Relevant to scientists and engineers as well as mathematicians, this introduction to basic theory and simpler approximation schemes covers systems with two degrees of freedom. It introduces the geometric aspects of the two-dimensional phase space, the importance of fixed points and how they can be classified, and the notion of a limit cycle. You'll develop schemes to approximate the solutions of autonomous and non-autonomous equations to understand how these solutions behave. Periodically forced nonlinear oscillators and nonlinear oscillators with periodically time-varying parameters leading to parametric resonances are discussed, along with the stability of these solutions and tests for obtaining stability.
Approximation theory is concerned with approximating functions of a given class, or data of a given type, using functions from another, usually more elementary, class. A simple example is the problem of approximating a function such as e by means of polynomial functions. The efficient solution of such problems is of great importance for computing such approximations, and this module will introduce the mathematical theory behind many approximation methods in common use. This intermediate-level module is based on the set book by M. J. D. Powell.
This free course is an introduction to differentiation. Section 1 looks at gradients of graphs and introduces differentiation from first principles. Section 2 looks at finding derivatives of simple functions. Section 3 introduces rates of change by looking at real life situations. Section 4 looks at using the derivative of a function to deduce useful facts for sketching its graph. Section 5 covers the second derivative test, used to determine the nature of stationary points and ends by looking at rates of change, including the real-life situation, using differentiation to find acceleration.