Define the curvature (\(\kappa\)) and torsion (\(\tau\)) of a twisted curve, explaining carefully any conventions of sign that you make, and prove from your definition that, with the usual notation, \[ s'^3\kappa^2\tau = \pm \begin{vmatrix} x' & y' & z' \\ x'' & y'' & z'' \\ x''' & y''' & z''' \end{vmatrix}, \] determining by means of your conventions which sign is to be taken.
A correspondence between points \((x,y,z)\) and points \((\xi, \eta, \zeta)\) is determined by the equations \[ \xi=\phi_1(x,y,z), \quad \eta=\phi_2(x,y,z), \quad \zeta=\phi_3(x,y,z), \] where the functions \(\phi_1, \phi_2, \phi_3\) are single-valued, continuous and differentiable over a certain range of points \((x,y,z)\). The image of a part of this range for which \[ a \le x \le a+h, \quad b \le y \le b+k, \quad c \le z \le c+l \] has a volume \(V\). Prove that, as \(h,k,l\) tend to zero, \[ \frac{V}{hkl} \to J(a,b,c), \] where \[ J(x,y,z) = \begin{vmatrix} \frac{\partial\xi}{\partial x} & \frac{\partial\xi}{\partial y} & \frac{\partial\xi}{\partial z} \\ \frac{\partial\eta}{\partial x} & \frac{\partial\eta}{\partial y} & \frac{\partial\eta}{\partial z} \\ \frac{\partial\zeta}{\partial x} & \frac{\partial\zeta}{\partial y} & \frac{\partial\zeta}{\partial z} \end{vmatrix}. \]
Prove from first principles that, if \(f(x)\) is continuous in \(a \le x \le b\) and differentiable in \(a
Establish the formula for change of variable in a simple integral, stating carefully what conditions you assume. The transformation \(\sin x = y\) apparently gives \[ \int_0^\pi \cos^2 x dx = \int_0^0 \sqrt{1-y^2} dy = 0. \] Explain this paradox.
Show how the number and approximate position of the real roots of an algebraic equation may be determined by means of the properties of a series of Sturm's functions. Show that the Legendre polynomials \(P_0(x), P_1(x), \dots, P_n(x)\) have the characteristic property of a series of Sturm's functions and state what information regarding the zeros of \(P_n(x)\) can be obtained from this fact.
The numbers \(v_0, v_1, \dots, v_n\) are positive and decrease. Prove that the ratio \[ \frac{a_0 v_0 + a_1 v_1 + \dots + a_n v_n}{b_0 v_0 + b_1 v_1 + \dots + b_n v_n} \] lies between the greatest and least of the ratios \(\dfrac{s_0}{t_0}, \dfrac{s_1}{t_1}, \dots, \dfrac{s_n}{t_n}\), where \[ s_r = a_0+a_1+\dots+a_r, \quad (r=0,1,\dots,n) \] \[ t_r = b_0+b_1+\dots+b_r, \] and it is assumed that none of \(t_0, t_1, \dots, t_n\) is zero or negative. Show that the series \[ a_0 v_0 + a_1 v_1 + \dots + a_n v_n + \dots \] converges if \(s_{n+p}v_n \to 0\) uniformly in \(p\), where \(p\) is any non-negative integer.
Solve the differential equations:
Assuming that the elliptic functions sn, cn, dn have the usual periods, zeros and poles, and behave in the usual way in the immediate neighbourhood of their zeros and poles, show that \[ \text{cn } u \text{ cn }(u+v) + \text{sn } u \text{ dn } v \text{ sn }(u+v) - \text{cn }v = 0, \] and deduce the addition theorem for the cn function.
Prove that, if \(2\omega\) is a period of \(\wp u\), then \[ \frac{\wp'(u+\omega)}{\wp'u} = -\left\{ \frac{\wp(u/2)-\wp\omega}{\wp u-\wp\omega} \right\}^2, \] and verify the formula by making \(u\to\omega\).