Vector Calculus UoR Home

Lecturer: Andrea Moiola
Phone: +44 (0) 118 378 4272
Room: 203 Maths building
Course webpage:
Lectures: Thursday 2-3 pm (29th September to 8th December, except 3rd November)
Friday 10-11 am (30th September to 9th December, except 4th November)
Tutorials: Friday 11-12 am (30th September to 9th December, except 4th November)
Room: all lectures and tutorials will take place in the JJ Thomson building, room DB
Teaching term: autumn term 2016
Module description:   link for MA2VC (part two students),   link for MA3VC (part three students)

Coursework Downloadable material Suggested books Content of the lectures


Assignment 1:
PDF file for MA2VC (part 2 students), PDF file for MA3VC (part 3 students).
Solutions and feedback for Assignment 1: MA2VC and MA3VC.

Assignment 2:
PDF file for MA2VC (part 2 students), PDF file for MA3VC (part 3 students).
Solutions and feedback for Assignment 2: MA2VC and MA3VC.

Assignment 3:
PDF file for MA3VC only (part 3 students).
Solutions and feedback for Assignment 3: MA3VC.

• Assignment 1 is due in the school office by Thursday 27th October 2016, 12 noon.
• Assignment 2 is due in the school office by Thursday 24th November 2016, 12 noon.
• Assignment 3, for MA3VC students only, is due in the school office by Thursday 8th December 2016, 12 noon.
Marked assignment will be returned at most within 15 term days.
Each assignment is worth 10% of the final mark. The final exam is worth 80% for MA2VC and 70% for MA3VC.

Downloadable material

Lecture notes:
This PDF file contains the lecture notes for the course.
Printed copies have been distributed in class, if you don't have a copy please ask me.
This document does not substitute the notes taken in class!
Please let me know any errors or typos you find.

This file contains the worked solutions of the exercises in section C of the notes.

This file summarises and compares the different kinds of integrals seen in the lectures and the main theorems encountered.

Assignments from previous years:
Try hard to solve them before checking the solutions! These are provided only to double check your results.
2011 assignment 1,
2011 assignment 2,
2011 assignment 3,
2011 assignment 4,
2012 assignment 1,
2012 assignment 2,
2012 assignment 3,
2012 assignment 4,
2013 assignment 1 (solutions),
2013 assignment 2 (solutions),
2013 assignment 3 (solutions),
2013 assignment 4 (solutions),
2014 assignment 1, MA2VC (solutions),
2014 assignment 1, MA3VC (solutions),
2014 assignment 2, MA2VC (solutions),
2014 assignment 2, MA3VC (solutions),
2015 assignment 1, MA2VC (solutions),
2015 assignment 1, MA3VC (solutions),
2015 assignment 2, MA2VC (solutions),
2015 assignment 2, MA3VC (solutions).
Results of the questions in previous assignments.
Older problem sheets and worked solutions.
Please note that some of the notation used in previous years is slightly different from that used in the notes and in class; for instance, double and triple integrals were denoted with a single integral sign.
Also note that old solutions refer to previous versions of the lecture notes, so the numbers of the equations, theorems, sections, etc., mentioned do not coincide with those in the current lecture notes.

Final exams from previous years:
MA2VC 2011-12,
MA3VC 2011-12,
MA2VC 2012-13,
MA3VC 2012-13,
MA2VC 2013-14,
MA3VC 2013-14,
MA2VC 2014-15,
MA3VC 2014-15.
MA2VC 2015-16,
MA3VC 2015-16.
In this file you find the final results of the questions in the old exams above. You can you use them to double check your results during revision.

Matlab files:
This ZIP file contains: (1) the Matlab/Octave routines used to make the figures in the lecture notes, and (2) the file VCplotter.m, which is a simple Matlab/Octave function that can be used to visualise curves, scalar fields, vector fields, domains and surfaces in different coordinate systems. It can easily be used even with no Matlab skills. The included pdf file briefly describes how to use it and how to install Octave on your computer.

Suggested books and other sources:

Content of the lectures:

I will summarise here the content of each lecture and the corresponding sections in the lecture notes.
In the "book" column you find some exercises of the book by Adams and Essex you may find useful (see also appendix E in the notes). Note that sometimes the notation and the language of the book are slightly different from what we use in class.
In the last column you find the questions of Appendix C of the notes that you can solve after each lecture. Some of these exercises will be discussed in the next tutorial, you are invited to try to solve them.
You should also try to solve the exercises in sections 1-3 of the lecture notes once the corresponding topics have been covered; the solutions are in appendix B.

Week Date Topics covered Notes sections Book sections
and exercises
1 29.09.2016

Introduction to the course. What is vector calculus about?

Quick refresh of vectors: notation, canonical basis, components, length, direction, unit vectors.
Vectors as elements of R³, as triples of real numbers, as points in space, as geometric objects with direction and length.



Operations with vectors. Vector addition and scalar-vector multiplication.

Scalar product: definition, use of scalar product to compute angles, magnitudes and lengths of projections, parallel and orthogonal vectors.

