From e2f6ff4487606f4052b9c912b9faa2c8d8f1ca10 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Sun, 18 Jun 2023 17:59:42 +0200 Subject: Improve knowledge base --- source/know/concept/capillary-action/index.md | 126 -------- source/know/concept/capillary-length/index.md | 79 +++++ source/know/concept/cartesian-coordinates/index.md | 4 +- .../cylindrical-parabolic-coordinates/index.md | 296 ------------------- .../concept/cylindrical-polar-coordinates/index.md | 296 ------------------- source/know/concept/drude-model/index.md | 6 +- source/know/concept/jurins-law/index.md | 78 +++++ .../orthogonal-curvilinear-coordinates/index.md | 6 +- .../parabolic-cylindrical-coordinates/index.md | 296 +++++++++++++++++++ .../concept/polar-cylindrical-coordinates/index.md | 296 +++++++++++++++++++ .../know/concept/sturm-liouville-theory/index.md | 321 +++++++++++---------- 11 files changed, 921 insertions(+), 883 deletions(-) delete mode 100644 source/know/concept/capillary-action/index.md create mode 100644 source/know/concept/capillary-length/index.md delete mode 100644 source/know/concept/cylindrical-parabolic-coordinates/index.md delete mode 100644 source/know/concept/cylindrical-polar-coordinates/index.md create mode 100644 source/know/concept/jurins-law/index.md create mode 100644 source/know/concept/parabolic-cylindrical-coordinates/index.md create mode 100644 source/know/concept/polar-cylindrical-coordinates/index.md (limited to 'source/know') diff --git a/source/know/concept/capillary-action/index.md b/source/know/concept/capillary-action/index.md deleted file mode 100644 index fea6ef8..0000000 --- a/source/know/concept/capillary-action/index.md +++ /dev/null @@ -1,126 +0,0 @@ ---- -title: "Capillary action" -sort_title: "Capillary action" -date: 2021-03-29 -categories: -- Physics -- Fluid mechanics -- Fluid statics -- Surface tension -layout: "concept" ---- - -**Capillary action** refers to the movement of liquid -through narrow spaces due to surface tension, often against gravity. -It occurs when the [Laplace pressure](/know/concept/young-laplace-law/) -from surface tension is much larger in magnitude than the -[hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity. - -Consider a spherical droplet of liquid with radius $$R$$. -The hydrostatic pressure difference -between the top and bottom of the drop -is much smaller than the Laplace pressure: - -$$\begin{aligned} - 2 R \rho g \ll 2 \frac{\alpha}{R} -\end{aligned}$$ - -Where $$\rho$$ is the density of the liquid, -$$g$$ is the acceleration due to gravity, -and $$\alpha$$ is the energy cost per unit surface area. -Rearranging the inequality yields: - -$$\begin{aligned} - R^2 \ll \frac{\alpha}{\rho g} -\end{aligned}$$ - -From the right-hand side we define the **capillary length** $$L_c$$, -so gravity is negligible if $$R \ll L_c$$: - -$$\begin{aligned} - \boxed{ - L_c - \equiv \sqrt{\frac{\alpha}{\rho g}} - } -\end{aligned}$$ - -In general, for a system with characteristic length $$L$$, -the relative strength of gravity compared to surface tension -is described by the **Bond number** $$\mathrm{Bo}$$ -or **Eötvös number** $$\mathrm{Eo}$$: - -$$\begin{aligned} - \boxed{ - \mathrm{Bo} - \equiv \mathrm{Eo} - \equiv \frac{L^2}{L_c^2} - = \frac{m g}{\alpha L} - } -\end{aligned}$$ - -The rightmost side gives an alternative way of understanding $$\mathrm{Bo}$$: -$$m$$ is the mass of a cube with side $$L$$, such that the numerator is the weight force, -and the denominator is the tension force of the surface. -In any case, capillary action can be observed when $$\mathrm{Bo \ll 1}$$. - -The most famous example of capillary action is **capillary rise**, -where a liquid "climbs" upwards in a narrow vertical tube with radius $$R$$, -apparently defying gravity. -Assuming the liquid-air interface is a spherical cap -with constant [curvature](/know/concept/curvature/) radius $$R_c$$, -then we know that the liquid is at rest -when the hydrostatic pressure equals the Laplace pressure: - -$$\begin{aligned} - \rho g h - \approx \alpha \frac{2}{R_c} - = 2 \alpha \frac{\cos\theta}{R} -\end{aligned}$$ - -Where $$\theta$$ is the liquid-tube contact angle, -and we are neglecting variations of the height $$h$$ due to the curvature -(i.e. the [meniscus](/know/concept/meniscus/)). -By isolating the above equation for $$h$$, -we arrive at **Jurin's law**, -which predicts the height climbed by a liquid in a tube with radius $$R$$: - -$$\begin{aligned} - \boxed{ - h - = 2 \frac{L_c^2}{R} \cos\theta - } -\end{aligned}$$ - -Depending on $$\theta$$, $$h$$ can be negative, -i.e. the liquid might descend below the ambient level. - - -An alternative derivation of Jurin's law balances the forces instead of the pressures. -On the right, we have the gravitational force -(i.e. the energy-per-distance to lift the liquid), -and on the left, the surface tension force -(i.e. the energy-per-distance of the liquid-tube interface): - -$$\begin{aligned} - \pi R^2 \rho g h - \approx 2 \pi R (\alpha_{sg} - \alpha_{sl}) -\end{aligned}$$ - -Where $$\alpha_{sg}$$ and $$\alpha_{sl}$$ are the energy costs -of the solid-gas and solid-liquid interfaces. -Thanks to the [Young-Dupré relation](/know/concept/young-dupre-relation/), -we can rewrite this as follows: - -$$\begin{aligned} - R \rho g h - = 2 \alpha \cos\theta -\end{aligned}$$ - -Isolating this for $$h$$ simply yields Jurin's law again, as expected. - - - -## References -1. B. Lautrup, - *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, - CRC Press. diff --git a/source/know/concept/capillary-length/index.md b/source/know/concept/capillary-length/index.md new file mode 100644 index 0000000..4dbb788 --- /dev/null +++ b/source/know/concept/capillary-length/index.md @@ -0,0 +1,79 @@ +--- +title: "Capillary length" +sort_title: "Capillary length" +date: 2021-03-29 +categories: +- Physics +- Fluid mechanics +- Fluid statics +- Surface tension +layout: "concept" +--- + +**Capillary action** refers to the movement of liquid +through narrow spaces due to surface tension, often against gravity. +It occurs when the [Laplace pressure](/know/concept/young-laplace-law/) +from surface tension is much larger in magnitude than the +[hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity. + +Consider a spherical droplet of liquid with radius $$R$$. +The hydrostatic pressure difference +between