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/

$\rho =\sqrt{x^2+y^2}$   ,     $\varphi =\arctan (y/x)$   ,     $z=z$Template:Padverified using mathworld

/Cartesian from cylindrical variable transformation/

$x=\rho \cos \varphi$   ,     $y=\rho \sin \varphi$   ,     $z=z$ Template:Padverified using mathworld

/Cartesian from spherical variable transformation/

$x=r\sin \theta \cos \varphi$   ,     $y=r\sin \theta \sin \varphi$   ,     $z=r\cos \theta$Template:Padverified using mathworld

/Cartesian from parabolic cylindrical variable transformation/

$x=\sigma \tau$   ,     $y=\frac{1}{2}({\tau }^2-{\sigma }^2)$   ,     $z=z$Template:Pad--no reference

/Spherical from Cartesian variable transformation/

$r=\sqrt{x^2+y^2+z^2}$   ,     $\theta =\arctan (\sqrt{x^2+y^2}/z)$   ,     $\varphi =\arctan (y/x)$Template:Padverified using mathworld

/Spherical from cylindrical variable transformation/

$r=\sqrt{{\rho }^2+z^2}$   ,     $\theta =\arctan (\rho /z)$   ,     $\varphi =\varphi$Template:Padno reference

/Cylindrical from spherical variable transformation/

$\rho =r\sin \theta$   ,     $\varphi =\varphi$   ,     $z=r\cos \theta$Template:Padno reference

/Cylindrical from parabolic cylindrical variable transformation/

$\rho \cos \varphi =\sigma \tau$   ,     $\rho \sin \varphi =\frac{1}{2}({\tau }^2-{\sigma }^2)$   ,     $z=z$Template:Padno reference

/Unit vectors/

/Cylindrical from Cartesian unit vectors/

$\begin{array}{cc}\hat{\mathit{\rho }}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \hat{\mathit{\varphi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$ Template:PadVerified, see page linked in title

/Cartesian from cylindrical unit vectors/

$\begin{array}{cc}\widehat{𝐱}& =\cos \varphi \hat{\mathit{\rho }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \varphi \hat{\mathit{\rho }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$ Template:PadVerified, see page linked in title

/Cartesian from spherical unit vectors/

$\begin{array}{cc}\widehat{𝐱}& =\sin \theta \cos \varphi \widehat{𝒓}+\cos \theta \cos \varphi \hat{\mathit{\theta }}-\sin \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐲}& =\sin \theta \sin \varphi \widehat{𝒓}+\cos \theta \sin \varphi \hat{\mathit{\theta }}+\cos \varphi \hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝒓}-\sin \theta \hat{\mathit{\theta }}\end{array}$ Template:PadVerified, see page linked in title

/Parabolic cylindrical from Cartesian unit vectors/

$\begin{array}{cc}\hat{\mathit{\sigma }}& =\frac{\tau \widehat{𝐱}-\sigma \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \hat{\mathit{\tau }}& =\frac{\sigma \widehat{𝐱}+\tau \widehat{𝐲}}{\sqrt{{\tau }^2+{\sigma }^2}}\\ \widehat{𝐳}& =\widehat{𝐳}\end{array}$

/Spherical from Cartesian unit vectors/

$\begin{array}{cc}\widehat{𝐫}& =\frac{x\widehat{𝐱}+y\widehat{𝐲}+z\widehat{𝐳}}{\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{xz\widehat{𝐱}+yz\widehat{𝐲}-(x^2+y^2)\widehat{𝐳}}{\sqrt{x^2+y^2}\sqrt{x^2+y^2+z^2}}\\ \hat{\mathit{\varphi }}& =\frac{-y\widehat{𝐱}+x\widehat{𝐲}}{\sqrt{x^2+y^2}}\end{array}$Template:PadVerified, see page linked in title

/Spherical from cylindrical unit vectors/

$\begin{array}{cc}\widehat{𝐫}& =\frac{\rho \hat{\mathit{\rho }}+z\widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\theta }}& =\frac{z\hat{\mathit{\rho }}-\rho \widehat{𝐳}}{\sqrt{{\rho }^2+z^2}}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\end{array}$

/Cylindrical from spherical unit vectors/

$\begin{array}{cc}\hat{\mathit{\rho }}& =\sin \theta \widehat{𝐫}+\cos \theta \hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}& =\hat{\mathit{\varphi }}\\ \widehat{𝐳}& =\cos \theta \widehat{𝐫}-\sin \theta \hat{\mathit{\theta }}\end{array}$

Vector and scalar fields

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

1. $A_x\widehat{𝐱}+A_y\widehat{𝐲}+A_z\widehat{𝐳}$
2. $A_{\rho }\hat{\mathit{\rho }}+A_{\varphi }\hat{\mathit{\varphi }}+A_z\widehat{𝐳}$
3. $A_r\widehat{𝒓}+A_{\theta }\hat{\mathit{\theta }}+A_{\varphi }\hat{\mathit{\varphi }}$
4. $A_{\sigma }\hat{\mathit{\sigma }}+A_{\tau }\hat{\mathit{\tau }}+A_{\varphi }\widehat{𝐳}$

