Categories: Mathematics, Physics.

Cartesian coordinates

This article is a supplement to the ones on orthogonal curvilinear systems, spherical coordinates, polar cylindrical coordinates, and parabolic cylindrical coordinates.

The well-known Cartesian coordinate system (x,y,z)(x, y, z) has trivial scale factors:

hx=hy=hz=1\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 d\dd{\vb{\ell}} for a curve is as follows:

d=e^xdx+e^ydy+e^zdz\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 dS\dd{\vb{S}} for a surface is given by:

dS=e^xdydz+e^ydxdz+e^zdxdy\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 dV\dd{V} takes the following form:

dV=dxdydz\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:

f=e^xfx+e^yfy+ezfz\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} V=Vxx+Vyy+Vzz\begin{aligned} \boxed{ \nabla \cdot \vb{V} = \pdv{V_x}{x} + \pdv{V_y}{y} + \pdv{V_z}{z} } \end{aligned} ×V=e^x(VzyVyz)+e^y(VxzVzx)+e^z(VyxVxy)\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} 2f=2fx2+2fy2+2fz2\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 (V)\nabla (\nabla \cdot \vb{V}), the gradient of a vector V\nabla \vb{V}, the advection of a vector (U)V(\vb{U} \cdot \nabla) \vb{V} with respect to U\vb{U}, the Laplacian of a vector 2V\nabla^2 \vb{V}, and the divergence of a 2nd-order tensor T\nabla \cdot \overline{\overline{\vb{T}}}:

(V)=e^x(2Vxx2+2Vyxy+2Vzxz)+e^y(2Vxyx+2Vyy2+2Vzyz)+e^z(2Vxzx+2Vyzy+2Vzz2)\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} V=e^xe^xVxx+e^xe^yVyx+e^xe^zVzx+e^ye^xVxy+e^ye^yVyy+e^ye^zVzy+e^ze^xVxz+e^ze^yVyz+e^ze^zVzz\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} (U)V=e^x(UxVxx+UyVxy+UzVxz)+e^y(UxVyx+UyVyy+UzVyz)+e^z(UxVzx+UyVzy+UzVzz)\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} 2V=e^x(2Vxx2+2Vxy2+2Vxz2)+e^y(2Vyx2+2Vyy2+2Vyz2)+e^z(2Vzx2+2Vzy2+2Vzz2)\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} T=e^x(Txxx+Tyxy+Tzxz)+e^y(Txyx+Tyyy+Tzyz)+e^z(Txzx+Tyzy+Tzzz)\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}


  1. M.L. Boas, Mathematical methods in the physical sciences, 2nd edition, Wiley.