the top and bottom of the drop +is much smaller than the Laplace pressure: + +$$\begin{aligned} + 2 R \rho g \ll 2 \frac{\alpha}{R} +\end{aligned}$$ + +Where $$\rho$$ is the density of the liquid, +$$g$$ is the acceleration due to gravity, +and $$\alpha$$ is the energy cost per unit surface area. +Rearranging the inequality yields: + +$$\begin{aligned} + R^2 \ll \frac{\alpha}{\rho g} +\end{aligned}$$ + +From this, we define the **capillary length** $$L_c$$ +such that gravity is negligible if $$R \ll L_c$$: + +$$\begin{aligned} + \boxed{ + L_c + \equiv \sqrt{\frac{\alpha}{\rho g}} + } +\end{aligned}$$ + +In general, for a system with characteristic length $$L$$, +the relative strength of gravity compared to surface tension +is described by the **Bond number** $$\mathrm{Bo}$$ +or **Eötvös number** $$\mathrm{Eo}$$: + +$$\begin{aligned} + \boxed{ + \mathrm{Bo} + \equiv \mathrm{Eo} + \equiv \frac{L^2}{L_c^2} + } +\end{aligned}$$ + +Capillary action is observed when $$\mathrm{Bo \ll 1}$$, +while for $$\mathrm{Bo} \gg 1$$ surface tension is negligible. + +For an alternative interpretation of $$\mathrm{Bo}$$, +let $$m \equiv \rho L^3$$ be the mass of a cube with side $$L$$ +such that its weight is $$m g$$. +The tension force on its face is $$\alpha L$$, +so $$\mathrm{Bo}$$ is simply the force ratio: + +$$\begin{aligned} + \mathrm{Bo} + = \frac{m g}{\alpha L} +\end{aligned}$$ + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/source/know/concept/cartesian-coordinates/index.md b/source/know/concept/cartesian-coordinates/index.md index d198e84..a0bfc39 100644 --- a/source/know/concept/cartesian-coordinates/index.md +++ b/source/know/concept/cartesian-coordinates/index.md @@ -11,8 +11,8 @@ layout: "concept" This article is a supplement to the ones on [orthogonal curvilinear systems](/know/concept/orthogonal-curvilinear-coordinates/), [spherical coordinates](/know/concept/spherical-coordinates/), -[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), -and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/). +[polar cylindrical coordinates](/know/concept/polar-cylindrical-coordinates/), +and [parabolic cylindrical coordinates](/know/concept/parabolic-cylindrical-coordinates/). The well-known Cartesian coordinate system $$(x, y, z)$$ has trivial **scale factors**: diff --git a/source/know/concept/cylindrical-parabolic-coordinates/index.md b/source/know/concept/cylindrical-parabolic-coordinates/index.md deleted file mode 100644 index 58358dd..0000000 --- a/source/know/concept/cylindrical-parabolic-coordinates/index.md +++ /dev/null @@ -1,296 +0,0 @@ ---- -title: "Cylindrical parabolic coordinates" -sort_title: "Cylindrical parabolic coordinates" -date: 2021-03-04 -categories: -- Mathematics -- Physics -layout: "concept" ---- - -**Cylindrical parabolic coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D, -by describing a point in space using the variables $$(\sigma, \tau, z)$$. -The $$z$$-axis is the same as in the Cartesian system, (hence the name *cylindrical*), -while the coordinate lines of $$\sigma$$ and $$\tau$$ are confocal parabolas. - -[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ -and this system $$(\sigma, \tau, z)$$ are related by: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - x - &= \frac{1}{2} (\tau^2 - \sigma^2) - \\ - y - &= \sigma \tau - \\ - z - &= z - \end{aligned} - } -\end{aligned}$$ - -Conversely, a point given in $$(x, y, z)$$ can be converted -to $$(\sigma, \tau, z)$$ using these formulae: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \sigma - &= \sgn(x) \sqrt{\sqrt{x^2 + y^2} - x} - \\ - \tau - &= \sqrt{\sqrt{x^2 + y^2} + x} - \\ - z - &= z - \end{aligned} - } -\end{aligned}$$ - -Cylindrical parabolic coordinates form -an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), -whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need. -To get those, we calculate the unnormalized local basis: - -$$\begin{aligned} - h_\sigma \vu{e}_\sigma - &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma} - \\ - &= - \vu{e}_x \sigma + \vu{e}_y \tau - \\ - h_\tau \vu{e}_\tau - &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau} - \\ - &= \vu{e}_x \tau + \vu{e}_y \sigma - \\ - h_\sigma \vu{e}_\sigma - &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} - \\ - &= \vu{e}_z -\end{aligned}$$ - -By normalizing the **local basis vectors** -$$\vu{e}_\sigma$$, $$\vu{e}_\tau$$ and $$\vu{e}_z$$, -we arrive at these expressions, -where we have defined the abbreviation $$\rho$$ for convenience: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - h_\sigma - &= \rho - \equiv \sqrt{\sigma^2 + \tau^2} - \\ - h_\tau - &= \rho - \equiv \sqrt{\sigma^2 + \tau^2} - \\ - h_z - &= 1 - \end{aligned} - } - \qquad\qquad - \boxed{ - \begin{aligned} - \vu{e}_\sigma - &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho} - \\ - \vu{e}_\tau - &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho} - \\ - \vu{e}_z - &= \vu{e}_z - \end{aligned} - } -\end{aligned}$$ - -Thanks to these scale factors, we can easily convert calculus from the Cartesian system -using the standard formulae for orthogonal curvilinear coordinates. - - - -## Differential elements - -For line integrals, -the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows: - -$$\begin{aligned} - \boxed{ - \dd{\vb{\ell}} - = \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z} - } -\end{aligned}$$ - -For surface integrals, -the normal vector element $$\dd{\vb{S}}$$ for a surface is given by: - -$$\begin{aligned} - \boxed{ - \dd{\vb{S}} - = \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau} - } -\end{aligned}$$ - -And for volume integrals, -the infinitesimal volume $$\dd{V}$$ takes the following form: - -$$\begin{aligned} - \boxed{ - \dd{V} - = \rho^2 \dd{\sigma} \dd{\tau} \dd{z} - } -\end{aligned}$$ - - - -## Common operations - -The basic vector operations (gradient, divergence, curl and Laplacian) are given by: - -$$\begin{aligned} - \boxed{ - \nabla f - = \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma} - + \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau} - + \vu{e}_z \pdv{f}{z} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \nabla \cdot \vb{V} - = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3} - + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3} - + \pdv{V_z}{z} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla \times \vb{V} - &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg) - \\ - &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3} - - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \nabla^2 f - = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z} - } -\end{aligned}$$ - - - -## Uncommon operations - -Uncommon operations include: -the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$, -the gradient of a vector $$\nabla \vb{V}$$, -the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$, -the Laplacian of a vector $$\nabla^2 \vb{V}$$, -and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla (\nabla \cdot \vb{V}) - &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau} - + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z} - \\ - &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau} - + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg) - \\ - &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} - + \frac{1}{\rho} \mpdv{V_z}{\tau}{z} - \\ - &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau} - - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z} - + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla \vb{V} - &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg) - + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg) - + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma} - \\ - &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg) - + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg) - + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau} - \\ - &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z} - + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z} - + \vu{e}_z \vu{e}_z \pdv{V_z}{z} - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - (\vb{U} \cdot \nabla) \vb{V} - &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z} - + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg) - \\ - &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z} - + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla^2 \vb{V} - &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z} - + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg) - \\ - &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z} - - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla \cdot \overline{\overline{\mathbf{T}}} - &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z} - + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3} - + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg) - \\ - &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z} - - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3} - + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg) - \\ - &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z} - + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg) - \end{aligned} - } -\end{aligned}$$ - - - -## References -1. M.L. Boas, - *Mathematical methods in the physical sciences*, 2nd edition, - Wiley. diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/cylindrical-polar-coordinates/index.md deleted file mode 100644 index cf227a6..0000000 --- a/source/know/concept/cylindrical-polar-coordinates/index.md +++ /dev/null @@ -1,296 +0,0 @@ ---- -title: "Cylindrical polar coordinates" -sort_title: "Cylindrical polar coordinates" -date: 2021-07-26 -categories: -- Mathematics -- Physics -layout: "concept" ---- - -**Cylindrical polar coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D, -by describing the location of a point in space -using the variables $$(r, \varphi, z)$$. -The $$z$$-axis is unchanged from the Cartesian system, -hence the name *cylindrical*. - -[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ -and the cylindrical system $$(r, \varphi, z)$$ are related by: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - x - &= r \cos{\varphi} - \\ - y - &= r \sin{\varphi} - \\ - z - &= z - \end{aligned} - } -\end{aligned}$$ - -Conversely, a point given in $$(x, y, z)$$ -can be converted to $$(r, \varphi, z)$$ using these formulae, -where $$\mathtt{atan2}$$ is the 2-argument arctangent, -which is needed to handle the signs correctly: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - r - &= \sqrt{x^2 + y^2} - \\ - \varphi - &= \mathtt{atan2}(y, x) - \\ - z - &= z - \end{aligned} - } -\end{aligned}$$ - -Cylindrical polar coordinates form -an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), -whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need. -To get those, we calculate the unnormalized local basis: - -$$\begin{aligned} - h_r \vu{e}_r - &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r} - \\ - &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi} - \\ - h_\varphi \vu{e}_\varphi - &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi} - \\ - &= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi} - \\ - h_z \vu{e}_z - &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} - \\ - &= \vu{e}_z -\end{aligned}$$ - -By normalizing the **local basis vectors** -$$\vu{e}_r$$, $$\vu{e}_\varphi$$ and $$\vu{e}_z$$, -we arrive at these expressions: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - h_r - &= 1 - \\ - h_\varphi - &= r - \\ - h_z - &= 1 - \end{aligned} - } - \qquad\qquad - \boxed{ - \begin{aligned} - \vu{e}_r - &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi} - \\ - \vu{e}_\varphi - &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi} - \\ - \vu{e}_z - &= \vu{e}_z - \end{aligned} - } -\end{aligned}$$ - -Thanks to these scale factors, we can easily convert calculus from the Cartesian system -using the standard formulae for orthogonal curvilinear coordinates. - - - -## Differential elements - -For line integrals, -the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows: - -$$\begin{aligned} - \boxed{ - \dd{\vb{\ell}} - = \vu{e}_r \dd{r} - + \: \vu{e}_\varphi \: r \dd{\varphi} - + \: \vu{e}_z \dd{z} - } -\end{aligned}$$ - -For surface integrals, -the normal vector element $$\dd{\vb{S}}$$ for a surface is given by: - -$$\begin{aligned} - \boxed{ - \dd{\vb{S}} - = \vu{e}_r \: r \dd{\varphi} \dd{z} - + \: \vu{e}_\varphi \dd{r} \dd{z} - + \: \vu{e}_z \: r \dd{r} \dd{\varphi} - } -\end{aligned}$$ - -And for volume integrals, -the infinitesimal volume $$\dd{V}$$ takes the following form: - -$$\begin{aligned} - \boxed{ - \dd{V} - = r \dd{r} \dd{\varphi} \dd{z} - } -\end{aligned}$$ - - - -## Common