YEAR 6: Mathematical methods
Projects utilizing computer systems, technical writing, research skills. Topics include vector analysis, complex analysis, linear algebra, physical applications of ordinary differential equations and partial differential equations.
Course Outline
Part I

Vector Analysis.

Vector Product. Orientation of a Surface. Vector of an Area. Properties of Vector Product.

Tangent Line to a Curve. Osculating Circle. Frenet Formulae.

Tangent Plane and Normal to a Surface. Curvature. Second Derivative Concepts.


Multiple Integrals.

Examples and Concepts. Basic Properties. Geometric Meaning.

Mass and Density. Distributions.

Integral Over a Rectangle. Integral Over an Arbitrary Plane Region. Integral Over an Arbitrary Surface.

Change of Variables in Multiple Integrals.

Polar Coordinates in Plane. Cylindrical and Spherical Coordinates.

Curvilinear Coordinates on a Plane and Surface.



Volumes of Solids. Areas of Plane Figures. Arc Length of a Curve.

Vector Fields.

Flux through a Surface. Divergence.

Line Integrals and Circulation.

Green's Formula. Gauss Formula. Stoke's Formula.

Potential Field. Hamiltonian.

Multivariable Power Series. Asymptotic Expansions. Extrema of Functions of Several Variables.

Part II

Matrices.

Linear Spaces. Concept of Linear Space. Examples. Dimension. Euclidean Space. Orthogonality.

Operations on Matrices. Inverse Matrix. Eigenvectors and Eigenvalues. Rank. Characteristic Equation of a Matrix.

Matrix of a Linear Mapping. Change of Basis. Matrix of a Mapping Relative to Basis of Eigenvectors.

Symmetric Matrices. Quadratic Forms. Equations of Quadratic Curves and Surfaces. Optical Properties.

Covariance and Correlation Matrices.


Higher Algebra.

Sturm's Theorem. Bounds of Roots.

Quadratic Forms. Law of Inertia.

Symmetric Polynomials. Resultants. Discriminant.


Linear Programming.

Inequalities of Averages. Systems of Inequalities.

Optimization.

The Transportation Problem.

Simplex Method. Dual Method.

Network Flow Problems.

Part III

Complex Variables.

Winding Number. Residue Formula. Complex Integrals.

Complex Logarithm and Exponential.

Applications to Oscillations.

Conformal Mapping.


Partial Differential Equations.

Fourier Series. Expansion of Periodic Functions. Parseval Relation.

Laplace Equation. Dirichlet Problem.

Wave Equation. Oscillations of String. D'Alembert Formula. Vibrations of a Membrane.

Heat Equation. Heat Propagation in a Rod.