/Gradient of a scalar field/

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

1. $\frac{\partial f}{\partial x}\widehat{𝐱}+\frac{\partial f}{\partial y}\widehat{𝐲}+\frac{\partial f}{\partial z}\widehat{𝐳}$
2. $\frac{\partial f}{\partial \rho }\hat{\mathit{\rho }}+\frac{1}{\rho }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}+\frac{\partial f}{\partial z}\widehat{𝐳}$
3. $\frac{\partial f}{\partial r}\widehat{𝒓}+\frac{1}{r}\frac{\partial f}{\partial \theta }\hat{\mathit{\theta }}+\frac{1}{r\sin \theta }\frac{\partial f}{\partial \varphi }\hat{\mathit{\varphi }}$
4. $\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \sigma }\hat{\mathit{\sigma }}+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial f}{\partial \tau }\hat{\mathit{\tau }}+\frac{\partial f}{\partial z}\widehat{𝐳}$

/Divergence of a vector field/*

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

1. $\frac{\partial A_x}{\partial x}+\frac{\partial A_y}{\partial y}+\frac{\partial A_z}{\partial z}$
2. $\frac{1}{\rho }\frac{\partial \left(\rho A_{\rho }\right)}{\partial \rho }+\frac{1}{\rho }\frac{\partial A_{\varphi }}{\partial \varphi }+\frac{\partial A_z}{\partial z}$
3. $\frac{1}{r^2}\frac{\partial \left(r^2{A}_r\right)}{\partial r}+\frac{1}{r\sin \theta }\frac{\partial }{\partial \theta }(A_{\theta }\sin \theta )+\frac{1}{r\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi }$
4. $\frac{1}{{\sigma }^2+{\tau }^2}(\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\sigma })}{\partial \sigma }+\frac{\partial (\sqrt{{\sigma }^2+{\tau }^2}{A}_{\tau })}{\partial \tau })+\frac{\partial A_z}{\partial z}$

/Curl of a vector field/

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

1. $(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\widehat{𝐱}+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\widehat{𝐲}+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\widehat{𝐳}$
2. $(\frac{1}{\rho }\frac{\partial A_z}{\partial \varphi }-\frac{\partial A_{\varphi }}{\partial z})\hat{\mathit{\rho }}+(\frac{\partial A_{\rho }}{\partial z}-\frac{\partial A_z}{\partial \rho })\hat{\mathit{\varphi }}+\frac{1}{\rho }(\frac{\partial \left(\rho A_{\varphi }\right)}{\partial \rho }-\frac{\partial A_{\rho }}{\partial \varphi })\widehat{𝐳}$
3. $\frac{1}{r\sin \theta }(\frac{\partial }{\partial \theta }(A_{\varphi }\sin \theta )-\frac{\partial A_{\theta }}{\partial \varphi })\widehat{𝒓}+\frac{1}{r}(\frac{1}{\sin \theta }\frac{\partial A_r}{\partial \varphi }-\frac{\partial }{\partial r}\left(r{A}_{\varphi }\right))\hat{\mathit{\theta }}+\frac{1}{r}(\frac{\partial }{\partial r}\left(r{A}_{\theta }\right)-\frac{\partial A_r}{\partial \theta })\hat{\mathit{\varphi }}$
4. $(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \tau }-\frac{\partial A_{\tau }}{\partial z})\hat{\mathit{\sigma }}-(\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}\frac{\partial A_z}{\partial \sigma }-\frac{\partial A_{\sigma }}{\partial z})\hat{\mathit{\tau }}$$+\frac{1}{\sqrt{{\sigma }^2+{\tau }^2}}(\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 })\widehat{𝐳}$

/Laplacian of a scalar field/

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

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

/Laplacian of a vector field/

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

1. $\mathrm{\Delta }{A}_x\widehat{𝐱}+\mathrm{\Delta }{A}_y\widehat{𝐲}+\mathrm{\Delta }{A}_z\widehat{𝐳}$
2. $(\mathrm{\Delta }{A}_{\rho }-\frac{A_{\rho }}{{\rho }^2}-\frac{2}{{\rho }^2}\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\rho }}$$+(\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{{\rho }^2}+\frac{2}{{\rho }^2}\frac{\partial A_{\rho }}{\partial \varphi })\hat{\mathit{\varphi }}$$+\mathrm{\Delta }{A}_z\widehat{𝐳}$
3. $(\mathrm{\Delta }{A}_r-\frac{2A_r}{r^2}-\frac{2}{r^2\sin \theta }\frac{\partial (A_{\theta }\sin \theta )}{\partial \theta }-\frac{2}{r^2\sin \theta }\frac{\partial A_{\varphi }}{\partial \varphi })\widehat{𝒓}$$+(\mathrm{\Delta }{A}_{\theta }-\frac{A_{\theta }}{r^2{\sin }^2\theta }+\frac{2}{r^2}\frac{\partial A_r}{\partial \theta }-\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\varphi }}{\partial \varphi })\hat{\mathit{\theta }}$$+(\mathrm{\Delta }{A}_{\varphi }-\frac{A_{\varphi }}{r^2{\sin }^2\theta }+\frac{2}{r^2\sin \theta }\frac{\partial A_r}{\partial \varphi }+\frac{2\cos \theta }{r^2{\sin }^2\theta }\frac{\partial A_{\theta }}{\partial \varphi })\hat{\mathit{\varphi }}$

