B_{\theta }\cot \theta }{r}}\right){\hat {\boldsymbol {\phi }}}$

Differential displacement

1. $d\mathbf {l} =dx\,{\hat {\mathbf {x} }}+dy\,{\hat {\mathbf {y} }}+dz\,{\hat {\mathbf {z} }}$
2. $d\mathbf {l} =d\rho \,{\hat {\boldsymbol {\rho }}}+\rho \,d\phi \,{\hat {\boldsymbol {\phi }}}+dz\,{\hat {\mathbf {z} }}$
3. $d\mathbf {l} =dr\,{\hat {\mathbf {r} }}+r\,d\theta \,{\hat {\boldsymbol {\theta }}}+r\,\sin \theta \,d\phi \,{\hat {\boldsymbol {\phi }}}$
4. $d\mathbf {l} ={\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,{\hat {\boldsymbol {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,{\hat {\boldsymbol {\tau }}}+dz\,{\hat {\mathbf {z} }}$

Differential normal areas

• These vector differentials cannot be integrated for curved surfaces. Click the title above to see why.

Differential normal area $d\mathbf {S}$

1. $dy\,dz{\hat {\mathbf {x} }}+dx\,dz{\hat {\mathbf {y} }}+dx\,dy{\hat {\mathbf {z} }}$
2. $\rho \,d\phi \,dz{\hat {\boldsymbol {\rho }}}+d\rho \,dz{\hat {\boldsymbol {\phi }}}+\rho \,d\rho \,d\phi {\hat {\mathbf {z} }}$
3. $r^{2}\sin \theta \,d\theta \,d\phi {\hat {\mathbf {r} }}+r\sin \theta \,dr\,d\phi {\hat {\boldsymbol {\theta }}}+r\,dr\,d\theta {\hat {\boldsymbol {\phi }}}$
4. ${\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\tau \,dz{\hat {\boldsymbol {\sigma }}}+{\sqrt {\sigma ^{2}+\tau ^{2}}}\,d\sigma \,dz{\hat {\boldsymbol {\tau }}}+\left(\sigma ^{2}+\tau ^{2}\right)\,d\sigma \,d\tau {\hat {\mathbf {z} }}$

Differential volume

1. $dV=dx\,dy\,dz$ verified
2. $dV=\rho \,d\rho \,d\phi \,dz$ verified
3. $dV=r^{2}\sin \theta \,dr\,d\theta \,d\phi$ verified
4. $dV=\left(\sigma ^{2}+\tau ^{2}\right)d\sigma \,d\tau \,dz$

nabla's on nabla's

Non-trivial calculation rules:

1. $\operatorname {div} \,\operatorname {grad} f\equiv \nabla \cdot \nabla f=\nabla ^{2}f\equiv \Delta f$
2. $\operatorname {curl} \,\operatorname {grad} f\equiv \nabla \times \nabla f=\mathbf {0}$
3. $\operatorname {div} \,\operatorname {curl} \mathbf {A} \equiv \nabla \cdot (\nabla \times \mathbf {A} )=0$
4. $\operatorname {curl} \,\operatorname {curl} \mathbf {A} \equiv \nabla \times (\nabla \times \mathbf {A} )=\nabla (\nabla \cdot \mathbf {A} )-\nabla ^{2}\mathbf {A}$ (Lagrange's formula for del)
5. $\Delta (fg)=f\Delta g+2\nabla f\cdot \nabla g+g\Delta f$

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