(^560) Partial derivatives
satisfies the Laplace equation over the entire plane but is, nevertheless, not
harmonic over regions containing the origin.
5 Prove that
au au
(1) xax+y ay=0
when u = y/x, when u = log (y/x), and when u = sin (y/x). Continue opera-
tions to prove that if f is
s
a differentiable function of one variable, then
Vx'f x/ =f, (z) x2' af OX) =f'
`(/
\XD k
and (1) still holds.
6 Prove that the wave equation
u
a2
azu z
axz

ate
is satisfied when
(a) u = (x + at)3 (b) u = (x - at)'
(c) u = es+at (d) u = sin (x - at)
(e) u = f(x + at) (.f) u = g(x - at),
it being supposed that f and g are twice-differentiable functions.
7 Show that the function defined by (11.11) satisfies the heat equation
a2a2uax2
au
at
8 Show that each of the following functions satisfies the equation written
opposite it:

(a) u=ax+by xaz-{- yau=u

Y

(b) u = sin (x sin y) cos y ax - sin yy = 0

a2u a2u a2u

(c) u

(x - x5)2 + (y - yo)2 + (z - zo)2 ax2

+ ay2 -F azz = 0

(3U au au

(d) u=(x-Y)(Y-z)(z-x) a x+ay+az-0

9 To simplify matters, we suppose that each function appearing in our work

is continuous and has continuous partial derivatives of first and second orders.

We begin acquaintance with the idea that if F or F(x,y,z) is a scalar function,

then the vector function VF defined by

(1) DF=al+ayi+azk

is called the gradient of F. If V is a vector function defined by

(2) V(x,y,z) = P(x,y,z)i + Q(x,y,z)j + R(x,y,z)k,

then the scalar function defined by

(3)

aP+8Q+ OR

ax ay az