The emphasis is on understanding and applying the covered techniques rather than on sheer memorization. Nevertheless, you do need to remember basic definitions such as gradient and Hessian, as well as important equations such as Euler's (for different types of variational problems).

The following topics are covered in the exam:

• Nonlinear Equations: bisection, regula falsi and its modified version, secant method, Newton's method.

• Polynomials: evaluation, Horner scheme, multiplication, FFT, root counting, root bounds, deflation, Newton's and Muller's methods.

• Singular Value Decomposition: null space, row and column spaces, singular values and their computation, SVD solution of a linear system, pseudoinverse.

• Data Fitting: basis function, least-squares fitting, normal equations.

• Orthogonal Polynomials: inner product on function space, recursive generation of orthogonal polynomials, least-squares approximation.

• Fourier Series: time and frequency domains, inner product for trigonometric functions, amplitude, frequency, period, phase angle, aliasing, sampling theorem.

• Nonlinear Optimization: Golden section search, gradient & Hessian, convex functions, steepest descent, line search, Q-orthogonality, conjugate gradient method, constrained optimization, Lagrange multipliers.

• Calculus of Variations: functional, varitional derivative, Euler's equation (and four special types of integrands), cases of variable end points, several variables, and multiple unknown functions.

• Linear Programming: Standard forms and conversions (slack, surplus, and free variables), basic feasible solutions.

Exam 1

The emphasis is on understanding and applying the covered techniques rather than on sheer memorization. Complex formulae and relatively insignificant theorems (such as reflection matrix, rotation matrix about an arbitrary axis, homogeneous projection matrix), if appear on the exam, will be available to you with adequate descriptions.

Nevertheless, you do need to have some basic knowledge, for example, rotation matrices about the three coordinate axes, rotation axis and angle from an orthogonal matrix with unit determinant, quaternion multiplication, quaternion rotation operator, the Frenet apparatus (including curvature & torsion) of a parametric curve, first and second fundamental forms, principal curvatures.

The exam will cover the following topics:

• Homogeneous Coordinates: definition, projective plane, points at infinity, visualization, coordinates of point, line, and plane,

• Plane and Space Transformations: translation, rotation, scaling, reflection, inverse transformation, all in Cartesian as well as homogeneous coordinates.

• Projections: perspective and parallel projections of plane and space, viewplane coordinates, foreshortening ratio, orthographic and oblique projections, vanishing point for perspective projection.

• Rotations: rotations about the fixed coordinate axes, about an arbitrary line, and about the body frame; Euler angles, solutions of rotation axis and angle.

• Quaternions: addition, multiplication, complex conjugates, inverse, rotation operator.

• Plane Curves: Velocity, speed, arc length, regularity, cusp, unit-speed reparametrization, tangent, normal, tangential angle, curvature, center of curvature, osculating circle, radius of curvature, total curvature, inflection, vertex, area of a simple closed curve.

• Space Curves (unit- and arbitrary-speed): Principal normal, binormal, torsion, Frenet formulas, osculating plane, approximation, spherical image.

• Algebraic Curves: singular points, local parametrization, curvature evaluation.

• Surfaces: surface patch, orientable surface, special surfaces, surface curves (length, geodesic curvature), the first & second fundamental forms, surface area.
• Surface Curvatures: normal curvature, principal curvatures and directions, Gaussian and mean curvatures, Gauss map, total Gaussian curvature.