/Material derivative of a vector field/

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

1. $(A_x\frac{\partial B_x}{\partial x}+A_y\frac{\partial B_x}{\partial y}+A_z\frac{\partial B_x}{\partial z})\widehat{𝐱}$$+(A_x\frac{\partial B_y}{\partial x}+A_y\frac{\partial B_y}{\partial y}+A_z\frac{\partial B_y}{\partial z})\widehat{𝐲}$$+(A_x\frac{\partial B_z}{\partial x}+A_y\frac{\partial B_z}{\partial y}+A_z\frac{\partial B_z}{\partial z})\widehat{𝐳}$
2. $(A_{\rho }\frac{\partial B_{\rho }}{\partial \rho }+\frac{A_{\varphi }}{\rho }\frac{\partial B_{\rho }}{\partial \varphi }+A_z\frac{\partial B_{\rho }}{\partial z}-\frac{A_{\varphi }{B}_{\varphi }}{\rho })\hat{\mathit{\rho }}$$+(A_{\rho }\frac{\partial B_{\varphi }}{\partial \rho }+\frac{A_{\varphi }}{\rho }\frac{\partial B_{\varphi }}{\partial \varphi }+A_z\frac{\partial B_{\varphi }}{\partial z}+\frac{A_{\varphi }{B}_{\rho }}{\rho })\hat{\mathit{\varphi }}$$+(A_{\rho }\frac{\partial B_z}{\partial \rho }+\frac{A_{\varphi }}{\rho }\frac{\partial B_z}{\partial \varphi }+A_z\frac{\partial B_z}{\partial z})\widehat{𝐳}$
3. $(A_r\frac{\partial B_r}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_r}{\partial \theta }+\frac{A_{\varphi }}{r\sin \theta }\frac{\partial B_r}{\partial \varphi }-\frac{A_{\theta }{B}_{\theta }+A_{\varphi }{B}_{\varphi }}{r})\widehat{𝒓}$$+(A_r\frac{\partial B_{\theta }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\theta }}{\partial \theta }+\frac{A_{\varphi }}{r\sin \theta }\frac{\partial B_{\theta }}{\partial \varphi }+\frac{A_{\theta }{B}_r}{r}-\frac{A_{\varphi }{B}_{\varphi }\cot \theta }{r})\hat{\mathit{\theta }}$$+(A_r\frac{\partial B_{\varphi }}{\partial r}+\frac{A_{\theta }}{r}\frac{\partial B_{\varphi }}{\partial \theta }+\frac{A_{\varphi }}{r\sin \theta }\frac{\partial B_{\varphi }}{\partial \varphi }+\frac{A_{\varphi }{B}_r}{r}+\frac{A_{\varphi }{B}_{\theta }\cot \theta }{r})\hat{\mathit{\varphi }}$

/Differential displacement/

1. $d𝐥=dx\widehat{𝐱}+dy\widehat{𝐲}+dz\widehat{𝐳}$
2. $d𝐥=d\rho \hat{\mathit{\rho }}+\rho d\varphi \hat{\mathit{\varphi }}+dz\widehat{𝐳}$
3. $d𝐥=dr\widehat{𝐫}+rd\theta \hat{\mathit{\theta }}+r\sin \theta d\varphi \hat{\mathit{\varphi }}$
4. $d𝐥=\sqrt{{\sigma }^2+{\tau }^2}d\sigma \hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\tau \hat{\mathit{\tau }}+dz\widehat{𝐳}$

/Differential normal areas/

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

Differential normal area $d𝐒$

1. $dydz\widehat{𝐱}+dxdz\widehat{𝐲}+dxdy\widehat{𝐳}$
2. $\rho d\varphi dz\hat{\mathit{\rho }}+d\rho dz\hat{\mathit{\varphi }}+\rho d\rho d\varphi \widehat{𝐳}$
3. $r^2\sin \theta d\theta d\varphi \widehat{𝐫}+r\sin \theta drd\varphi \hat{\mathit{\theta }}+rdrd\theta \hat{\mathit{\varphi }}$
4. $\sqrt{{\sigma }^2+{\tau }^2}d\tau dz\hat{\mathit{\sigma }}+\sqrt{{\sigma }^2+{\tau }^2}d\sigma dz\hat{\mathit{\tau }}+({\sigma }^2+{\tau }^2)d\sigma d\tau \widehat{𝐳}$

/Differential volume/

1. $dV=dxdydz$Template:Padverified
2. $dV=\rho d\rho d\varphi dz$Template:Padverified
3. $dV=r^2\sin \theta drd\theta d\varphi$Template:Padverified
4. $dV=({\sigma }^2+{\tau }^2)d\sigma d\tau dz$

/nabla's on nabla's/

Non-trivial calculation rules:

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

References

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Backup copy from Wikipedia

Copy or read but never change /Original Copy from Wikipedia/