- Vector-Valued Functions and Space Curves
- Polar, Cylindrical and Spherical Coordinates
- Quadric Surfaces and Their Plane Sections
- Functions of Several Variables, Partial Derivatives
- Tangent Planes and Linear Approximations
- Directional Derivatives and the Gradient Vector
- Maxima and Minima, Lagrange Multipliers
- Double Integrals Over Rectangles and over General Regions
- Double Integrals in Polar Coordinates
- Triple Integrals in Cylindrical and Spherical Coordinates
- Change of Variables, Jacobians

ATTENTION: in contrast to the last year's exam paper, there will be no formulas and hints given this year. Please make sure you remember main formulas given in class.

A collection of useful formulas and facts from the lectures

Niall Redmond solved questions 1,2 and 4. Question 1 was also solved by Michael Tierney, and there were good though incomplete solutions by Conor Byrne and Jacub Jadwiszczak. Question 2 was also solved by Darragh Keane and Lua Koenig, and there were several partial solutions too. In Question 4 the best results were obtained again by Michael Tierney and Darragh Keane, though many students computed either the volume or the surface area. Question 3 appeared to be the hardest, the highest grades there were 20%. I think the exam was difficult, and these are really good results. My regards, and all the best!

Every student is assigned to one of the three groups A, B or C. The times for tutorials are

(A) Wednesday, 14:00-15:00 in MGLT

(B) Tuesday, 11:00-12:00 in M17

(C) Monday, 9:00-10:00 in Syn2

Homeworks are due to hand in during the corresponding classes; late assignments are not accepted. Marking scheme: each exercise is marked with 1 point. Exercises with (*) are optional, they are not counted for your final mark. Though it is recommended to try them, it might be also helpful to prepare for the scholarship exam. Exercises with (**) are for curious people only, the matherial will not appear in any of the exam papers.

- Homework 1, due on October 10 solutions (thanks James!)
- Homework 2, due on October 17 solutions
- Homework 3, due on October 24 solutions
- Homework 4, due on October 31 solutions
- Homework 5, due on November 14 solutions
- Homework 6, due on November 21 solutions
- Homework 7, due on November 28 solutions
- Homework 8, due on December 5 solutions
- Homework 9, due on December 12 solutions
- solutions of exercises from the last week (see Homework 9)

- Parametric plane curves.
- Parabola, ellipse and hyperbola.
- Shifts and rotations of rectangular coordinate system.
- Quadric curves.
- Planes and lines in 3-space. Distance problems.
- Quadric surfaces and their plane sections.
- Parametric space curves.
- Cylindrical and spherical coordinates in 3-space.
- Differentiation and integration of vector-valued functions.
- Length of a portion of a curve.
- Change of the parameter. Parametrization by the length.
- Unit tangent, normal and binormal vectors.
- Curvature.
- (*) Kepler's Laws of Planetary Motion.

- Limits and continuity.
- Partial and directional derivatives.
- Linear approximation and planes tangent to the graph.
- Partial derivatives of higher order.
- Gradient vectors and gradient curves.
- Chain rule for partial derivatives.
- Gradient vectors and level curves.
- Chain rule and change of coordinates in differential operators.
- Maxima and minima of functions of two variables.
- Extremum problems.
- Finding global extrema over a closed and bounded set.
- Method of least squares.
- Constrained extremum principle: the method of Lagrange multipliers.

- Double integrals as iterated integrals. Fubini's theorem.
- Double integrals in polar coordinates.
- Using double integrals to compute volumes of solids.
- Area of a portion of a surface.
- Triple integrals in rectangular, cylindrical and spherical coordinates.
- Change of variables in multiple integrals. Jacobians.
- Applications: averages, center of gravity and centroid.

- write equations of planes, lines and quadric surfaces in the 3-space
- determine the type of conic section and write change of coordinates turning a quadratic equation into its standard form
- use cylindrical and spherical coordinate systems
- write equations of a tangent line, compute unit tangent, normal and binormal vectors and curvature at a given point on a parametric curve; compute the length of a portion of a curve
- apply above concepts to describe motion of a particle in the space
- calculate limits and partial derivatives of functions of several variables
- write local linear and quadratic approximations of a function of several variables, write equation of the plane tangent to its graph at a given point
- compute directional derivatives and determine the direction of maximal growth of a function using its gradient vector
- use the method of Lagrange multipliers to find local maxima and minima of a function
- compute double and triple integrals by application of Fubini's theorem or use change of variables
- use integrals to find quantities defined via integration in a number of contexts (such as average, area, volume, mass)