---
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/),
[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**:

$$\begin{aligned}
    \boxed{
        h_x
        = h_y
        = h_z
        = 1
    }
\end{aligned}$$

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.