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### Course Overview

Matrices and matrix algebra are introduced in order to solve systems of linear equations by Gaussian elimination. Differentiation is generalized from functions of one variable to functions of several variables, through the concept of partial differentiation.  A study of systems describable by linear differential equations is begun by looking at some simple examples of engineering interest. The solution of ordinary differential equations using the methods of complementary function and particular integral.

The course is only available to students on the Graduate Apprenticeship in BEng in Civil Engineering.

### Course Details

Study Type Level Undergraduate 2 First Sub Session 15 credits (7.5 ECTS credits) Aberdeen No Dr Richard Hepworth

### What courses & programmes must have been taken before this course?

• Bachelor Of Engineering In Engineering (Civil Grad App)
• Engineering (EG)
• ED15M1 Graduate Apprenticeship Engineering Mathematics 1 (Passed)

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### Course Description

1. Matrices:

Basic definitions and notation. Algebra of matrices: multiplication by scalars, addition and subtraction of matrices, multiplication. Zero matrix, identity matrix, transpose, symmetric & anti-symmetric matrices.

The meaning of matrix inversion. Inverse of 2x2 matrix. Determinants, with some work on row & column operations together with general expansion formula. Systems of linear equations. Geometrical interpretation. Discussion of various possibilities: unique solutions, no solution, infinitely many solutions. Gaussian reduction. Solution of systems of linear equations by formal Gaussian reduction with partial pivoting down to upper triangular form followed by back-substitution. The idea of approximating one function by another.

1. Ordinary Differential Equations:

Revision of integration: Integrals (definition and geometric interpretation), anti-derivatives, integration by parts, polynomials, exponential and logarithm, trigonometric functions.

Examples of differential equations; linearity. First Order: Separation of variables and integrating factors.  Second Order: Theory and applications of linear equations with constant coefficients.  First and second order linear differential equations with constant coefficients: initial value conditions; solution of homogeneous equations and investigation of the form of the solution; solution of non-homogeneous equations using complementary function and particular integral; forced oscillations and resonance.

1. Partial Differentiation:

Introduction to partial differentiation; the heat equation and wave equation as examples of two-variable (space and time) problems; partial differentiation as linear approximation; representation of a function of two variables by a surface; estimation of small errors; the chain rule; 2nd  order approximation for a function of two variables; maxima, minima and saddle-points; application of the chain rule to solve the wave equation.

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

### Summative Assessments

First Attempt

1x Coursework (30%)
1x Timed Online Test (30%)
1x Timed Online Test (40%)

Alternative Resit Arrangements

Resit of only the failed assessment component(s)

### Formative Assessment

There are no assessments for this course.

### Course Learning Outcomes

Knowledge LevelThinking SkillOutcome
FactualRememberILO’s for this course are available in the course guide.

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