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Department of Mathematical Sciences

# MATH1551 Mathematics for Scientists and Engineers

Please note: This module is not available to Mathematics students.

This module is intended to supply the basic mathematical needs for students in Engineering and other sciences.

There are revision classes during the first two weeks of term where you can practise problems and ask questions. They are based on a wide range of A-level mathematics material. The purpose is to help you brush-up on any material you have forgotten or did not cover in great detail at A-level (as not everyone has the same mathematical background.) It does not count in any way towards your final mark for this module.

There are 3 lectures each week and fortnightly tutorials. The tutorials start in Week 3. Problems will be set to be handed in each week and there is a Collection examination in December to test your understanding of the first term material. All these form an integral part of the module, and the homework is summative, constituting 10 led by a member of staff based on the material of the course that allow you to consolidate your knowledge.

## Outline of Course

Aim: Supply basic mathematical needs for students in Engineering and other sciences.

### Term 1

• Elementary Functions (Practical): Their graphs, trigonometric identities and 2D Cartesian geometry. To include polynomials, trigonometric functions, inverse trigonometric functions, ℯx; x; log(x); sin(x+y), sine and cosine formulae. Line, circle, ellipse,parabola, hyperbola.
• Differentiation (Practical): Definition of the derivative of a function as slope of tangent line to graph. Local maxima, minima and stationary points. Differentiation of elementary functions. Rules for differentiation of sums, products, quotients and function of a function.
• Integration (Practical): Definition of integration as reverse of differentiation and as area under a graph. Integration by partial fractions, substitution and parts. Reduction formula, e.g. for ∫sinn(x) dx.
• Complex Numbers: Addition, subtraction, multiplication, division, complex conjugate. Argand diagram, modulus, argument. Complex exponential, trigonometric and hyperbolic functions. Polar coordinates. de Moivre's theorem. Positive integer powers of sin(u) and cos(u) in terms of multiple angles.
• Differentiation: Limits, continuity and differentiability. L'Hopital's rule. Leibniz rule. Newton-Raphson method for roots of f(x)=0. Power series, Taylor's and MacLaurin's theorem, and applications.
• Vectors: Addition, subtraction and multiplication by a scalar. Applications in mechanics. Lines and planes. Distance apart of skew lines. Scalar and vector products. Triple scalar product, determinant notation. Moments about point and line. Differentiation with respect to a scalar. Velocity and acceleration.

### Terms 2 & 3

• Partial Differentiation: Functions of several variables. Chain rule. Level curves and surfaces. Gradient of a scalar function. Div and curl. Normal lines and tangent planes to surfaces. Local maxima, minima, and saddle points.
• Linear Algebra: Matrices and determinants, solution of simultaneous linear equations. Gaussian elimination for Ax=b. Gaussian elimination with pivoting. Iterative methods: Jacobi, Gauss-Seidel, SOR. Eigenvalues in matrices.
• Ordinary Differential Equations: First order differential equations: separable, homogeneous, exact, linear. Second order linear equations: superposition principle, complementary function and particular integral for equations with constant coefficients, fitting initial conditions, application to circuit theory and mechanical vibrations.

### Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.