operations - -The basic vector operations (gradient, divergence, curl and Laplacian) are given by: - -$$\begin{aligned} - \boxed{ - \nabla f - = \vu{e}_r \pdv{f}{r} - + \vu{e}_\varphi \frac{1}{r} \pdv{f}{\varphi} - + \mathbf{e}_z \pdv{f}{z} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \nabla \cdot \vb{V} - = \pdv{V_r}{r} + \frac{V_r}{r} - + \frac{1}{r} \pdv{V_\varphi}{\varphi} - + \pdv{V_z}{z} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla \times \vb{V} - &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \bigg) - \\ - &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_r}{z} - \pdv{V_z}{r} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \pdv{V_\varphi}{r} + \frac{V_\varphi}{r} - \frac{1}{r} \pdv{V_r}{\varphi} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \nabla^2 f - = \pdvn{2}{f}{r} + \frac{1}{r} \pdv{f}{r} - + \frac{1}{r^2} \pdvn{2}{f}{\varphi} - + \pdvn{2}{f}{z} - } -\end{aligned}$$ - - - -## Uncommon operations - -Uncommon operations include: -the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$, -the gradient of a vector $$\nabla \vb{V}$$, -the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$, -the Laplacian of a vector $$\nabla^2 \vb{V}$$, -and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$: - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla (\nabla \cdot \vb{V}) - &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\varphi}{r}{\varphi} + \mpdv{V_z}{r}{z} - + \frac{1}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg) - \\ - &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} - + \frac{1}{r} \mpdv{V_z}{\varphi}{z} + \frac{1}{r^2} \pdv{V_r}{\varphi} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \mpdv{V_r}{z}{r} + \frac{1}{r} \mpdv{V_\varphi}{z}{\varphi} + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_r}{z} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla \vb{V} - &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r} - + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r} - + \vu{e}_r \vu{e}_z \pdv{V_z}{r} - \\ - &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg) - + \vu{e}_\varphi \vu{e}_\varphi \bigg( \frac{1}{r} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} \bigg) - + \vu{e}_\varphi \vu{e}_z \frac{1}{r} \pdv{V_z}{\varphi} - \\ - &\quad\: + \vu{e}_z \vu{e}_r \pdv{V_r}{z} - + \vu{e}_z \vu{e}_\varphi \pdv{V_\varphi}{z} - + \vu{e}_z \vu{e}_z \pdv{V_z}{z} - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - (\vb{U} \cdot \nabla) \vb{V} - &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\varphi}{r} \pdv{V_r}{\varphi} + U_z \pdv{V_r}{z} - - \frac{U_\varphi V_\varphi}{r} \bigg) - \\ - &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\varphi}{r} \pdv{V_\varphi}{\varphi} + U_z \pdv{V_\varphi}{z} - + \frac{U_\varphi V_r}{r} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( U_r \pdv{V_z}{r} + \frac{U_\varphi}{r} \pdv{V_z}{\varphi} + U_z \pdv{V_z}{z} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla^2 \vb{V} - &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z} - + \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg) - \\ - &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z} - + \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} - + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg) - \end{aligned} - } -\end{aligned}$$ - -$$\begin{aligned} - \boxed{ - \begin{aligned} - \nabla \cdot \overline{\overline{\mathbf{T}}} - &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\varphi r}}{\varphi} + \pdv{T_{zr}}{z} - + \frac{T_{rr}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg) - \\ - &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\varphi \varphi}}{\varphi} + \pdv{T_{z \varphi}}{z} - + \frac{T_{r \varphi}}{r} + \frac{T_{\varphi r}}{r} \bigg) - \\ - &\quad\: + \vu{e}_z \bigg( \pdv{T_{rz}}{r} + \frac{1}{r} \pdv{T_{\varphi z}}{\varphi} + \pdv{T_{zz}}{z} - + \frac{T_{rz}}{r} \bigg) - \end{aligned} - } -\end{aligned}$$ - - - -## References -1. M.L. Boas, - *Mathematical methods in the physical sciences*, 2nd edition, - Wiley. -2. B. Lautrup, - *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, - CRC Press. diff --git a/source/know/concept/drude-model/index.md b/source/know/concept/drude-model/index.md index c4faf81..0026d90 100644 --- a/source/know/concept/drude-model/index.md +++ b/source/know/concept/drude-model/index.md @@ -11,7 +11,7 @@ layout: "concept" The **Drude model**, also known as the **Drude-Lorentz model** due to its analogy -to the *Lorentz oscillator model* +to the [Lorentz oscillator model](/know/concept/lorentz-oscillator-model/) classically predicts the [dielectric function](/know/concept/dielectric-function/) and electric conductivity of a gas of free charges, as found in metals and doped semiconductors. @@ -59,7 +59,7 @@ $$\begin{aligned} = - \frac{N q^2}{m (\omega^2 + i \gamma \omega)} \vb{E}(t) \end{aligned}$$ -The electric displacement field $$\vb{D}$$ is then as follows, +The electric displacement field $$\vb{D}(t)$$ is then as follows, where the parenthesized expression is the dielectric function $$\varepsilon_r$$ of the material: @@ -180,7 +180,7 @@ We must replace the carriers' true mass $$m$$ with their *effective mass* $$m^*$ found from the material's electronic band structure. Furthermore, semiconductors already have a high intrinsic dielectric function $$\varepsilon_{\mathrm{int}}$$ -before being doped, so the displacement field $$\vb{D}$$ becomes: +before being doped, so the displacement field $$\vb{D}(t)$$ becomes: $$\begin{aligned} \vb{D} diff --git a/source/know/concept/jurins-law/index.md b/source/know/concept/jurins-law/index.md new file mode 100644 index 0000000..6214477 --- /dev/null +++ b/source/know/concept/jurins-law/index.md @@ -0,0 +1,78 @@ +--- +title: "Jurin's law" +sort_title: "Jurin's law" +date: 2023-06-15 +categories: +- Physics +- Fluid mechanics +- Fluid statics +- Surface tension +layout: "concept" +--- + +A well-known example of *capillary action* is +when a liquid climbs up a narrow vertical tube with radius $$R$$, +apparently defying gravity. +Indeed, this occurs when the liquid's surface tension can overpower gravity; +specifically, when the [capillary length](/know/concept/capillary-length/) $$L_c > R$$. + +Let us assume that the liquid-air interface has a spherical shape, +which may point up or down depending on the liquid. +This