Elliptic Cylindrical Coordinates - Scale Factors

Scale Factors

The scale factors for the elliptic cylindrical coordinates and are equal

h_{\mu} = h_{\nu} = a\sqrt{\sinh^{2}\mu + \sin^{2}\nu}

whereas the remaining scale factor . Consequently, an infinitesimal volume element equals

dV = a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right) d\mu d\nu dz

and the Laplacian equals

\nabla^{2} \Phi = \frac{1}{a^{2} \left( \sinh^{2}\mu + \sin^{2}\nu \right)} \left( \frac{\partial^{2} \Phi}{\partial \mu^{2}} + \frac{\partial^{2} \Phi}{\partial \nu^{2}} \right) + \frac{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.

Read more about this topic:  Elliptic Cylindrical Coordinates

Famous quotes containing the words scale and/or factors:

    “We honor motherhood with glowing sentimentality, but we don’t rate it high on the scale of creative occupations.”
    —Leontine Young (20th century)

    “Girls tend to attribute their failures to factors such as lack of ability, while boys tend to attribute failure to specific factors, including teachers’ attitudes. Moreover, girls avoid situations in which failure is likely, whereas boys approach such situations as a challenge, indicating that failure differentially affects self-esteem.”
    —Michael Lewis (late–20th-century)