## ENGSCI 211

Mathematical Modelling 2

Semester 1, 2019

Staff

#### Contents

Calendar notes

First and second order ordinary differential equations and solutions. Laplace transforms. Taylor series and series in general. Multivariable and vector calculus including divergence, gradient and curl. Further linear algebra. Eigenvalues and eigenvectors. Fourier series. Application of the techniques through appropriate modelling examples. Introductory data analysis and statistics.
Prerequisite: ENGGEN 150, or ENGSCI 111, or a B+ grade or higher in MATHS 108 or 110 or 150 or 153, or a B+ grade or higher in MATHS 120 and 130 Restriction: ENGSCI 213

#### Outcome mapping

##### Intended learning outcomes

Data Analysis: students will have a comprehension of hypothesis testing as related to data analysis, and to use these ideas in the context of simple linear regression and one-way analysis of variance. They will be able to apply these techniques to analyse the type of data that arises in engineering practice.

Multivariable Calculus: Describe a spatial motion problem in terms of a paraemterised position vector and determine the associated velocity vector and acceleration vectors. Compute the unit tangent vector for a parameterised path by normalising the velocity vector. Compute the tangential and normal components of acceleration via the projection of vectors (using the dot product). Identify motion that is parameterised in terms of arc length. Calclulate the arc length travelled along a path via a line integral. Calculate the work done by a force applied along a path by integrating the dot product of a force and velocity vector along a line. Compute partial derivatives. Compute rates of change for multivariable functions by application of the multivariable chain rule. Use computer software to visualise multivariable functions of 2 variables as surfaces. Apply multivariable optimisation techniques to find and classify critical points for functions of 2 variables. Explain the physical significance of grad, div and curl. Compute the Gradient of a scalar field to determine the direction of maximum rate of change and hence be able to calculate the directional derivative. Compute the Divergence of a given vector field, to determine whether it is zero and hence incompressible. Compute the Curl of a given vector field, to see if it is zero and hence whether it is irrotational. Describe the volume beneath a surface over a region using multivariable integrals. Sketch the region of integration for a multivariable integral. Compute multivariable integrals using appropriate techniques, including change of order and change of variable. Compute the Jacobian for a given change of variable

Linear Algebra: Derive and apply the Jacobi and Gauss-Seidel iteration methods. Solve a system of linear equations using iterative methods. Determine the convergence properties of iterative methods. Use mathematical modelling to formulate a system of linear equations. Solve a system of linear equations using LU factorisation. Compute the determinant of a matrix using LU factorisation. Determine the linear independence of a matrix using LU factorisation. Determine the eigenvectors and eigenvalues of a square matrix. Understand and apply normalising methods to eigenvectors. Use eigenvectors and eigenvalues for the diagonalisation of a square matrix. Demonstrate the orthogonality of a matrix of normalised eigenvectors. Apply the diagonalisation of a matrix to solve a system of linear ODEs. Use symmetric matrices to represent ellipses in a quadratic form. Apply linear algebra for problem analysis.

Fourier Series: Apply orthogonality principles to determine the coefficients of the Fourier series for a periodic function. Determine the coefficients of the Fourier series for a odd or even function. Use the periodic nature of sine and cosine functions to simplify trigonometric expressions in a Fourier series. Understand and apply the definition of a half range Fourier series. Comprehension of how Fourier series can be used for function approximation, problem analysis and mathematical modelling.

Ordinary Differential Equations : Application of initial conditions to a general solution to an ODE and to find the specific solution to an initial value problems. Application of Laplace transforms to solve non-homogenous ODEs. Application of physical laws as related to mathematical model systems, to derive an ODE representation. Application of Newton's 2nd law to derive an ODE which describes a physical system. Application of certain second-order ordinary differential equations. Ability to find the particular integral for certain non-homogeneous ODEs. Application to solve certain linear, constant coefficient second order ODEs.

#### Assessment

Coursework

Basic Skills Quiz (4% of final grade).
Weekly Quizzes (10% of final grade).
Assignment 1 (8% of final grade).
Assignment 2 (8% of final grade).
Test 1 (10% of final grade).
Test 2 (10% of final grade).

Exam rules

The exam is worth 50% of the final grade. It is a 3 hour, closed-book and no calculator exam.

Inclusive learning

Students are urged to discuss privately any impairment-related requirements face-to-face and/or in written form with the course convenor/lecturer and/or tutor.

Other assessment rules

A 10% rule will operate. Specifically, the final mark may not exceed the exam mark by more than 10%.
Final grades in this course will be determined using the standard University of Auckland grade boundaries after rounding your final mark to the nearest integer.
All coursework mark queries must be lodged before the day of the final exam.