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+---
+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.