interface then has a constant [curvature radius](/know/concept/curvature/) $$r$$ +determined by the contact angle $$\theta$$ of the liquid to the tube: +$$r = R / \cos{\theta}$$. We know that the liquid is at rest +when the [hydrostatic pressure](/know/concept/hydrostatic-pressure/) +equals the resulting [Laplace pressure](/know/concept/young-laplace-law/): + +$$\begin{aligned} + \rho g h + = \alpha \frac{2}{r} + = 2 \alpha \frac{\cos{\theta}}{R} +\end{aligned}$$ + +Note that $$h$$ is the height of interface's highest/lowest point; +we neglect the [meniscus](/know/concept/meniscus/). +By isolating the above equation for $$h$$, we arrive at **Jurin's law**: + +$$\begin{aligned} + \boxed{ + h + = \frac{2 \alpha \cos{\theta}}{\rho g R} + = 2 \frac{L_c^2}{R} \cos{\theta} + } +\end{aligned}$$ + +Where $$L_c \equiv \sqrt{\alpha / \rho g}$$. +This predicts the height climbed by a liquid in a narrow tube. +If $$\theta > 90\degree$$, then $$h$$ is negative, +i.e. the liquid descends below the ambient level. + +An alternative derivation of Jurin's law balances the forces instead of the pressures. +On the right, we have the gravitational force +(i.e. the energy-per-distance to lift the liquid), +and on the left, the surface tension force +(i.e. the energy-per-distance of the liquid-tube interface): + +$$\begin{aligned} + \pi R^2 \rho g h + \approx 2 \pi R (\alpha_{sg} - \alpha_{sl}) +\end{aligned}$$ + +Where $$\alpha_{sg}$$ and $$\alpha_{sl}$$ are the energy costs +of the solid-gas and solid-liquid interfaces. +Thanks to the [Young-Dupré relation](/know/concept/young-dupre-relation/), +we can rewrite this as follows: + +$$\begin{aligned} + R \rho g h + = 2 \alpha \cos\theta +\end{aligned}$$ + +Isolating this for $$h$$ simply yields Jurin's law again, as expected. + + + +## References +1. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/source/know/concept/orthogonal-curvilinear-coordinates/index.md b/source/know/concept/orthogonal-curvilinear-coordinates/index.md index 669358c..4714a95 100644 --- a/source/know/concept/orthogonal-curvilinear-coordinates/index.md +++ b/source/know/concept/orthogonal-curvilinear-coordinates/index.md @@ -18,10 +18,10 @@ where at least one of the coordinate surfaces is curved: e.g. in cylindrical coordinates, the coordinate line of $$r$$ and $$z$$ is a circle. Here we limit ourselves to **orthogonal** systems, where the coordinate surfaces are always perpendicular. -Examples of such orthogonal curvilinear systems include +Examples of such orthogonal curvilinear systems are [spherical coordinates](/know/concept/spherical-coordinates/), -[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), -[cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/), +[polar cylindrical coordinates](/know/concept/polar-cylindrical-coordinates/), +[parabolic cylindrical coordinates](/know/concept/parabolic-cylindrical-coordinates/), and (trivially) [Cartesian coordinates](/know/concept/cartesian-coordinates/). diff --git a/source/know/concept/parabolic-cylindrical-coordinates/index.md b/source/know/concept/parabolic-cylindrical-coordinates/index.md new file mode 100644 index 0000000..6ba19f5 --- /dev/null +++ b/source/know/concept/parabolic-cylindrical-coordinates/index.md @@ -0,0 +1,296 @@ +--- +title: "Parabolic cylindrical coordinates" +sort_title: "Parabolic cylindrical coordinates" +date: 2021-03-04 +categories: +- Mathematics +- Physics +layout: "concept" +--- + +**Parabolic cylindrical coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D, +by describing a point in space using the variables $$(\sigma, \tau, z)$$. +The $$z$$-axis is the same as in the Cartesian system, (hence the name *cylindrical*), +while the coordinate lines of $$\sigma$$ and $$\tau$$ are confocal parabolas. + +[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ +and this system $$(\sigma, \tau, z)$$ are related by: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + x + &= \frac{1}{2} (\tau^2 - \sigma^2) + \\ + y + &= \sigma \tau + \\ + z + &= z + \end{aligned} + } +\end{aligned}$$ + +Conversely, a point given in $$(x, y, z)$$ can be converted +to $$(\sigma, \tau, z)$$ using these formulae: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \sigma + &= \sqrt{\sqrt{x^2 + y^2} - x} + \\ + \tau + &= \sgn(y) \sqrt{\sqrt{x^2 + y^2} + x} + \\ + z + &= z + \end{aligned} + } +\end{aligned}$$ + +Parabolic cylindrical coordinates form +an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), +whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need. +To get those, we calculate the unnormalized local basis: + +$$\begin{aligned} + h_\sigma \vu{e}_\sigma + &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma} + \\ + &= - \vu{e}_x \sigma + \vu{e}_y \tau + \\ + h_\tau \vu{e}_\tau + &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau} + \\ + &= \vu{e}_x \tau + \vu{e}_y \sigma + \\ + h_\sigma \vu{e}_\sigma + &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} + \\ + &= \vu{e}_z +\end{aligned}$$ + +By normalizing the **local basis vectors** +$$\vu{e}_\sigma$$, $$\vu{e}_\tau$$ and $$\vu{e}_z$$, +we arrive at these expressions, +where we have defined the abbreviation $$\rho$$ for convenience: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + h_\sigma + &= \rho + \equiv \sqrt{\sigma^2 + \tau^2} + \\ + h_\tau + &= \rho + \equiv \sqrt{\sigma^2 + \tau^2} + \\ + h_z + &= 1 + \end{aligned} + } + \qquad\qquad + \boxed{ + \begin{aligned} + \vu{e}_\sigma + &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho} + \\ + \vu{e}_\tau + &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho} + \\ + \vu{e}_z + &= \vu{e}_z + \end{aligned} + } +\end{aligned}$$ + +Thanks to these scale factors, we can easily convert calculus from the Cartesian system +using the standard formulae for orthogonal curvilinear coordinates. + + + +## Differential elements + +For line integrals, +the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows: + +$$\begin{aligned} + \boxed{ + \dd{\vb{\ell}} + = \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z} + } +\end{aligned}$$ + +For surface integrals, +the normal vector element $$\dd{\vb{S}}$$ for a surface is given by: + +$$\begin{aligned} + \boxed{ + \dd{\vb{S}} + = \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau} + } +\end{aligned}$$ + +And for volume integrals, +the infinitesimal volume $$\dd{V}$$ takes the