3.3 Multivariable Differential and Integral Calculus 169for somek≤n. Multiply this equality byn, then apply the operatorx∂x∂ +y∂y∂ to both
sides. The left-hand side becomes
∑
j(
k
j)(
x∂
∂x+y∂
∂y)
xjyk−j∂kz
∂xj∂yk−j=∑
jj(
k
j)
xjyk−j
∂kz
∂xj∂yk−j+
∑
j(
k
j)
xj+^1 yk−j
∂k+^1 z
∂xj+^1 ∂yk−j+
∑
j(k−j)(
k
j)
xjyk−j∂kz
∂xj∂yk−j+
∑
j(
k
j)
xjyk−j+^1∂k+^1 z
∂xj∂yk−j+^1=k∑
j(
k
j)
xjyk−j
∂kz
∂xj∂yk−j+
∑
j((
k
j− 1)
+
(
k
j))
xjyk+^1 −j
∂k+^1 z
∂xjyk+^1 −j=k·n(n− 1 )···(n−k+ 1 )z+∑
j(
k+ 1
j)
xjyk+^1 −j∂k+^1 z
∂xjyk+^1 −j.
The base casek=1 implies that the right side equalsn·n(n− 1 )···(n−k+ 1 )z.
Equating the two, we obtain
∑j(
k+ 1
j)
xjyk+^1 −j∂k+^1 z
∂xjyk+^1 −j
=n(n− 1 )···(n−k+ 1 )(n−k)z,completing the induction. This proves the formula.
498.Prove that if the functionu(x, t)satisfies the equation
∂u
∂t=
∂^2 u
∂x^2,(x,t)∈R^2 ,then so does the functionv(x, t)=1
√
te−
x 42 t
u(xt−^1 ,−t−^1 ), x∈R,t> 0.499.Assume that a nonidentically zero harmonic functionu(x, y)isn-homogeneous for
some real numbern. Prove thatnis necessarily an integer. (The functionuis called
harmonic if∂
(^2) u
∂x^2 +
∂^2 u
∂y^2 =0.)
500.LetP(x, y)be a harmonic polynomial divisible byx^2 +y^2. Prove thatP(x, y)is
identically equal to zero.
501.Letf :R^2 →Rbe a differentiable function with continuous partial derivatives
and withf( 0 , 0 )=0. Prove that there exist continuous functionsg 1 ,g 2 :R^2 →R
such that
f (x, y)=xg 1 (x, y)+yg 2 (x, y).