The purpose of this module is to ensure the learner can visualise or describe physical systems using the basic elements of mathematics. The leaner must then be able to adapt these descriptions as needed using the fundamental skills of algebra and manipulation of equations. The learner will use vector algebra and differential and integral calculus to describe Engineering systems, solve the resulting mathematical problems and interpret the results. These skills and concepts will enable the learner to use mathematics fluently during their career as an engineer.
The basics of algebra and the grammar of number – The laws of indices, the distributive law, basic algebraic operations, transposition of equations. The concept of functions and the common notation for the argument of a function. The basic functions: the logarithm and the exponential functions and their connection. The sine and cosine as functions, their angular frequency, phase, amplitude, time displacement and other parameters.
The idea of an equation and the solutions of an equation. The solution of equations as geometric problems in the x-y plane. Simultaneous equations, Quadratic equations and Cubic equations and the remainder theorem.
The x-y plane. The behaviour of common functions and their illustration by graphs in the plane, including simple linear functions, power functions, the exponential, logarithm, sine and cosine and other trigonometric functions. The use of the common functions to model the behaviour of physical systems such as the discharge of a capacitor, the voltage and current from a dynamo, waves and other examples.
The Trigonometric Functions as geometric ideas. The Unit Circle and polar coordinates. The sine and cosine rule and the resolution of triangles.
The definition of a vector as a quantity with direction and magnitude. Physical quantities such as velocity, acceleration and force defined as vectors and their derivatives. The addition of vectors; resolution of two or more vectors.
Complex numbers in Cartesian, Polar and exponential forms, their representation as Phasors, and the use of trigonometric knowledge to carry out arithmetic operations on complex numbers. De Moivre’s theorem, the Euler equations and the roots of unity.
The Concept of the Derivative
The concept of the Derivative as a rate-of-change, as the slope of a curve. Defining and calculating the derivative of a function by first principles. Establishing the derivatives of simple functions, such as linear, quadratic and cubic functions using first principles. The derivatives of common functions encountered in the physical sciences and engineering, including polynomials, the logarithm and exponential functions and the basic trigonometric functions.
The product and quotient rules and their application to finding derivatives. Composite functions and the Chain rule. The derivatives of common functions found in engineering formed from the basic functions. Finding the maxima and minima and the turning points of a range of functions and interpreting the results.
Integration as the inverse operation to the derivative. Identifying the integrals of common functions from this concept. The definite integral and its interpretation as the area below a curve. The mean square and root mean square of functions and the use of these ideas in electrical science. The average power of an electrical circuit.
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