Question 2.30: a) Let M be a point on a surface, and let the point N be the......
a) Let M be a point on a surface, and let the point N be the intersection of the xy-plane and surface’s perpendicular line through M. Find the surfaces which have the property that for every point M on the surface the segment M N is equal to a given number a > 0.
b) Find the Cauchy integral through the circle z=b, x^2+y^2=R^2. for given b,
o < b < a, and R > 0.
a) Let z = z(x,y) be the sought after surface and M(x,y,z) its arbitrary point.
Then the equation of the surface’s perpendicular line through M is
\frac{X-x}{p}=\frac{Y-y}{q}=\frac{Z-z}{-1} (2.47)
Here X, Y and Z are the coordinates on the perpendicular line. If N\left(X_0, Y_0, 0\right) is its intersection with the xy-plane, then by supposition it holds
\overline{M N}^2=\left(x-X_0\right)^2+\left(y-Y_0\right)^2+(z-0)^2=a^2 .
From (2.47) we get X_0=p z+x \text { and } Y_0=q z+y, which gives the PDE
a^2=z^2\left(p^2+q^2+1\right)
As in Example 2.29 a), we obtain that a first integral is q = sp, for an arbitrary constant s. Thus the complete integral is
s \mp \sqrt{a^2-z^2}=x \cos t+y \sin t
where t is another arbitrary constant.
Clearly, the singular integrals are z = a and z = -a (give a geometric explanation).
Finally, putting s = f(t), where f is an arbitrary function of the class C^1(R), we obtain the general solution given with the system of equations
f(t) \mp \sqrt{a^2-z^2}=x \cos t+y \sin t, \quad f^{\prime}(t)=-x \sin t+y \cos t .
b) By supposition we have
f(t)=\sqrt{a^2-b^2}+R \sin (s+t)
hence the Cauchy integral is
\left(R \pm \sqrt{a^2-b^2} \mp \sqrt{a^2-z^2}\right)^2=x^2+y^2 .