Unit 39 Further Mathematics is a great milestone in HND engineering qualification which is tailored to provide learners the advanced recognition of the principles of mathematics. These principles are significant to tackle the difficult problems of engineering. The forms of mathematics are a significant component of the disciplines of engineering. This unit has the main aim to generate fundamental knowledge needed in earlier studies including the initial unit of Engineering Mathematics. By getting an introduction to advanced concepts of mathematics the learners with increase their potential to engage with the applied subjects of Engineering efficiently supporting their future studies and career development.
The significant aim of the unit is to provide learners with the tools of mathematics required to model and analyse the scenarios of real-world engineering. The topics including complex numbers, number theory, linear equations and matrix theory are identified for developing the skills of problem-solving. The students will also deal with the methods of numeric sets as differentiation and integration, and graphical techniques for curve estimation to assist them in preparing the solution in the context of practical engineering. These methods are advanced components for addressing the sophisticated challenges with the progress of a student through the curriculum of engineering.
Apart from this, the focus of unit 39 further mathematics is on expanding the knowledge of students in calculus which is initially first application and second-order equation differentials to solve and model the problems of engineering. On the successful ending of the unit, the student will have the potential to apply the theories of numbers to the task of engineering, solving the systems of linear equations utilising the matrix method and use of numerical and graphical methods for approximate solutions. These skills are crucial for improving and reviewing the engineering systems models. It also prepares learners for practical engineering and academic advancement roles.
Unit objectives
The main objectives in the HND engineering subject particularly in unit 39 further mathematics are demonstrated below.
- To equip learners with the potential to model and analyse engineering problems utilising techniques of advanced mathematics.
- To introduce learners to additional topics including matrix theory, number theory, complex numbers and versatile equations wider their recognition.
- To make learners able to utilise differentiation, and integration techniques for the solution of estimation in the context of engineering.
- To strengthen the foundation of students in mathematical knowledge for supporting their progress to more difficult engineering concepts in the studies of future.
Learning outcomes
The learning outcomes of unit 39 further mathematics are given below containing the main learning agendas for the engineering students.
LO1: Use applications of number theory in practical engineering situations.
The focus of the learning outcome is on the applications of the theory of numbers for scenarios of practical engineering. They get the introduction to categories of numbers such as rational, natural, complex, real and integer. All these systems will deepen the recognition of learners for the classification of mathematics. The complex number has a crucial role in the studying of learners in these conjugates, arguments and moduli with exponential and polar forms. The theorem of De Moivre gets a great introduction as a simplifying tool of power and complex numbers roots. It will assist in tasks such as electric circuits. The practical applications of the complex number in the engineering Include the analysis of electrical systems and circuits for energy and information control. It also demonstrates the way through which the concept of mathematics is directly connected to the challenges of real-world engineering.
LO2: Solve systems of linear equations relevant to engineering applications using matrix methods.
This learning outcome has an emphasis on solving the linear equation systems relevant to the application of engineering utilising the matrix method. The students increase their learning of matrix operations and notations including subtractions multiplication and addition which are crucial for manipulating and representing data in the problems of engineering. They will identify the determinant concept which starts with the way through which it calculates for the matrix 2x2. Moreover, it teaches the way through which the determinants have a crucial character in the properties of the matrix. This square, matrix, inverse is the introduction as a strong tool for solving the equations of linear particular in the scenarios of engineering like network system and structure analysis. Moreover, the learners will utilise the elimination of Gaussian for solving the linear equation systems up to 3x3 which is a method sometimes applied in the systems of modelling engineering.
LO3: Approximate solutions of contextualised examples with graphical and numerical methods.
The focus of the learning outcomes is on the approximating solution of the problems of contextualized and generally utilising the numerical and graphical methods. The students increased by identifying the curve standards for the functions including cubic, quadratic, exponential and logarithmic and curve practical systems sketching emphasized on their equations. It will permit them to approximate the solution visually to the quotations of Engineering by interpreting the curve`s behaviour. In terms of numeric, they will also have the introduction to the Newton-Raphson method and bisection method which will provide the solutions iterative to the context of Engineering with non-linear equations. The approximating areas under the curves make learners acquire the numerical techniques of integration such as the trapezium rule, Simpson’s rule and the mid-ordinate rule. These methods provide great application to integral estimation which is common in versatile analysis of engineering such as displacement determination from calculating areas or velocity data in structural engineering.
LO4: Review models of engineering systems using ordinary differential equations.
The focus of this learning outcome is reviewing the engineering system models utilising the odes. Students will learn to solve and form the first-order differential equations that are broadly applied in the contacts of engineering like RL and RC electric circuits, the law of cooling of newton and discharging and charging of capacitors. They will also identify the second-order equations differentials utilising the complex systems such as energy control systems, second-order equations of differential such as mass, RCL circuits and energy control systems. By having mastery of odes and multiple applications. Thus, learners will increase the potential to analyse, solve and model the complex problems of engineering.
Assessment Criteria
Here are the assessment criteria for unit 39 further mathematics which is connected with each of the learning outcomes.
LO1: Use applications of number theory in practical engineering situations.
- 1.1 Apply the multiplication and addition methods to the numbers which are expressed in multiple base systems.
- 1.2 Solve the problems of engineering utilising the complex number theory.
- 1.3 Perform the operations of arithmetic utilising the exponential and polar categories of complex numbers.
- 1.4 Deduce the problem solution utilising de Moivre’s theorem.
LO2: Solve systems of linear equations relevant to engineering applications using matrix methods.
- 2.1 Discover the determination of the provided 3X3 matrix.
- 2.2 Solve the linear equation systems of three utilising Gaussian elimination.
- 2.3 Identify the solutions to a linear equation set utilising the method of inverse matrix.
LO3: Approximate solutions of contextualised examples with graphical and numerical methods.
- 3.1 Estimate the sketched function solutions utilising the method of graphical estimation.
- 3.2 Calculate the equation roots utilising two versatile iterative techniques.
- 3.3 Determine the numerical integral engineering function utilising two versatile methods.
- 3.4 Solve the problems of engineering and formulate the models of mathematics utilising numerical and graphical integration.
LO4: Review models of engineering systems using ordinary differential equations.
- 4.1 Determine the initial order versatile equations and their engineering systems application utilising the methods of analytics.
- 4.2 Determine the non-homogeneous and homogeneous second-order differential equations and their engineering system application utilising analytical methods.
- 4.3 Calculate the linear ordinary solutions for versatile equations utilising Laplace transforms.
- 4.4 Identify the way through which versatile engineering system models utilise first-order differential to the problems of equation solving for engineering.
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