Prove that there are six normals from a point \((f,g,h)\) to the ellipsoid \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\), and that at least two of them are real. Prove further that the six normals are generators of the cone \[ \frac{f(b^2-c^2)}{x-f} + \frac{g(c^2-a^2)}{y-g} + \frac{h(a^2-b^2)}{z-h} = 0. \]
Illustrate the use of a spherical indicatrix in the differential geometry of a twisted curve. Prove that the radius of curvature of the spherical indicatrix of the binormals of a curve is \(\rho(\rho^2+\sigma^2)^{-\frac{1}{2}}\) where \(\rho, \sigma\) are the radii of curvature and torsion of the curve.
Prove that any point \(P\) of the conic \[ z=0, \quad \frac{x^2}{a^2-c^2}+\frac{y^2}{b^2-c^2}=1 \quad (a>b>c) \] is a focus of the section of the conicoid \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \] by the plane through \(P\) perpendicular to the tangent at \(P\) to the conic.
A developable surface is commonly defined
If \(f(x)\) has a derivative at \(x=\xi\) prove that \[ \frac{f(\xi+h)-f(\xi+k)}{h-k} \to f'(\xi) \] as \(h\to 0, k\to 0\), provided that \(h\) and \(k\) have opposite signs. Shew by an example that the condition that \(h\) and \(k\) have opposite signs cannot be omitted.
Starting from the definition of the continuity of a function at a point, state carefully the sequence of definitions and theorems required to prove the following result: If \(f(x)\) is continuous in \(a \le x \le b\), then there exists a unique function \(F(x)\) continuous in \((a,b)\) such that
Define uniform convergence and prove that the sum of a uniformly convergent series of continuous functions is continuous. Examine for uniform convergence the series \[ \sum_{n=1}^\infty \frac{x}{1+n^2x^2}. \]
If \(f(z)=a_0+a_1z+a_2z^2+\dots\) is regular (holomorphic) inside and on the circle \(C\) with centre \(z=0\) and radius \(r\), and if \(|f(z)| \le M(r)\) on \(C\), prove that \[ |a_n|r^n \le M(r). \] Deduce Liouville's theorem that a bounded function regular in the whole \(z\)-plane is a constant.
Solve in series the equation \[ x^2\frac{d^2y}{dx^2}+x\frac{dy}{dx}+k^2y=0, \] giving special consideration to odd integral values of \(h\). If \(h\) is an even integer \(2m\), prove that the solution is \[ y = \left(x\frac{d}{dx}\right)^m (A\cos kx + B\sin kx). \] % Note: The question seems to have a typo, mixing h and k. I will assume k in the equation, and h in the condition.
Let \(f_n(x)\) be a polynomial defined by the equations \[ f_0(x)=1, f_1(x)=x, f_n(x)=(a_nx+b_n)f_{n-1}(x)-c_nf_{n-2}(x), \quad (n=2,3,\dots) \] where \(a_n, b_n, c_n\) are functions of \(n\) and \(a_n>0, c_n>0\). Prove that all the roots of \(f_n(x)=0\) are real and that between any two consecutive roots of \(f_n(x)=0\) there lies a root of \(f_{n-1}(x)=0\). Defining Legendre's \(n\)th polynomial \(P_n(x)\) as the coefficient of \(h^n\) in the expansion of \[ \frac{1}{\sqrt{1-2xh+h^2}} \] prove that all the roots of \(P_n(x)=0\) are real.