184 



EXAMPLES OF THE CHANGE 





But by means of the three equations between x, y, z, 

 0, <f>, r, we can determine the values of 



dx ' dy' dz' dx' dy ' dz' dx' dy' dz' 



du du , du . 

 and hence the above equations express -y- , -y- , and -y- , in 



f du du . du 

 terms of -y^ , -yj , and ,- . 

 do a<p dr 



Also by solving the above equations we can express 



du du du . du du , du , . , 



-ja, -T7, -y-, m terms of -y-, -y-, and -y-, which can also 



dv d<f> dr dx dy dz 



be found by the equations 



du du dx du du du dz 



du _ du dx du dy du dz 

 d<j> dx d<j> dy d$> dz d<f) 

 du _ du dx du dy du dz 

 dr dx dr dy dr dz dr _ 



207. Suppose, to exemplify the above, we put 



x = r sin 9 cos (j>, y = r sin 9 sin <, z = r cos 9. 



Hence, to apply equations (2) of Art. 206, we have 



(2). 



dx 



cos 



d9 

 dy 



/j dz 

 = r cos sm <, ~JQ~ 



= r sin 9 cos 6, -=-, = 0, 



dx . a . 



-JT rsmu sm <f>, 



Ci(f> I*Y 



dx . n dy . ., . 



-y- = sin 9 cos <f>. -~ sin 9 sin 



dr dr ar 



therefore 



du n , du n , du . n du, \ 



--= = r cos o cos <p -y- + T cos a sin 0-7 r sm 9 -y- 



d9 ^ dx T . dy dz I 



du . . . , du . n . du 



-j- = r sm 9 sin <f> -=- + r sin ^ cos -y- 

 09 r aa; r ay 



<?M ..- . cZw . ,. . , du n du 



-y- = sm 9 cos 6 -y- + sin ^ sm <f> -y- + cos 9 -y- 

 ar r die r dy 



y 



dz 



(1). 



