Spherical Coordinates

A general vector is written

(C.55) |

where , , and . Of course, the unit vectors , , and are mutually orthogonal, so , et cetera.

As is easily demonstrated, an element of length (squared) in the spherical coordinate system takes the form

(C.56) |

Hence, comparison with Equation (C.6) reveals that the scale factors for this system are

(C.57) | ||

(C.58) | ||

(C.59) |

Thus, surface elements normal to , , and are written

(C.60) | ||

(C.61) | ||

(C.62) |

respectively, whereas a volume element takes the form

(C.63) |

According to Equations (C.13), (C.15), and (C.18), gradient, divergence, and curl in the spherical coordinate system are written

(C.64) | ||

(C.65) | ||

(C.66) |

respectively. Here, is a general scalar field, and a general vector field.

According to Equation (C.19), when expressed in spherical coordinates, the Laplacian of a scalar field becomes

(C.67) |

Moreover, from Equation (C.23), the components of in the spherical coordinate system are

(C.68) | ||

(C.69) | ||

(C.70) |

Now, according to Equation (C.26), the components of in the spherical coordinate system are

(C.71) | ||

(C.72) | ||

(C.73) | ||

(C.74) | ||

(C.75) | ||

(C.76) |

Finally, from Equation (C.28), the components of in the spherical coordinate system are

(C.77) | ||

(C.78) | ||

(C.79) |