The course is a further development of calculus at a more advanced level. We will start with some vector geometry and matrices (later to be used in analysing surfaces), then cover power series (including Taylor series), geometry of surfaces (quadric surfaces, ruled surfaces, level sets), and then start with functions of several variables - partial derivatives, critical points and the Hessian matrix, volumes as double integrals.

AN ADVISORY NOTE: This course contains some material that would perhaps be more likely to be found in a Calculus III class in many American universities. This is because some of our degree-seeking students require this material for their higher-level classes. Nevertheless, anyone with a good background in a university level Calculus I class should be able to do well in this class if they are prepared to study. Do think carefully though before signing up if you are a transfer student with a particularly high GPA requirement - this class will be challenging.

Attendance is mandatory for the course.

You cannot make-up a major exam (midterm or final) without the permission of the Dean’s Office. The Dean’s Office will grant such permission only when the absence was caused by a serious impediment, such as a documented illness, hospitalization or death in the immediate family (in which you must attend the funeral) or other situations of similar gravity. Absences due to other meaningful conflicts, such as job interviews, family celebrations, travel difficulties, student misunderstandings or personal convenience, will not be excused. Students who will be absent from a major exam must notify the Dean’s Office prior to that exam. Absences from class due to the observance of a religious holiday will normally be excused. Individual students who will have to miss class to observe a religious holiday should notify the instructor by the end of the Add/Drop period to make prior arrangements for making up any work that will be missed.

DURING THE SEMESTER, MAKE-UP EXAMS WILL NOT BE GIVEN UNDER ANY CIRCUMSTANCES. If you need to miss an assessment, the weight of that assessment will be shifted to the final. If you have to miss the final, if you are in good standing with the course, you can ask for an incomplete. If you are not in good standing and miss the final, I will be unable to assign a grade for the course, other than by considering the class tests you have already completed (this means you will be assigned a D or an F as appropriate).

• Sequences of real numbers and their limits; in particular we investigate monotonic, bounded sequences and prove that they always converge.
• Infinite series, the definition of convergence (partial sums) and some standard examples (harmonic, geometric, p-series).  Convergence tests: the comparison test, the ratio test (the proof of this depends on the convergence property of monotonic sequences given above), the integral test, the alternating series test.
• Taylor series, definition and basic properties, standard examples.

• Definition and basic properties of functions of two or more variables.
• Level curves for functions of two variables, which we will then use to help us make rough sketches of surfaces.
• The ϵ-δ definition of the limit of a function of two variables with examples; limit laws and continuity.

• Partial derivatives, first and higher order.
• Here we will do a brief recap of some necessary material from linear algebra - Euclidean vectors, unit vectors, the scalar and cross product, lines and planes in 3D space, the determinant of a matrix.
• Differentiability and the chain rule. We will spend some time on the concept of differentiability, giving the precise definition (with examples) and the characterisation with continuous partial derivatives (with proof).
• Directional derivatives, tangent lines and tangent planes.
• The gradient vector - definition and basic properties.
• Critical points and classifying them using the Hessian matrix. Here, to develop the second derivative test using the Hessian matrix, we will need the Taylor series introduced at the start of term, and ideally some more advanced linear algebra (positive/negative definite matrices) to give a full explanation of the topic.
• Optimisation with Lagrange multipliers.

• The definition of the integral as a volume and Fubini’s Theorem. 
• Integrals over rectangles and more complicated regions in the x-y plane, integrals using polar co-ordinates.
• The Jacobian and changing variables.