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 dℓ\dd{\vb{\ell}} for a curve is as follows:

dℓ=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^x∂f∂x+e^y∂f∂y+ez∂f∂z\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=∂Vx∂x+∂Vy∂y+∂Vz∂z\begin{aligned} \boxed{ \nabla \cdot \vb{V} = \pdv{V_x}{x} + \pdv{V_y}{y} + \pdv{V_z}{z} } \end{aligned} ∇×V=e^x(∂Vz∂y−∂Vy∂z) +e^y(∂Vx∂z−∂Vz∂x) +e^z(∂Vy∂x−∂Vx∂y)\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=∂2f∂x2+∂2f∂y2+∂2f∂z2\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(∂2Vx∂x2+∂2Vy∂x∂y+∂2Vz∂x∂z) +e^y(∂2Vx∂y∂x+∂2Vy∂y2+∂2Vz∂y∂z) +e^z(∂2Vx∂z∂x+∂2Vy∂z∂y+∂2Vz∂z2)\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^x∂Vx∂x+e^xe^y∂Vy∂x+e^xe^z∂Vz∂x +e^ye^x∂Vx∂y+e^ye^y∂Vy∂y+e^ye^z∂Vz∂y +e^ze^x∂Vx∂z+e^ze^y∂Vy∂z+e^ze^z∂Vz∂z\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(Ux∂Vx∂x+Uy∂Vx∂y+Uz∂Vx∂z) +e^y(Ux∂Vy∂x+Uy∂Vy∂y+Uz∂Vy∂z) +e^z(Ux∂Vz∂x+Uy∂Vz∂y+Uz∂Vz∂z)\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(∂2Vx∂x2+∂2Vx∂y2+∂2Vx∂z2) +e^y(∂2Vy∂x2+∂2Vy∂y2+∂2Vy∂z2) +e^z(∂2Vz∂x2+∂2Vz∂y2+∂2Vz∂z2)\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(∂Txx∂x+∂Tyx∂y+∂Tzx∂z) +e^y(∂Txy∂x+∂Tyy∂y+∂Tzy∂z) +e^z(∂Txz∂x+∂Tyz∂y+∂Tzz∂z)\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.