State and prove Leibnitz's Theorem for finding the \(n\)th differential coefficient of the product of two functions of the variable. Prove that, if \(y=e^{\tan^{-1}x}\), \[ (1+x^2)\frac{d^{n+1}y}{dx^{n+1}} = (1-2nx)\frac{d^ny}{dx^n}-n(n-1)\frac{d^{n-1}y}{dx^{n-1}}. \]
Conics through four fixed points or touching four fixed lines.
The analysis of vector fields.
Limits and bounds of functions of a real variable.
Mean-value theorems and Taylor's theorem.
The Jacobian of \(n\) functions of \(n\) independent variables.
Partial differential equations.
Spherical harmonics or Fourier series.
Develop the formulae expressing the acceleration of a point in terms of its coordinates referred to moving rectangular axes. Use these formulae to find the equations of motion of a non-viscous fluid referred to cylindrical coordinates.
The application of conjugate functions to the solution of problems in electrostatics or current electricity.