following form: + +$$\begin{aligned} + \boxed{ + \dd{V} + = \rho^2 \dd{\sigma} \dd{\tau} \dd{z} + } +\end{aligned}$$ + + + +## Common operations + +The basic vector operations (gradient, divergence, curl and Laplacian) are given by: + +$$\begin{aligned} + \boxed{ + \nabla f + = \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma} + + \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau} + + \vu{e}_z \pdv{f}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{V} + = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3} + + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3} + + \pdv{V_z}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \times \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3} + - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \nabla^2 f + = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z} + } +\end{aligned}$$ + + + +## Uncommon operations + +Uncommon operations include: +the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$, +the gradient of a vector $$\nabla \vb{V}$$, +the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$, +the Laplacian of a vector $$\nabla^2 \vb{V}$$, +and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla (\nabla \cdot \vb{V}) + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau} + + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z} + \\ + &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau} + + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + + \frac{1}{\rho} \mpdv{V_z}{\tau}{z} + \\ + &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau} + - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z} + + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \vb{V} + &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg) + + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg) + + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma} + \\ + &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg) + + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg) + + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau} + \\ + &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z} + + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z} + + \vu{e}_z \vu{e}_z \pdv{V_z}{z} + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + (\vb{U} \cdot \nabla) \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z} + + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z} + + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla^2 \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z} + + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z} + - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \cdot \overline{\overline{\mathbf{T}}} + &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z} + + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3} + + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg) + \\ + &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z} + - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3} + + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg) + \\ + &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z} + + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg) + \end{aligned} + } +\end{aligned}$$ + + + +## References +1. M.L. Boas, + *Mathematical methods in the physical sciences*, 2nd edition, + Wiley. diff --git a/source/know/concept/polar-cylindrical-coordinates/index.md b/source/know/concept/polar-cylindrical-coordinates/index.md new file mode 100644 index 0000000..2223996 --- /dev/null +++ b/source/know/concept/polar-cylindrical-coordinates/index.md @@ -0,0 +1,296 @@ +--- +title: "Polar cylindrical coordinates" +sort_title: "Polar cylindrical coordinates" +date: 2021-07-26 +categories: +- Mathematics +- Physics +layout: "concept" +--- + +**Polar cylindrical coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D, +by describing the location of a point in space +using the variables $$(r, \varphi, z)$$. +The $$z$$-axis is unchanged from the Cartesian system, +hence the name *cylindrical*. + +[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$ +and the cylindrical system $$(r, \varphi, z)$$ are related by: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + x + &= r \cos{\varphi} + \\ + y + &= r \sin{\varphi} + \\ + z + &= z + \end{aligned} + } +\end{aligned}$$ + +Conversely, a point given in $$(x, y, z)$$ +can be converted to $$(r, \varphi, z)$$ using these formulae, +where $$\mathtt{atan2}$$ is the 2-argument arctangent, +which is needed to handle the signs correctly: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + r + &= \sqrt{x^2 + y^2} + \\ + \varphi + &= \mathtt{atan2}(y, x) + \\ + z + &= z + \end{aligned} + } +\end{aligned}$$ + +Polar cylindrical coordinates form +an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), +whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need. +To get those, we calculate the unnormalized local basis: + +$$\begin{aligned} + h_r \vu{e}_r + &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r} + \\ + &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi} + \\ + h_\varphi \vu{e}_\varphi + &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi} + \\ + &= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi} + \\ + h_z \vu{e}_z + &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} + \\ + &= \vu{e}_z +\end{aligned}$$ + +By normalizing the **local basis vectors** +$$\vu{e}_r$$, $$\vu{e}_\varphi$$ and $$\vu{e}_z$$, +we arrive at these expressions: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + h_r + &= 1 + \\ + h_\varphi + &= r + \\ + h_z + &= 1 + \end{aligned} + } + \qquad\qquad + \boxed{ + \begin{aligned} + \vu{e}_r + &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi} + \\ + \vu{e}_\varphi + &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi} + \\ + \vu{e}_z + &= \vu{e}_z + \end{aligned} + } +\end{aligned}$$ + +Thanks to these scale factors, we can easily convert calculus from the Cartesian system +using the standard formulae for orthogonal curvilinear coordinates. + + + +## Differential elements + +For line integrals, +the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows: + +$$\begin{aligned} + \boxed{ + \dd{\vb{\ell}} + = \vu{e}_r \dd{r} + + \: \vu{e}_\varphi \: r \dd{\varphi} + + \: \vu{e}_z \dd{z} + } +\end{aligned}$$ + +For surface integrals, +the normal vector element $$\dd{\vb{S}}$$ for a surface is given