Prove the formula \[ \frac{(s')^6}{\rho^2\sigma} = \begin{vmatrix} x' & y' & z' \\ x'' & y'' & z'' \\ x''' & y''' & z''' \end{vmatrix}^2 \] for a twisted curve. A curve traced on a sphere cuts the meridians at a constant angle; show that \(\rho^2\sigma\) varies as the square of the sine of the co-latitude.
The coordinates of any point of a surface are expressed in terms of two parameters \(u, v\), the element of length on the surface being given by \[ ds^2 = E du^2 + 2F du\,dv + G dv^2. \] Prove that the measure of curvature depends only on E, F, G, and their differential coefficients with respect to \(u\) and \(v\). Prove that if \(F=0\) for all values of \(u\) and \(v\), and G is a function of \(v\) alone, the curves \(u=\) constant are geodesics.
Prove that if \(f(x)\) is continuous at every point of an interval ab, then, given any positive \(\epsilon\), there exists a positive \(\delta\), independent of \(x, x'\), such that
\[ |f(x') - f(x)| < \epsilon \]
for all \(x, x'\) of ab satisfying \(|x-x'|<\delta\).
Show by an example that this result does not necessarily hold when the finite interval \(a
State any tests that you know for the convergence of series that are not absolutely convergent. Discuss completely the convergence of \[ \Sigma \frac{x^n}{n^\alpha \log n}, \] where \(\alpha\) is real, and \(x\) real or complex, distinguishing between absolute and conditional convergence.
Prove that for all values of \(x\), real or complex, \[ \sin x = x \prod_{n=1}^\infty \left(1-\frac{x^2}{n^2\pi^2}\right). \]
If P, Q, R are functions of \(x\) only, and one solution of \begin{equation} \frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy = 0 \tag{1} \end{equation} is known, prove that the complete solution of \[ \frac{d^2y}{dx^2} + P\frac{dy}{dx} + Qy = R \] can be obtained by quadratures. If (1) has two (unknown) solutions \(y_1\) and \(y_2\) connected by \(y_1=y_2^2\), find the complete solution, and show that \[ \left(\frac{dQ}{dx}+2PQ\right)^2 = 18Q^3. \]
If \(\phi(z) \to 0\) uniformly as \(|z|\to\infty\), prove that \[ \int_\Gamma e^{iz}\phi(z)\,dz \to 0 \] as \(R\to\infty\), where \(\Gamma\) is the semicircle \(z=Re^{i\theta}, 0 \le \theta < \pi\). Hence, or otherwise, prove that \[ \int_0^\infty \frac{\sin x}{x}\,dx = \frac{1}{2}\pi. \]
Prove Liouville's theorem, that a bounded function regular at every point is necessarily a constant. Prove that \[ \wp(u-a) - \wp(u+a) = \frac{\wp'(u)\wp'(a)}{(\wp(u)-\wp(a))^2}. \]
Show that if \(\omega\) is one of the imaginary cube roots of unity, then the other is \(\omega^2\); and that \[ x^2+y^2+z^2-yz-zx-xy = (x+\omega y+\omega^2 z)(x+\omega^2 y+\omega z). \] Prove that if \((y-z)^n+(z-x)^n+(x-y)^n\) is divisible by \(\Sigma x^2 - \Sigma yz\), \(n\) must be an integer which is not a multiple of 3. Prove also that if it is divisible by \((\Sigma x^2 - \Sigma yz)^2\), \(n\) must exceed by unity a multiple of 3.
Find the number of homogeneous products of degree \(r\) in \(n\) letters, and show that if there are three letters \(a, b, c\), the sum of these products is \[ \Sigma a^{r+2}(b-c)/\Sigma a^2(b-c). \]