# MULTIVARIABLE CALCULUS - 2025/6

Module code: MAT1043

## Module Overview

This module introduces students to Multivariable Calculus in two and three dimensions, and selected topics such as differential operators and line integrals in Vector Calculus. This extends students’ knowledge developed in MAT1030 Calculus on differentiation and integration of single variable functions to partial differentiation, and double and triple integration of two and three-variable functions. This module provides the necessary ground work for modules such as MAT2047 Curves and Surfaces, MAT2050 Inviscid Fluid Dynamics, and MAT2054 Functions of a Complex Variable.

### Module provider

Mathematics & Physics

PRINSLOO Andrea (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 42

Lecture Hours: 44

Seminar Hours: 5

Guided Learning: 15

Captured Content: 44

Semester 2

None.

## Module content

Indicative content includes:

• Curves and Surfaces: Review of lines and planes in two and three dimensions. Conic sections and quadric surfaces.

• Multivariable Differential Calculus: Continuity and differentiability of two and three-variable functions. Partial derivatives. Tangent planes and tangent hyperplanes. Coordinate transformations and the chain rule. Polar coordinates, spherical polar coordinates, and cylindrical polar coordinates. Directional derivatives and curves of steepest descent. Taylor expansions. Classifying stationary points.

• Multivariable Integral Calculus: Double and Triple integrals. Coordinate transformations in double and triple integrals. Double integrals in polar coordinates. Triple integrals in spherical polar and cylindrical polar coordinates.

• Vector Calculus: Scalar and vector fields in two and three dimensions. Differential operators div, grad and curl. Line integrals. Green’s theorem.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (50 mins) 25
Examination End-of-Semester Examination (2 hours) 75

NA

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of key subject knowledge in multivariable differential and integral calculus, and differential operators and line integrals in vector calculus.

• The ability to identify and use the appropriate methods to solve unseen problems relating to multivariable calculus and vector calculus.

Thus, the summative assessment for this module consists of:

• One in-semester test (50 minutes), worth 25% of the module mark, corresponding to Learning Outcomes 1, 2 and 3.

• A synoptic examination (2 hours), worth 75% of the module mark, corresponding to all Learning Outcomes 1 to 6.

Formative assessment

There are three formative unassessed courseworks over an 11 week period, designed to consolidate student learning.  Students will also receive formative feedback at biweekly seminars on problem sheets, provided to students in advance and designed to consolidate student learning.

Feedback

Students will receive feedback on both the unassessed courseworks and the in-semester test. The feedback is timed such that feedback from the first coursework will assist students with preparation for the in-semester test. The feedback from all three courseworks and the in-semester test will assist students with preparation for the synoptic examination. This feedback is complemented by verbal feedback at biweekly seminars and at office hours.

## Module aims

• Provide students with a brief introduction to curves and surfaces in two and three dimensions, including revisiting lines and planes, and introducing conic sections and quadric surfaces.
• Introduce students to multivariable differential calculus, and enable them to find partial derivatives of two and three-variable functions.
• Introduce students multivariable integral calculus and enable them to calculate double and triple integrals.
• Develop students understanding of the differential operators div, grad and curl, and their properties. Enable them to calculus grad of a scalar field, and div and curl of a vector field.
• Enable students to calculate line integrals in two and three dimensions, and apply Green's theorem.

## Learning outcomes

 Attributes Developed 001 Students will be able to find partial derivatives of functions of two and three variables, and find tangent planes to surfaces in three dimensions. KC 002 Students will understand coordinate transformations for two and three-variable functions, and will be able to apply the chain rule. KC 003 Students will be able to classify the stationary points of functions of two variables. KC 004 Students will be able to calculate double and triple integrals, and perform coordinate transformations in double and triple integrals. KC 005 Students will be able to calculate grad of scalar fields, and div and curl of vector fields. KC 006 Students will be able to calculate line integrals in two and three dimensions. KC

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Provide students with a thorough introduction to multivariable differential and integral calculus.

• Introduce students to differential operators and line integrals in vector calculus.

• Provide students with experience of methods used to understand and solve problems in multivariable calculus and vector calculus.

The learning and teaching methods include:

• Four one-hour lectures for eleven weeks, with module notes provided to complement the lectures. These lectures provide a structured learning environment and opportunities for students to ask questions and to practice methods taught.

• Five seminars for guided discussion of solutions to problem sheets (provided to students in advance) to reinforce their understanding and guide their learning.

• Three assessed courseworks to provide students with further opportunity to consolidate learning. Students receive feedback on these courseworks as guidance on their progress and understanding.

• Lectures may be recorded or equivalent recordings of lecture material provided. These recordings are intended to give students an opportunity to review parts of lectures which they may not fully have understood and should not be seen as an alternative to attending lectures.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MAT1043

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Digital Capabilities, Employability, Global and Cultural Capabilities, Resourceness and Resilience, and Sustainability. This module is designed to allow students to develop knowledge, skills and capabilities in the following areas:

Digital Capabilities: The SurreyLearn page for MAT1043 features a dynamic discussion forum where students can pose questions and engage with others using e.g. LaTeX and MathML tools. This enhances their digital competencies while facilitating collaborative learning and information sharing.

Employability: The module MAT1043 equips students with skills which significantly enhance their employability. The mathematical proficiency gained hones critical thinking and problem-solving abilities. Students learn to evaluate complex geometric problems and calculations, break them into manageable components, and apply logical reasoning to arrive at solutions — these are highly sought after skills in any profession.

Global and Cultural Capabilities: Student enrolled in MAT1043 originate from a variety of countries and have a wide range of cultural backgrounds. Students are encouraged to work together during problem-solving teaching activities in seminars and lectures, which naturally facilitates the sharing of different cultures.

Resourcefulness and ResilienceMAT1043 is a module which demands the ability to visualise geometric problems and perform lengthy unseen calculations accurately. Students will gain skills in analysing geometric problems using creative and lateral thinking, and will complete assessments which challenge them and build resilience.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Data Science BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 2 Compulsory A weighted aggregate mark of 40% is required to pass the module
Economics and Mathematics BSc (Hons) 2 Compulsory A weighted aggregate mark of 40% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.