Calculus: Analytic Geometry and Calculus, with Vectors

11.3 Formulas involving partial derivatives 581

Tau__au au sin 4,

apcos 0 -a4, P

On au au cos

aY

=

aP

sin }

To P

Then show that the formulas

axe=

Ca49

p(az)] ax+ [aO CaxJJ ax

49 a

ay2=Cap ayay + TOCaYU aye

can be put in the forms

(6) a2u =a2u cost4, - 2 a2u sin ¢ cos 4,

axe ape aP a P

+ au sins t + a2u sins + 2 au sin 0 cos 4,

ap p P2 To p2

(7) a2u=

192U

sins -I- 2 a2u sin ¢ cos 4,

aY2 a p2 ap ao P

+ au cost + a2u cost _ 2au sin ¢ cos 4,

ap p a4,2 p2 To p2

Show finally that

(8)

and hence

(9)

a2u a2u a2u 1 On 1 a2u

axe+ 0y2= apt + p oP+ pt 84,2

a2u a2u a2u 492U 1 au 1 a2u a2u

axe+8y2+az2

=apt+Pap+p'a2+az2

For use in the next remark, we note also the formula

(10) l auy aY l au + cos 4, au

P aP p2 sin ¢ 90

which comes from (5) and (1).

5 This remark can be very helpful to those who will study brands of physics

and engineering in which the Laplace equation, the heat equation, and the wave

equation appear. While the operation might be tedious and need not be per-

formed, we could copy all of Problem 4 with x, y, p, and 0 respectively replaced

by z, p, r, and 0. This shows us that if u is a function of z and p and if

z=rcos0, p = r sin 0,

(3)

a2u a2u atu 1 au

+

I a2u

az2 + apt are + r Or r2 002

lau__lau

+

cos0 au

p ap r ar 0 T O *