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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.