PARTIAL DIFFERENTIATION 
149 
21. Extend the theorem of Question 19 to the case of derivatives 
of the third order. 
22. If u and v are two functions of x and y satisfying the relations : 
du _dv du dv 
dx dy dy dx 
show that, on introducing polar coordinates : 
x — r cos (f>, y = r sin (f>, 
we have 
du _ 1 dv 1 d>bt_ _ dv 
dr rdy>’ r d<f> dr 
23. Under the hypotheses of the preceding question, show that 
d 2 u 1 du . 1 d 2 u _ q 
dr 2 r dr r 2 d<f> 2 
24. If 
/(*, y) = 0 and cf>(x, z) = 0, 
show that 
d$dfcly_dfd$' 
dx dy dz dxdz 
25. If 
<H'P, v, t) = 0, 
show that 
dp dt dv _ 
dt dv dp 
Explain the meaning of each of the partial derivatives. 
26. If a is a function of x, y, z and x, y, z are connected by a 
single relation, is it true that 
dy dz dy 
27. If u=f(x, y) and v = <f>(x, y) are two functions which sat 
isfy the relations 
du _dv du__dv 
dx dy’ dy dx’ 
and if V is any third function, show that 
frV dW = I“ fdu\ 2 
dx 2 dy 2 
du 2 dv 2 ) 
28. If 
X — r sin cfi cos 6, 
y = r sin (f> sin 6, 
Z = r COS cj>,