Mathematical Methods 1

Overview

Review of A-level calculus: elementary functions and their graphs, domains and ranges, trigonometric functions, derivatives and differentials, integration. Maclaurin expansion. Complex numbers and Euler’s formula.

Differential equations (DE); first-order DE: variable separable, linear; second-order linear DE with constant coefficients: homogeneous and inhomogeneous.

Vectors in 3D, definitions and notation, operations on vectors, scalar and vector products, triple products, 2x2 and 3x3 determinants, applications to geometry, equations of a plane and straight line. Rotations and linear transformations in 2D, 2x2 and 3x3 matrices, eigenvectors and eigenvalues.

Newtonian mechanics: kinematics, plane polar coordinates, projectile motion, Newton’s laws, momentum, types of forces, simple pendulum, oscillations (harmonic, forced, damped), planetary motion (universal law of gravity, angular momentum, conic sections, Kepler’s problem).

Curves in 3D (length, curvature, torsion). Functions of several variables, derivatives in 2D and 3D, Taylor expansion, total differential, gradient (nabla operator), stationary points for a function of two variables. Vector functions; div, grad and curl operators and vector operator identities. Line integrals, double integrals, Green's theorem. Surfaces (parametric form, 2nd-degree surfaces). Curvilinear coordinates, spherical and cylindrical coordinates, orthogonal curvilinear coordinates, Lame coefficients. Volume and surface integrals, Gauss's theorem, Stokes's theorem. Operators div, grad, curl and Laplacian in orthogonal curvilinear coordinates.

Learning Objectives

On completion of the module, the students are expected to be able to:
• Sketch graphs of standard and other simple functions;
• Use of the unit circle to define trigonometric functions and derive their properties;
• Integrate and differentiate standard and other simple functions;
• Expand simple functions in Maclaurin series and use them;
• Perform basic operations with complex numbers, derive and use Euler's formula;
• Solve first-order linear and variable separable differential equations;
• Solve second-order linear differential equations with constant coefficients (both homogeneous and inhomogeneous), identify complementary functions and particular integrals, and find solutions satisfying given initial conditions;
• Perform operations on vectors in 3D, including vector products, and apply vectors to solve a range of geometrical problems; derive and use equations of straight lines and planes in 3D;
• Calculate 2x2 and 3x3 determinants;
• Use matrices to describe linear transformations in 2D, including rotations, and find eigenvalues and eigenvectors for 2x2 matrices.
• Define basis quantities in mechanics, such as velocity, acceleration and momentum, and state Newton’s laws;
• Use calculus for solving a range of problems in kinematics and dynamics, including projectile motion, oscillations and planetary motion;
• Define and recognise the equations of conics, in Cartesian and polar coordinates;
• Investigate curves in 3D, find their length, curvature and tension;
• Find partial derivatives for a function of several variables;
• Expand functions of one and two variables in the Taylor series and investigate their stationary points;
• Find the partial differential operators div, grad and curl for scalar and vector fields;
• Calculate line integrals along curves;
• Calculate double and triple integrals, including surface and volume integrals;
• Transform between Cartesian, spherical and cylindrical coordinate systems;
• Investigate simple surfaces in 3D and evaluate surface for the shapes such as the cube, sphere, hemisphere or cylinder;
• State and apply Green's theorem, Gauss's divergence theorem, and Stokes's theorem

Skills

• Proficiency in calculus and its application to a range of problems.
• Constructing and clearly presenting mathematical and logical arguments.
• Mathematical modelling and problem solving.
• Ability to manipulate precise and intricate ideas.
• Analytical thinking and logical reasoning.

Assessment

Computer Test