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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. |