Overview

WARNING

The information in this section is being updated.

Mathematical Analysis 2 is a 9 CFU course, part of the Engineering Sciences bachelor course. These pages contain the practical details related to the course together with the lecture notes.

Instructors:

• Prof Florin Radulescu
• Prof Jacopo Bassi

Teams code: [TBA] (use this to join on Microsoft Teams)

Classroom: Aula 8

Classes:

• Monday 14:00 - 15:45
• Wednesday 09:30 - 11:15
• Friday 11:30 - 13:15

Notice. The classes scheduled for Monday 21 September and Wednesday 23 September 2026 are cancelled. The course will start on Friday 25 September. The missed teaching hours will be recovered in additional exercise sessions.

Lecture notes and key skills learnt in the course

See here for a list of the key skills learnt in this course.

A pdf of lecture notes is available for download. If you wish to have a reference book, we recommend Mathematical Analysis II by Canuto and Tabacco.

General advice

• Develop your intuition, it's a powerful skill - But don't trust it completely
• Don't aim to memorize but rather seek to understand - It is easy to remember anything when you understand it.
• Question always, be skeptical of all statements presented to you. Don't accept them until you are sure they are believable.
• Observe, question how everything fits together, notice all the details.
• Part of the process of mathematical reasoning is creative - to be creative we must drop our inhibitions and be ready to be wrong, repeatedly.

Schedule

The material of the course is divided into the parts as listed below. Mathematically the parts are intimately linked.

Topic
Mathematical reasoning
Higher dimension
Extrema
Line integrals
Multiple integrals
Surface integrals

See the lesson diary for further details.

What is MA2?

Much of what we do in this course builds on ideas established in Mathematical Analysis 1. In particular many of the ideas are extended to the higher dimensional setting.

Mathematical Analysis 1	Mathematical Analysis 2
(Functions)$f:\mathbb{R}\to \mathbb{R}$	$f:{\mathbb{R}}^n\to \mathbb{R}$ (Scalar fields)
	$\mathbf{F}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ (Vector fields)
	$\mathit{\alpha }:\mathbb{R}\to {\mathbb{R}}^n$ (Paths)
(Derivative)$f^{\prime }(x)=\frac{df}{dx}(x)$	$\frac{\partial f}{\partial x_j}(x_1,\dots ,x_n)$ (Partial derivatives)
	$\mathrm{\nabla }f$ (Gradient)
	$D_{v}f$ (Directional derivative)
	${\mathit{\alpha }}^{\prime }$ (Derivative of path)
	$Df$ (Jacobian matrix)
	$\mathrm{\nabla }\cdot \mathbf{F}$ (Divergence)
	$\mathrm{\nabla }\times \mathbf{F}$ (Curl)
(Extrema)$\underset{x\in \mathbb{R}}{sup}f(x)$	$\underset{x\in {\mathbb{R}}^n}{sup}f(x)$ (Extrema)
	Lagrange multiplier method
Integral${\int }_a^{b}f(x)dx$	Multiple integral
	Line integral
	Surface integral

Additional info

The online version of the notes and exercises is recommended, but a PDF version can also be downloaded (updated as the notes are updated).

Course material last year is available (including past exams from that year and the years before):

• 2025/26 Butterley