by: + +$$\begin{aligned} + \boxed{ + \dd{\vb{S}} + = \vu{e}_r \: r \dd{\varphi} \dd{z} + + \: \vu{e}_\varphi \dd{r} \dd{z} + + \: \vu{e}_z \: r \dd{r} \dd{\varphi} + } +\end{aligned}$$ + +And for volume integrals, +the infinitesimal volume $$\dd{V}$$ takes the following form: + +$$\begin{aligned} + \boxed{ + \dd{V} + = r \dd{r} \dd{\varphi} \dd{z} + } +\end{aligned}$$ + + + +## Common operations + +The basic vector operations (gradient, divergence, curl and Laplacian) are given by: + +$$\begin{aligned} + \boxed{ + \nabla f + = \vu{e}_r \pdv{f}{r} + + \vu{e}_\varphi \frac{1}{r} \pdv{f}{\varphi} + + \mathbf{e}_z \pdv{f}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{V} + = \pdv{V_r}{r} + \frac{V_r}{r} + + \frac{1}{r} \pdv{V_\varphi}{\varphi} + + \pdv{V_z}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \times \vb{V} + &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \bigg) + \\ + &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_r}{z} - \pdv{V_z}{r} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdv{V_\varphi}{r} + \frac{V_\varphi}{r} - \frac{1}{r} \pdv{V_r}{\varphi} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \nabla^2 f + = \pdvn{2}{f}{r} + \frac{1}{r} \pdv{f}{r} + + \frac{1}{r^2} \pdvn{2}{f}{\varphi} + + \pdvn{2}{f}{z} + } +\end{aligned}$$ + + + +## Uncommon operations + +Uncommon operations include: +the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$, +the gradient of a vector $$\nabla \vb{V}$$, +the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$, +the Laplacian of a vector $$\nabla^2 \vb{V}$$, +and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla (\nabla \cdot \vb{V}) + &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\varphi}{r}{\varphi} + \mpdv{V_z}{r}{z} + + \frac{1}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg) + \\ + &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + + \frac{1}{r} \mpdv{V_z}{\varphi}{z} + \frac{1}{r^2} \pdv{V_r}{\varphi} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \mpdv{V_r}{z}{r} + \frac{1}{r} \mpdv{V_\varphi}{z}{\varphi} + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_r}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \vb{V} + &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r} + + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r} + + \vu{e}_r \vu{e}_z \pdv{V_z}{r} + \\ + &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg) + + \vu{e}_\varphi \vu{e}_\varphi \bigg( \frac{1}{r} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} \bigg) + + \vu{e}_\varphi \vu{e}_z \frac{1}{r} \pdv{V_z}{\varphi} + \\ + &\quad\: + \vu{e}_z \vu{e}_r \pdv{V_r}{z} + + \vu{e}_z \vu{e}_\varphi \pdv{V_\varphi}{z} + + \vu{e}_z \vu{e}_z \pdv{V_z}{z} + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + (\vb{U} \cdot \nabla) \vb{V} + &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\varphi}{r} \pdv{V_r}{\varphi} + U_z \pdv{V_r}{z} + - \frac{U_\varphi V_\varphi}{r} \bigg) + \\ + &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\varphi}{r} \pdv{V_\varphi}{\varphi} + U_z \pdv{V_\varphi}{z} + + \frac{U_\varphi V_r}{r} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( U_r \pdv{V_z}{r} + \frac{U_\varphi}{r} \pdv{V_z}{\varphi} + U_z \pdv{V_z}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla^2 \vb{V} + &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z} + + \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg) + \\ + &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z} + + \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} + + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \cdot \overline{\overline{\mathbf{T}}} + &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\varphi r}}{\varphi} + \pdv{T_{zr}}{z} + + \frac{T_{rr}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg) + \\ + &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\varphi \varphi}}{\varphi} + \pdv{T_{z \varphi}}{z} + + \frac{T_{r \varphi}}{r} + \frac{T_{\varphi r}}{r} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdv{T_{rz}}{r} + \frac{1}{r} \pdv{T_{\varphi z}}{\varphi} + \pdv{T_{zz}}{z} + + \frac{T_{rz}}{r} \bigg) + \end{aligned} + } +\end{aligned}$$ + + + +## References +1. M.L. Boas, + *Mathematical methods in the physical sciences*, 2nd edition, + Wiley. +2. B. Lautrup, + *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, + CRC Press. diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md index bff57af..d7984b5 100644 --- a/source/know/concept/sturm-liouville-theory/index.md +++ b/source/know/concept/sturm-liouville-theory/index.md @@ -8,14 +8,15 @@ categories: layout: "concept" --- -**Sturm-Liouville theory** defines the analogue of Hermitian matrix -eigenvalue problems for linear second-order ODEs. +**Sturm-Liouville theory** extends +the concept of Hermitian matrix eigenvalue problems +to linear second-order ordinary differential equations. -It states that, given suitable boundary conditions, any linear -second-order ODE can be rewritten using the **Sturm-Liouville operator**, -and that the corresponding eigenvalue problem, known as a -**Sturm-Liouville problem**, will give real eigenvalues and a complete set -of eigenfunctions. +It states that, given suitable boundary conditions, +any such equation can be rewritten using the **Sturm-Liouville operator**, +and that the corresponding eigenvalue problem, +known as a **Sturm-Liouville problem**, +will give real eigenvalues and a complete set of eigenfunctions. @@ -23,18 +24,19 @@ of eigenfunctions. Consider the most general form of a second-order linear differential operator $$\hat{L}$$, where $$p_0(x)$$, $$p_1(x)$$, and $$p_2(x)$$ -are real functions of $$x \in [a,b]$$ which are nonzero for all $$x \in ]a, b[$$: +are real functions of $$x \in [a,b]$$ and are nonzero for all $$x \in \,\,]a, b[$$: $$\begin{aligned} - \hat{L} \{u(x)\} = p_0(x) u''(x) + p_1(x) u'(x) + p_2(x) u(x) + \hat{L} \{u(x)\} + \equiv p_2(x) \: u''(x) + p_1(x) \: u'(x) + p_0(x) \: u(x) \end{aligned}$$ -We now define the **adjoint** or **Hermitian** operator -$$\hat{L}^\dagger$$ analogously to matrices: +Analogously to matrices, +we now define its **adjoint** operator $$\hat{L}^\dagger$$ as follows: $$\begin{aligned} - \inprod{f}{\hat{L} g} - = \inprod{\hat{L}^\dagger f}{g} + \inprod{\hat{L}^\dagger f}{g} + \equiv \inprod{f}{\hat{L} g} \end{aligned}$$ What is $$\hat{L}^\dagger$$, given the above definition of $$\hat{L}$$? @@ -43,146 +45,155 @@ We start from the inner product $$\inprod{f}{\hat{L} g}$$: $$\begin{aligned} \inprod{f}{\hat{L} g} &= \int_a^b f^*(x) \hat{L}\{g(x)\} \dd{x} - = \int_a^b (f^* p_0) g'' + (f^* p_1) g' + (f^* p_2) g \dd{x} + = \int_a^b (f^* p_2) g'' + (f^* p_1) g' + (f^* p_0) g \dd{x} \\ - &= \big[ (f^* p_0) g' + (f^* p_1) g \big]_a^b - \int_a^b (f^* p_0)' g' + (f^* p_1)' g - (f^* p_2) g \dd{x} + &= \Big[ (f^* p_2) g' + (f^* p_1) g \Big]_a^b - \int_a^b (f^* p_2)' g' + (f^* p_1)' g - (f^* p_0) g \dd{x} \\ - &= \big[ f^* \big( p_0 g' \!