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This course covers material in multivariable integral and differential calculus, linear algebra and statistics which is applicable to the engineering professions.
Mathematics underpins almost every aspect of modern engineering. This is reflected by the fact that all first professional year students must take EMTH210. With the centrality of this course to your professional development in mind, considerable effort has gone into selecting mathematical and statistical topics which will provide the groundwork for you to appropriately mathematise your engineering work. Throughout the course, your lecturers will also endeavour to relate the rigour of the mathematics to the practicality of the situations in which it will be applied, as we concentrate on your ability to apply the techniques to realistic situations.The following topics will be covered, subject to the time available:• Partial differentiation, chain rule, gradient, directional derivatives, tangent planes, Jacobian, differentials, line integrals, divergence and curl, extreme values and Lagrange multipliers.• Second order linear differential equations and their applications.• Fourier series.• Double and triple integrals: elements of area, change of order of integration, polar coordinates, volume elements, cylindrical and spherical coordinates.• Eigenvalues and eigenvectors and their applications.• Laplace transforms.• Statistics: approximating expectations, characteristic functions, random vectors (joint distributions, marginal distributions, expectations, independence, covariance), linking data to probability models (sample mean and variance, order statistics and the empirical distribution function, convergence of random variables, law of large numbers and point estimation, the central limit theorem, error bounds and confidence intervals, sample size calculations, likelihood).
A student achieving total mastery of this course will be able to: Show proficiency in multivariable calculus, including partial differentiation, implicit partial differentiation, the multidimensional chain rule, gradient, directional derivative, tangent planes, Jacobians, differentials, line integrals (exact and inexact), divergence, curl, and Lagrange multipliers. Solve homogeneous constant coefficient ODEs and also inhomogeneous constant coefficient ODEs using undetermined coefficients. This includes ODEs of order other than two. Solve elementary second order boundary value problems, and appreciate some applications of BVPs in engineering. Calculate real Fourier series of arbitrary period, and employ them to solve ODEs with periodic driving functions. The student will also be knowledgeable of concepts such as harmonics and Gibbs phenomenon in Fourier series analysis. Integrate in multiple dimensions using Cartesian, polar and spherical polar coordinate systems. Calculate the eigenpairs of matrices. Familiar with orthogonal decomposition, and use it to find the principal axes of an ellipse. Proficient in the solution of systems of first and second order ODEs via eigenvalues and eigenvectors, and familiar with the implications defective matrices in such situations. Apply Laplace transforms to differential and some integral equations, including those with piecewise functions via the Heaviside step function. Approximate expectations. Be cognizant of characteristic functions, random vectors, joint and marginal distributions, independence and covariance. Link data to probability models, sample mean, variance, order statistics, and the empirical distribution function. Be familiar with convergence of random variables, the law of large numbers, point estimation, the central limit theorem, likelihood, error bounds, and confidence intervals. Do sample size calculations.
Subject to approval of the Dean of Engineering and Forestry
EMTH202, EMTH204, MATH201, MATH261, MATH262, MATH264
Chris Price (MATH)
and Blair Robertson
Recommended Reading:•“Advanced Engineering Mathematics” by Erwin Kreyszig. (This text also covers the statistics material.)•“Advanced Engineering Mathematics” by Zill and Wright.•“Advanced Engineering Mathematics” by Zill and Cullen.
General information for students
Domestic fee $919.00
International fee $5,000.00
* Fees include New Zealand GST and do not include any programme level discount or additional course related expenses.
For further information see
Mathematics and Statistics.