19, 20] VARIABLES AND FUNCTIONS. 35 



element of arc is 



^ 2 _^i 2 ,^ 2 , ^ 3 2 

 = I7 + ~V + 1?~' 



the elements of area of the surfaces q lt q z , q s are respectively 



and the element of volume is 



Examples. Rectangular coordinates #, y, z 



hx h y = h z = 1, 



dS x = dy dz, dS y = dz dx, dS z = dx dy, dr = da dy dz. 

 Polar coordinates r, 0, </>, 



dS r r 2 sin 0d0d(f> element of area of sphere, 

 dS e = r sin 0drd(f> element of area of cone, 

 dS<f> rdrdd element of area of plane, 



Cylindrical coordinates, z t p, co, 



h z = h p = I, A w = -, 



dS z = pdpdco element of area of plane, 

 dS p = pdcodz element of area of cylinder, 

 dS a = dpdz element of area of meridian plane, 



dr = pdpdcodz. 

 Elliptic coordinates, X, //,, v. 



, _ _ d/mdv (/Lt v)(fji- _ . 



~ 4 V 



dvd\ (v -\)(v- /A) (X - /A) (X - i/) , , ., 



/\ r'/y A"/\ / hyperboloid, 



__ 

 4 (a 2 + !/) (6 2 + v) (c 2 + v) (a 2 + \) (6 2 + X) (c 2 + X) 



- _ hyp e r boloid, 



a 2 + X) (6 2 + X) (c 2 -f X) (a 2 + /*) (6 2 + ^) (c 2 + ft) 



, _ d\dfj,dv(\ 



~ 



^)(c 2 + 

 32 