+\! p_1 g \big) \!-\! (f^* p_0)' g \big]_a^b + \int_a^b \! \big( (f p_0)'' - (f p_1)' + (f p_2) \big)^* g \dd{x} + &= \Big[ f^* (p_2 g' + p_1 g) - (f^* p_2)' g \Big]_a^b + \int_a^b \! \Big( (f p_2)'' - (f p_1)' + (f p_0) \Big)^* g \dd{x} \\ - &= \big[ f^* \big( p_0 g' + (p_1 - p_0') g \big) - (f^*)' p_0 g \big]_a^b + \int_a^b \big( \hat{L}^\dagger\{f\} \big)^* g \dd{x} -\end{aligned}$$ - -We now have an expression for $$\hat{L}^\dagger$$, but are left with an -annoying boundary term: - -$$\begin{aligned} - \inprod{f}{\hat{L} g} - &= \big[ f^* \big( p_0 g' + (p_1 - p_0') g \big) - (f^*)' p_0 g \big]_a^b + \inprod{\hat{L}^\dagger f}{g} + &= \Big[ f^* \big( p_2 g' + (p_1 - p_2') g \big) - (f^*)' p_2 g \Big]_a^b + \int_a^b \Big( \hat{L}^\dagger\{f\} \Big)^* g \dd{x} \end{aligned}$$ -To fix this, -let us demand that $$p_1(x) = p_0'(x)$$ and that -$$[p_0(f^* g' - (f^*)' g)]_a^b = 0$$, leaving: +The newly-formed operator on $$f$$ must be $$\hat{L}^\dagger$$, +but there is an additional boundary term. +To fix this, we demand that $$p_1(x) = p_2'(x)$$ +and that $$\big[ p_2 (f^* g' - (f^*)' g) \big]_a^b = 0$$, leaving: $$\begin{aligned} \inprod{f}{\hat{L} g} - &= \big[ p_0 \big( f^* g' - (f^*)' g \big) \big]_a^b + \inprod{\hat{L}^\dagger f}{g} - = \inprod{\hat{L}^\dagger f}{g} + &= \Big[ f^* \big( p_2 g' + (p_1 - p_2') g \big) - (f^*)' p_2 g \Big]_a^b + \inprod{\hat{L}^\dagger f}{g} + \\ + &= \Big[ p_2 \big( f^* g' - (f^*)' g \big) \Big]_a^b + \inprod{\hat{L}^\dagger f}{g} + \\ + &= \inprod{\hat{L}^\dagger f}{g} \end{aligned}$$ -Using the aforementioned restriction $$p_1(x) = p_0'(x)$$, -we then take a look at the definition of $$\hat{L}^\dagger$$: +Let us look at the expression for $$\hat{L}^\dagger$$ we just found, +with the restriction $$p_1 = p_2'$$ in mind: $$\begin{aligned} \hat{L}^\dagger \{f\} - &= (p_0 f)'' - (p_1 f)' + (p_2 f) + &= (p_2 f)'' - (p_1 f)' + (p_0 f) \\ - &= p_0 f'' + (2 p_0' - p_1) f' + (p_0'' - p_1' + p_2) f + &= (p_2'' f + 2 p_2' f' + p_2 f'') - (p_1' f + p_1 f') + (p_0 f) \\ - &= p_0 f'' + p_0' f' + p_2 f + &= p_2 f'' + (2 p_2' - p_1) f' + (p_2'' - p_1' + p_0) f \\ - &= (p_0 f')' + p_2 f + &= p_2 f'' + p_1 f' + p_0 f + \\ + &= \hat{L}\{f\} \end{aligned}$$ -The original operator $$\hat{L}$$ reduces to the same form, -so it is **self-adjoint**: +So $$\hat{L}$$ is **self-adjoint**, i.e. $$\hat{L}^\dagger$$ is the same as $$\hat{L}$$! +Indeed, every such second-order linear operator is self-adjoint +if it satisfies the constraints $$p_1 = p_2'$$ and $$\big[ p_2 (f^* g' - (f^*)' g) \big]_a^b = 0$$. + +But what if $$p_1 \neq p_2'$$? +Let us multiply $$\hat{L}$$ by an unknown $$p(x) \neq 0$$ +and divide by $$p_2(x) \neq 0$$: $$\begin{aligned} - \hat{L} \{f\} - &= p_0 f'' + p_0' f' + p_2 f - = (p_0 f')' + p_2 f - = \hat{L}^\dagger \{f\} + \frac{p}{p_2} \hat{L} \{u\} + = p u'' + p \frac{p_1}{p_2} u' + p \frac{p_0}{p_2} u \end{aligned}$$ -Consequently, every such second-order linear operator $$\hat{L}$$ is self-adjoint, -as long as it satisfies the constraints $$p_1(x) = p_0'(x)$$ and $$[p_0 (f^* g' - (f^*)' g)]_a^b = 0$$. - -Let us ignore the latter constraint for now (it will return later), -and focus on the former: what if $$\hat{L}$$ does not satisfy $$p_0' \neq p_1$$? -We multiply it by an unknown $$p(x) \neq 0$$, and divide by $$p_0(x) \neq 0$$: +We now demand that the derivative $$p'(x)$$ of the unknown $$p(x)$$ satisfies: $$\begin{aligned} - \frac{p(x)}{p_0(x)} \hat{L} \{u\} = p(x) u'' + p(x) \frac{p_1(x)}{p_0(x)} u' + p(x) \frac{p_2(x)}{p_0(x)} u + p'(x) + = p(x) \frac{p_1(x)}{p_2(x)} + \quad \implies \quad + \frac{p_1(x)}{p_2(x)} \dd{x} + = \frac{1}{p(x)} \dd{p} \end{aligned}$$ -We now define $$q(x)$$, -and demand that the derivative $$p'(x)$$ of the unknown $$p(x)$$ satisfies: +Taking the indefinite integral of this differential equation +yields an expression for $$p(x)$$: $$\begin{aligned} - q(x) = p(x) \frac{p_2(x)}{p_0(x)} - \qquad - p'(x) = p(x) \frac{p_1(x)}{p_0(x)} + \int \frac{p_1(x)}{p_2(x)} \dd{x} + = \int \frac{1}{p} \dd{p} + = \ln\!\big( p(x) \big) + \quad \implies \quad + \boxed{ + p(x) + = \exp\!\bigg( \int \frac{p_1(x)}{p_2(x)} \dd{x} \bigg) + } \end{aligned}$$ -The latter is a differential equation for $$p(x)$$, which we solve by integration: +We define an additional function $$q(x)$$ +based on the last term of $$(p / p_2) \hat{L}$$ shown above: $$\begin{aligned} - \frac{p_1(x)}{p_0(x)} \dd{x} - &= \frac{1}{p(x)} \dd{p} - \\ - \implies \quad - \int \frac{p_1(x)}{p_0(x)} \dd{x} - &= \int \frac{1}{p} \dd{p} - = \ln\!\big( p(x) \big) - \\ - \implies \qquad\qquad - p(x) - &= \exp\!\bigg( \int \frac{p_1(x)}{p_0(x)} \dd{x} \bigg) + \boxed{ + q(x) + \equiv p(x) \frac{p_0(x)}{p_2(x)} + } + = \frac{p_0(x)}{p_2(x)} \exp\!\bigg( \int \frac{p_1(x)}{p_2(x)} \dd{x} \bigg) \end{aligned}$$ -Now that we have $$p(x)$$ and $$q(x)$$, we can define a new operator $$\hat{L}_p$$ as follows: +When rewritten using $$p$$ and $$q$$, +the modified operator $$(p / p_2) \hat{L}$$ looks like this: $$\begin{aligned} - \hat{L}_p \{u\} - = \frac{p}{p_0} \hat{L} \{u\} + \frac{p}{p_2} \hat{L} \{u\} = p u'' + p' u' + q u = (p u')' + q u \end{aligned}$$ This is the self-adjoint form from earlier! -So even if $$p_0' \neq p_1$$, any second-order linear operator with $$p_0(x) \neq 0$$ -can easily be put in self-adjoint form. - -This general form is known as the **Sturm-Liouville operator** $$\hat{L}_{SL}$$, -where $$p(x)$$ and $$q(x)$$ are nonzero real functions of the variable $$x \in [a,b]$$: +So even if $$p_1 \neq p_2'$$, any second-order linear operator +with $$p_2(x) \neq 0$$ can easily be made self-adjoint. +The resulting general form is called the **Sturm-Liouville operator** $$\hat{L}_\mathrm{SL}$$, +for nonzero $$p(x)$$: $$\begin{aligned} \boxed{ - \hat{L}_{SL} \{u(x)\} - = \fr