The function \(F(x,y)\) is continuous in \((x,y)\) in a neighbourhood of a certain point \((a,b)\) and \[ F(a,b)=0. \] Investigate conditions under which the equation \[ F(x,y)=0 \] determines, in some neighbourhood of \(a\), a function \(y=\phi(x)\) which reduces to \(b\) when \(x=a\). Find also conditions for \(\phi(x)\) to be
Explain what is meant by saying that the series \[ u_1+u_2+\dots+u_n+\dots \] is convergent. Prove that if this is so, then, as \(n\to\infty\), \[ u_n\to 0. \] Prove further that, if, for each \(n\), \[ u_{n-1}\ge u_n > 0, \] then the series cannot be convergent unless \[ nu_n\to 0. \] Construct a series of positive terms to show that, if the condition \(u_{n-1}\ge u_n\) does not hold, then this result may not be true. State what you consider to be the sum of the series \[ 0+0+\dots+0+\dots \] and give carefully the reasons which support your statement.
Explain what is meant by the uniform convergence of a series and give an example of a series which converges in \(0 < x\le 1\) and yet is not uniformly convergent in any interval \(0\le x \le \delta\), however small the positive number \(\delta\) may be. It is given that \[ f(x)=f_1(x)+f_2(x)+\dots+f_n(x)+\dots \] for all values of \(x\) in some neighbourhood of \(a\), and that \(f_1(x),f_2(x),\dots\) are differentiable in this neighbourhood. Prove that, if the series \[ f'_1(x)+f'_2(x)+\dots+f'_n(x)+\dots \] converges uniformly in some neighbourhood of \(a\), then \(f(x)\) is differentiable at \(a\) and \[ f'(a)=f'_1(a)+f'_2(a)+\dots+f'_n(a)+\dots. \]
Evaluate the integrals:
\[ \int_0^\infty \frac{x^{p-1}}{1+x+x^2}\,dx \quad (0
Explain how to distinguish the two "sides" of a bilateral surface. Define \(\iint f(x,y,z)dydz\) taken over a specified side of a given bilateral surface and show how to calculate it in terms of \(u\) and \(v\) when the equations to the surface are \[ x=\theta(u,v), \quad y=\phi(u,v), \quad z=\psi(u,v). \] Find the value of \[ \iint x^3y^3z^5dydz \] taken over the outer side of the octant of the surface of the ellipsoid \[ \frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1 \] for which \(x,y,z\) are all positive.
The Green's function \(G(x,y,z)\) associated with a given closed surface \(S\) and origin \((a,b,c)\) in its interior is defined by the properties:
Develop the Lagrange-Charpit method of solving a partial differential equation and illustrate by considering \[ p^2+q^2-2px-2qy+1=0. \]
Prove that two elliptic functions with the same periods, zeros and poles have a constant ratio. Assuming that the functions sn \(u\), cn \(u\), dn \(u\) have the usual periods, zeros and poles, apply this method to compare \[ \text{cn } u \text{ cn }(u+v) - \text{cn } v, \quad \text{sn } u \text{ sn }(u+v), \] proving that \[ \text{cn } u \text{ cn }(u+v) + \text{sn } u \text{ dn } v \text{ sn }(u+v) - \text{cn } v = 0. \] Deduce addition theorems for the functions sn, cn.
Find the condition that the equations \[ ax^2+2bx+c=0, \quad a'x^2+2b'x+c'=0 \] may have a common root. Prove that, if \(a, c, a', b', c'\) are given so that \(b'^2 > a'c'\), two real values \(b_1, b_2\) of \(b\) can be found to ensure a common root; and form the equation whose roots are the other roots of the equations \[ ax^2+2b_1x+c=0, \quad a'x^2+2b_2x+c'=0. \]
Prove that, if \(A-x^2=u\), (\(x>0, u>0\)), then \(\sqrt{A}\) lies between \(x\) and \(x+u/2x\). Hence prove that, if \(P = N^2+\frac{2}{3}n\) and \(Q = N^2+\frac{1}{3}n\), the difference between \((N^2+n)^{1/2}\) and \(N\sqrt{P/Q}\) is less than \(\frac{5}{36}n^3N^{-1}P^{-1}Q^{-1}\) when \(N\) and \(n\) are positive. Apply the method to express \(\sqrt{53}\) as a decimal, estimating the degree of accuracy obtained.