Categories: Mathematics, Physics.

# Cartesian 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.