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diff --git a/source/know/concept/cartesian-coordinates/index.md b/source/know/concept/cartesian-coordinates/index.md new file mode 100644 index 0000000..d198e84 --- /dev/null +++ b/source/know/concept/cartesian-coordinates/index.md @@ -0,0 +1,200 @@ +--- +title: "Cartesian coordinates" +sort_title: "Cartesian coordinates" +date: 2023-06-09 +categories: +- Mathematics +- Physics +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/). + +The well-known Cartesian coordinate system $$(x, y, z)$$ +has trivial **scale factors**: + +$$\begin{aligned} + \boxed{ + h_x + = h_y + = h_z + = 1 + } +\end{aligned}$$ + +With these, we can use the standard formulae for orthogonal curvilinear coordinates +to write out various vector calculus operations. + + + +## Differential elements + +For line integrals, +the tangent vector element $$\dd{\vb{\ell}}$$ for a curve is as follows: + +$$\begin{aligned} + \boxed{ + \dd{\vb{\ell}} + = \vu{e}_x \dd{x} + + \: \vu{e}_y \dd{y} + + \: \vu{e}_z \dd{z} + } +\end{aligned}$$ + +For surface integrals, +the normal vector element $$\dd{\vb{S}}$$ for a surface is given by: + +$$\begin{aligned} + \boxed{ + \dd{\vb{S}} + = \vu{e}_x \dd{y} \dd{z} + + \: \vu{e}_y \dd{x} \dd{z} + + \: \vu{e}_z \dd{x} \dd{y} + } +\end{aligned}$$ + +And for volume integrals, +the infinitesimal volume $$\dd{V}$$ takes the following form: + +$$\begin{aligned} + \boxed{ + \dd{V} + = \dd{x} \dd{y} \dd{z} + } +\end{aligned}$$ + + + +## Common operations + +The basic vector operations (gradient, divergence, curl and Laplacian) are given by: + +$$\begin{aligned} + \boxed{ + \nabla f + = \vu{e}_x \pdv{f}{x} + + \vu{e}_y \pdv{f}{y} + + \mathbf{e}_z \pdv{f}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{V} + = \pdv{V_x}{x} + \pdv{V_y}{y} + \pdv{V_z}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \times \vb{V} + &= \quad \vu{e}_x \bigg( \pdv{V_z}{y} - \pdv{V_y}{z} \bigg) + \\ + &\quad\: + \vu{e}_y \bigg( \pdv{V_x}{z} - \pdv{V_z}{x} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdv{V_y}{x} - \pdv{V_x}{y} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \nabla^2 f + = \pdvn{2}{f}{x} + \pdvn{2}{f}{y} + \pdvn{2}{f}{z} + } +\end{aligned}$$ + + + +## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla (\nabla \cdot \vb{V}) + &= \quad \vu{e}_x \bigg( \pdvn{2}{V_x}{x} + \mpdv{V_y}{x}{y} + \mpdv{V_z}{x}{z} \bigg) + \\ + &\quad\: + \vu{e}_y \bigg( \mpdv{V_x}{y}{x} + \pdvn{2}{V_y}{y} + \mpdv{V_z}{y}{z} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \mpdv{V_x}{z}{x} + \mpdv{V_y}{z}{y} + \pdvn{2}{V_z}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \vb{V} + &= \quad \vu{e}_x \vu{e}_x \pdv{V_x}{x} + + \vu{e}_x \vu{e}_y \pdv{V_y}{x} + + \vu{e}_x \vu{e}_z \pdv{V_z}{x} + \\ + &\quad\: + \vu{e}_y \vu{e}_x \pdv{V_x}{y} + + \vu{e}_y \vu{e}_y \pdv{V_y}{y} + + \vu{e}_y \vu{e}_z \pdv{V_z}{y} + \\ + &\quad\: + \vu{e}_z \vu{e}_x \pdv{V_x}{z} + + \vu{e}_z \vu{e}_y \pdv{V_y}{z} + + \vu{e}_z \vu{e}_z \pdv{V_z}{z} + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + (\vb{U} \cdot \nabla) \vb{V} + &= \quad + \vu{e}_x \bigg( U_x \pdv{V_x}{x} + U_y \pdv{V_x}{y} + U_z \pdv{V_x}{z} \bigg) + \\ + &\quad\: + \vu{e}_y \bigg( U_x \pdv{V_y}{x} + U_y \pdv{V_y}{y} + U_z \pdv{V_y}{z} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( U_x \pdv{V_z}{x} + U_y \pdv{V_z}{y} + U_z \pdv{V_z}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla^2 \vb{V} + &= \quad \vu{e}_x \bigg( \pdvn{2}{V_x}{x} + \pdvn{2}{V_x}{y} + \pdvn{2}{V_x}{z} \bigg) + \\ + &\quad\: + \vu{e}_y \bigg( \pdvn{2}{V_y}{x} + \pdvn{2}{V_y}{y} + \pdvn{2}{V_y}{z} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{x} + \pdvn{2}{V_z}{y} + \pdvn{2}{V_z}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \cdot \overline{\overline{\mathbf{T}}} + &= \quad \vu{e}_x \bigg( \pdv{T_{xx}}{x} + \pdv{T_{yx}}{y} + \pdv{T_{zx}}{z} \bigg) + \\ + &\quad\: + \vu{e}_y \bigg( \pdv{T_{xy}}{x} + \pdv{T_{yy}}{y} + \pdv{T_{zy}}{z} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdv{T_{xz}}{x} + \pdv{T_{yz}}{y} + \pdv{T_{zz}}{z} \bigg) + \end{aligned} + } +\end{aligned}$$ + + + +## 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 index 3c54ef8..fe7d7c1 100644 --- a/source/know/concept/cylindrical-polar-coordinates/index.md +++ b/source/know/concept/cylindrical-polar-coordinates/index.md @@ -257,14 +257,14 @@ $$\begin{aligned} \boxed{ \begin{aligned} \nabla^2 \vb{V} - &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \pdv{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} - + \pdvn{2}{V_r}{z} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg) + &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z} + + \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg) \\ - &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} - + \pdvn{2}{V_\varphi}{z} + \frac{2}{r^2} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r^2} \bigg) + &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z} + + \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg) \\ - &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r} \pdv{V_z}{r} - + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} + \pdvn{2}{V_z}{z} \bigg) + &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} + + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg) \end{aligned} } \end{aligned}$$ diff --git a/source/know/concept/material-derivative/index.md b/source/know/concept/material-derivative/index.md index d11287d..6bb83c5 100644 --- a/source/know/concept/material-derivative/index.md +++ b/source/know/concept/material-derivative/index.md @@ -88,6 +88,9 @@ $$\begin{aligned} } \end{aligned}$$ +To evaluate this in various coordinate systems, +see [orthogonal curvilinear coordinates](/know/concept/orthogonal-curvilinear-coordinates/). + ## References diff --git a/source/know/concept/spherical-coordinates/index.md b/source/know/concept/spherical-coordinates/index.md index 01c5a61..7f6d111 100644 --- a/source/know/concept/spherical-coordinates/index.md +++ b/source/know/concept/spherical-coordinates/index.md @@ -170,7 +170,7 @@ $$\begin{aligned} \boxed{ \nabla \cdot \vb{V} = \pdv{V_r}{r} + \frac{2}{r} V_r - + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{\cot{\theta}}{r} V_\theta + + \frac{1}{r} \pdv{V_\theta}{\theta} + \frac{V_\theta}{r \tan{\theta}} + \frac{1}{r \sin\theta} \pdv{V_\varphi}{\varphi} } \end{aligned}$$ @@ -179,7 +179,7 @@ $$\begin{aligned} \boxed{ \begin{aligned} \nabla \times \vb{V} - &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{\cot{\theta}}{r} V_\varphi + &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_\varphi}{\theta} + \frac{V_\varphi}{r \tan{\theta}} - \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} @@ -195,7 +195,7 @@ $$\begin{aligned} \boxed{ \nabla^2 f = \pdvn{2}{f}{r} + \frac{2}{r} \pdv{f}{r} - + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{\cot{\theta}}{r^2} \pdv{f}{\theta} + + \frac{1}{r^2} \pdvn{2}{f}{\theta} + \frac{1}{r^2 \tan{\theta}} \pdv{f}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{f}{\varphi} } \end{aligned}$$ @@ -219,12 +219,12 @@ $$\begin{aligned} + \frac{2}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\theta}{\theta} \\ &\qquad\qquad - \frac{1}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi} - + \frac{\cot{\theta}}{r} \pdv{V_\theta}{r} - \frac{2}{r^2} V_r - \frac{\cot{\theta}}{r^2} V_\theta \bigg) + + \frac{1}{r \tan{\theta}} \pdv{V_\theta}{r} - \frac{2 V_r}{r^2} - \frac{V_\theta}{r^2 \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg( \frac{1}{r} \mpdv{V_r}{\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\varphi}{\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} \\ - &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta} + &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta} - \frac{\cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r \sin{\theta}} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2 \sin{\theta}} \mpdv{V_\theta}{\varphi}{\theta} @@ -246,10 +246,10 @@ $$\begin{aligned} + \vu{e}_\theta \vu{e}_\varphi \frac{1}{r} \pdv{V_\varphi}{\theta} \\ &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg) - + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{\cot{\theta}}{r} V_\varphi \bigg) + + \vu{e}_\varphi \vu{e}_\theta \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\theta}{\varphi} - \frac{V_\varphi}{r \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \vu{e}_\varphi - \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{\cot{\theta}}{r} V_\theta \bigg) + \bigg( \frac{1}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} + \frac{V_\theta}{r \tan{\theta}} \bigg) \end{aligned} } \end{aligned}$$ @@ -262,10 +262,10 @@ $$\begin{aligned} + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_r}{\varphi} - \frac{U_\theta V_\theta}{r} - \frac{U_\varphi V_\varphi}{r} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg( U_r \pdv{V_\theta}{r} + \frac{U_\theta}{r} \pdv{V_\theta}{\theta} - + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{\cot{\theta}}{r} U_\varphi V_\varphi \bigg) + + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\theta}{\varphi} + \frac{U_\theta V_r}{r} - \frac{U_\varphi V_\varphi}{r \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\theta}{r} \pdv{V_\varphi}{\theta} - + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{\cot{\theta}}{r} U_\varphi V_\theta \bigg) + + \frac{U_\varphi}{r \sin{\theta}} \pdv{V_\varphi}{\varphi} + \frac{U_\varphi V_r}{r} + \frac{U_\varphi V_\theta}{r \tan{\theta}} \bigg) \end{aligned} } \end{aligned}$$ @@ -275,22 +275,22 @@ $$\begin{aligned} \begin{aligned} \nabla^2 \vb{V} &= \quad\: \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_r}{\varphi} - + \frac{2}{r} \pdv{V_r}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_r}{\theta} + + \frac{2}{r} \pdv{V_r}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_r}{\theta} \\ &\qquad\qquad - \frac{2}{r^2} \pdv{V_\theta}{\theta} - \frac{2}{r^2 \sin{\theta}} \pdv{V_\varphi}{\varphi} - - \frac{2}{r^2} V_r - \frac{2 \cot{\theta}}{r^2} V_\theta \bigg) + - \frac{2 V_r}{r^2} - \frac{2 V_\theta}{r^2 \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg( \pdvn{2}{V_\theta}{r} + \frac{1}{r^2} \pdvn{2}{V_\theta}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\theta}{\varphi} + \frac{2}{r^2} \pdv{V_r}{\theta} + \frac{2}{r} \pdv{V_\theta}{r} \\ - &\qquad\qquad + \frac{\cot{\theta}}{r^2} \pdv{V_\theta}{\theta} + &\qquad\qquad + \frac{1}{r^2 \tan{\theta}} \pdv{V_\theta}{\theta} - \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\varphi}{\varphi} - \frac{V_\theta}{r^2 \sin^2{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\theta} + \frac{1}{r^2 \sin^2{\theta}} \pdvn{2}{V_\varphi}{\varphi} + \frac{2}{r^2 \sin{\theta}} \pdv{V_r}{\varphi} \\ &\qquad\qquad + \frac{2 \cos{\theta}}{r^2 \sin^2{\theta}} \pdv{V_\theta}{\varphi} - + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{\cot{\theta}}{r^2} \pdv{V_\varphi}{\theta} + + \frac{2}{r} \pdv{V_\varphi}{r} + \frac{1}{r^2 \tan{\theta}} \pdv{V_\varphi}{\theta} - \frac{V_\varphi}{r^2 \sin^2{\theta}} \bigg) \end{aligned} } @@ -302,19 +302,20 @@ $$\begin{aligned} \nabla \cdot \overline{\overline{\mathbf{T}}} &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\theta r}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi r}}{\varphi} \\ - &\qquad\qquad + \frac{2}{r} T_{rr} + \frac{\cot{\theta}}{r} T_{\theta r} - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg) + &\qquad\qquad + \frac{2 T_{rr}}{r} + \frac{T_{\theta r}}{r \tan{\theta}} + - \frac{T_{\theta \theta}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg) \\ &\quad\: + \vu{e}_\theta \bigg(\pdv{T_{r \theta}}{r} + \frac{1}{r} \pdv{T_{\theta \theta}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \theta}}{\varphi} \\ - &\qquad\qquad + \frac{2}{r} T_{r \theta} + \frac{T_{\theta r}}{r} - + \frac{\cot{\theta}}{r} T_{\theta \theta} - \frac{\cot{\theta}}{r} T_{\varphi \varphi} \bigg) + &\qquad\qquad + \frac{2 T_{r \theta}}{r} + \frac{T_{\theta r}}{r} + + \frac{T_{\theta \theta}}{r \tan{\theta}} - \frac{T_{\varphi \varphi}}{r \tan{\theta}} \bigg) \\ &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\theta \varphi}}{\theta} + \frac{1}{r \sin{\theta}} \pdv{T_{\varphi \varphi}}{\varphi} \\ - &\qquad\qquad + \frac{2}{r} T_{r \varphi} + \frac{\cot{\theta}}{r} T_{\theta \varphi} - + \frac{T_{\varphi r}}{r} + \frac{\cot{\theta}}{r} T_{\varphi \theta} \bigg) + &\qquad\qquad + \frac{2 T_{r \varphi}}{r} + \frac{T_{\theta \varphi}}{r \tan{\theta}} + + \frac{T_{\varphi r}}{r} + \frac{T_{\varphi \theta}}{r \tan{\theta}} \bigg) \end{aligned} } \end{aligned}$$ diff --git a/source/know/concept/sturm-liouville-theory/index.md b/source/know/concept/sturm-liouville-theory/index.md index 0ac7476..bff57af 100644 --- a/source/know/concept/sturm-liouville-theory/index.md +++ b/source/know/concept/sturm-liouville-theory/index.md @@ -18,6 +18,7 @@ and that the corresponding eigenvalue problem, known as a of eigenfunctions. + ## General operator Consider the most general form of a second-order linear @@ -65,7 +66,7 @@ $$[p_0(f^* g' - (f^*)' g)]_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} + &= \big[ p_0 \big( f^* g' - (f^*)' g \big) \big]_a^b + \inprod{\hat{L}^\dagger f}{g} = \inprod{\hat{L}^\dagger f}{g} \end{aligned}$$ @@ -115,18 +116,19 @@ $$\begin{aligned} The latter is a differential equation for $$p(x)$$, which we solve by integration: -$$\begin{gathered} - \frac{p_1(x)}{p_0(x)} = \frac{1}{p(x)} \dv{p}{x} - \quad \implies \quad - \frac{p_1(x)}{p_0(x)} \dd{x} = \frac{1}{p(x)} \dd{p} +$$\begin{aligned} + \frac{p_1(x)}{p_0(x)} \dd{x} + &= \frac{1}{p(x)} \dd{p} \\ \implies \quad - \int_a^x \frac{p_1(\xi)}{p_0(\xi)} \dd{\xi} = \int_{p(a)}^{p(x)} \frac{1}{f} \dd{f} - = \ln\!\Big( \frac{p(x)}{p(a)} \Big) + \int \frac{p_1(x)}{p_0(x)} \dd{x} + &= \int \frac{1}{p} \dd{p} + = \ln\!\big( p(x) \big) \\ - \implies \quad - p(x) = p(a) \exp\!\Big( \int_a^x \frac{p_1(\xi)}{p_0(\xi)} \dd{\xi} \Big) -\end{gathered}$$ + \implies \qquad\qquad + p(x) + &= \exp\!\bigg( \int \frac{p_1(x)}{p_0(x)} \dd{x} \bigg) +\end{aligned}$$ Now that we have $$p(x)$$ and $$q(x)$$, we can define a new operator $$\hat{L}_p$$ as follows: @@ -153,6 +155,7 @@ $$\begin{aligned} \end{aligned}$$ + ## Eigenvalue problem A **Sturm-Liouville problem** (SLP) is analogous to a matrix eigenvalue problem, @@ -166,7 +169,7 @@ $$\begin{aligned} \end{aligned}$$ Necessarily, $$w(x) > 0$$ except in isolated points, where $$w(x) = 0$$ is allowed; -the point is that any inner product $$\Inprod{f}{w g}$$ may never be zero due to $$w$$'s fault. +the point is that any inner product $$\inprod{f}{w g}$$ may never be zero due to $$w$$'s fault. Furthermore, the convention is that $$u(x)$$ cannot be trivially zero. In our derivation of $$\hat{L}_{SL}$$, @@ -212,7 +215,7 @@ $$\begin{aligned} &= (\lambda_m^* - \lambda_n) \int_a^b u_n u_m^* w \:dx \\ \inprod{u_m}{\hat{L}_{SL} u_n} - \inprod{\hat{L}_{SL} u_m}{u_n} - &= (\lambda_m^* - \lambda_n) \Inprod{u_m}{w u_n} + &= (\lambda_m^* - \lambda_n) \inprod{u_m}{w u_n} \end{aligned}$$ The operator $$\hat{L}_{SL}$$ is self-adjoint by definition, @@ -220,17 +223,17 @@ so the left-hand side vanishes, leaving us with: $$\begin{aligned} 0 - &= (\lambda_m^* - \lambda_n) \Inprod{u_m}{w u_n} + &= (\lambda_m^* - \lambda_n) \inprod{u_m}{w u_n} \end{aligned}$$ -When $$m = n$$, the inner product $$\Inprod{u_n}{w u_n}$$ is real and positive +When $$m = n$$, the inner product $$\inprod{u_n}{w u_n}$$ is real and positive (assuming $$u_n$$ is not trivially zero, in which case it would be disqualified anyway). In this case we thus know that $$\lambda_n^* = \lambda_n$$, i.e. the eigenvalue $$\lambda_n$$ is real for any $$n$$. When $$m \neq n$$, then $$\lambda_m^* - \lambda_n$$ may or may not be zero, depending on the degeneracy. If there is no degeneracy, we -see that $$\Inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal. +see that $$\inprod{u_m}{w u_n} = 0$$, i.e. the eigenfunctions are orthogonal. In case of degeneracy, manual orthogonalization is needed, but as it turns out, this is guaranteed to be doable, using e.g. the [Gram-Schmidt method](/know/concept/gram-schmidt-method/). @@ -240,8 +243,8 @@ and all the corresponding eigenfunctions $$u(x)$$ are mutually orthogonal**: $$\begin{aligned} \boxed{ - \Inprod{u_m(x)}{w(x) u_n(x)} - = \Inprod{u_n}{w u_n} \delta_{nm} + \inprod{u_m(x)}{w(x) u_n(x)} + = \inprod{u_n}{w u_n} \delta_{nm} = A_n \delta_{nm} } \end{aligned}$$ @@ -257,6 +260,7 @@ in other words, there always exists a *lowest* eigenvalue $$\lambda_0 > -\infty$ known as the **ground state**. + ## Completeness Not only are the eigenfunctions $$u_n(x)$$ of an SLP orthogonal, they @@ -287,18 +291,18 @@ By integrating we get inner products on both the left and the right: $$\begin{aligned} \int_a^b f(x) w(x) u_m^*(x) \dd{x} - &= \int_a^b \Big(\sum_{n = 0}^\infty a_n u_n(x) w(x) u_m^*(x)\Big) \dd{x} + &= \int_a^b \bigg( \sum_{n = 0}^\infty a_n u_n(x) w(x) u_m^*(x) \bigg) \dd{x} \\ - \Inprod{u_m}{w f} - &= \sum_{n = 0}^\infty a_n \Inprod{u_m}{w u_n} + \inprod{u_m}{w f} + &= \sum_{n = 0}^\infty a_n \inprod{u_m}{w u_n} \end{aligned}$$ Because the eigenfunctions of an SLP are mutually orthogonal, the summation disappears: $$\begin{aligned} - \Inprod{u_m}{w f} - &= \sum_{n = 0}^\infty a_n \Inprod{u_m}{w u_n} + \inprod{u_m}{w f} + &= \sum_{n = 0}^\infty a_n \inprod{u_m}{w u_n} = \sum_{n = 0}^\infty a_n A_n \delta_{nm} = a_m A_m \end{aligned}$$ @@ -310,8 +314,8 @@ function $$f(x)$$ onto the normalized eigenfunctions $$u_n(x) / A_n$$: $$\begin{aligned} \boxed{ a_n - = \frac{\Inprod{u_n}{w f}}{A_n} - = \frac{\Inprod{u_n}{w f}}{\Inprod{u_n}{w u_n}} + = \frac{\inprod{u_n}{w f}}{A_n} + = \frac{\inprod{u_n}{w f}}{\inprod{u_n}{w u_n}} } \end{aligned}$$ @@ -321,10 +325,10 @@ after inserting the expression for $$a_n$$: $$\begin{aligned} f(x) - &= \sum_{n = 0}^\infty \frac{1}{A_n} \Inprod{u_n}{w f} u_n(x) - = \int_a^b \Big(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) f(\xi) u_n(x) \Big) \dd{\xi} + &= \sum_{n = 0}^\infty \frac{1}{A_n} \inprod{u_n}{w f} u_n(x) + = \int_a^b \bigg(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) f(\xi) u_n(x) \bigg) \dd{\xi} \\ - &= \int_a^b f(\xi) \Big(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) u_n(x) \Big) \dd{\xi} + &= \int_a^b f(\xi) \bigg(\sum_{n = 0}^\infty \frac{1}{A_n} u_n^*(\xi) w(\xi) u_n(x) \bigg) \dd{\xi} \end{aligned}$$ Upon closer inspection, the parenthesized summation |