Vector product: definition, geometric interpretation, anticommutativity, non-associativity.
Triple product: definition, volume of a parallelepiped.

Open sets, definition, intuitive description as sets containing spheres centred at each point.




10.2 (p.577)
Ex. 2-3, 16-31

10.3 (p.584)
Ex. 1-28


Limits of sequences of vectors. Closed sets: definition, intuitive description as sets containing limits.

Scalar fields: definition, definition of continuous fields, level sets, representation of two-dimensional fields with graph surfaces, example f=x²+y².

Vector fields: definition, quiver representation, examples (F=i+yj and F=i+xj).

Curves: definition, path of a curve, example a(t)=cos(t)i+sin(t)j.




12.1 (p.676)
Ex. 11-42

15.1 (p.848)
Ex. 1-8


2 6.10.2016

Recall definitions of curve, path and loop. Interpretation of a curve as trajectory of a moving point. Different parametrisations of the same path.
How to draw the path of a curve? Example: circle.
How to compute the parametrisation of a path? Examples: segments and graphs (Remark 1.24).

Partial derivatives: definition, interpretation, examples. Linearity, product rule and chain rule.

Gradient: definition, examples, linearity, product rule and chain rule.
Directional derivatives.



11.3 (p.641)
Ex. 1-12

12.3 (p.687)
Ex. 1-12, 25-31

12.7 (p.723)
Ex. 1-6 (part a),



The gradient of f is perpendicular to the level lines of f.
The gradient of f points in the direction of steepest ascent of f.
Examples: the field f=x²+y² and the "height above sea level" scalar field.

Jacobian matrix of a vector field.

Second order partial derivatives, Hessian of a scalar field, Laplacian, harmonic functions. Clairaut's Theorem, examples.

Divergence of a vector field. Definition, examples, physical interpretation.




12.4 (p.692)
Ex. 1-14

16.1 (p.896)
Ex. 1-8


Fri (tut)

Exercises C.2, C.3, C.4.

3 13.10.2016

Curl of a vector field. Definition, example, interpretation as rotation.
Summary of the differential operators encountered so far.

Vector differential identities: second-order vector differential operators, vector product rules. Proof of identities (24), (26), (29) and (31).




16.2 (p.902)
Ex. 8, 11, 13, 14

C.8, C.10,


Two proofs of Laplacian's product rule (32).

Special fields: definition of irrotational, solenoidal and conservative fields, scalar and vector potentials.
Non-uniqueness of potentials.
Minimal examples of fields which are only solenoidal, only irrotational, both or none (yi, xi, i or xi-yj, (x+y)i).
A field is irrotational if and only if its Jacobian is symmetric.
Conservative implies irrotational, existence of vector potential implies solenoidal; what about the converse? it depends on "holes" in the domain (no details on this).
Every 1D function is derivative of another one but not all vector fields are gradients or curls.

How to compute potentials? Integration of a differential equation; example: computation of the scalar potential for xi+yj.


15.2 (p.857)

16.2 (p.902)
Ex. 9, 10, 12, 15, 16


Fri (tut)

Exercises C.5, C.6, C.7.

4 20.10.2016

Example: compute vector potential for -yi+xj.

Total derivatives of curves. Linearity and product rules for derivatives of curves.
Total derivative is tangent to the path of a curve; its magnitude is velocity. Intrinsic parametrisations and arc length.

Recall chain rules already seen: for composition of real functions and for composition of a real function with a scalar field.
Chain rule for the evaluation of a scalar field along a curve.
Example: total derivative of the field f=xy constrained to the curve a(t) = cos t i + sin t i; two solution procedures.



11.1 (p.627)
Ex. 1-14, 27-32



Chain rule for the partial derivatives of the composition of a scalar field with a vector field.
Example 1.85: derivative of a scalar field constrained to a graph surface.

Summary of the programme covered so far, structure of chapter one of the notes.

General introduction to vector integration.
Interpretation of integrals as measures (lengths, areas, volumes) of geometric objects or as computation of a mass from a given density.
Three fundamental properties of integrals: linearity, additivity, integral of constant 1 is measure of domain of integration.

Line integrals of scalar fields: formula for the length of a path and for the line integral of a scalar field.


12.5 (p.702)
Ex. 1-12, 15-22


Fri (tut)

Exercises C.8, C.9, C.10, C.12.

5 27.10.2016

Recall line integrals of scalar fields and their interpretation.
Measure of the length of a curve and of the graph of a scalar function.
Independence of parametrisation.
Example: computation of line integral of f=y³ along two different parametrisations of half circle.

Line integrals of vector fields: formula, unit tangent vector, interpretation of line integral as integral of tangential component and relation with integral of scalar fields.
Example: integral of F=r over segment from i to i+j, and over the same segment in the opposite direction.
Oriented paths, changing path orientation changes sign of the integral.
Circulation, notation for integrals over loops.


11.3 (p.641)
Ex. 13-21

15.3 (p.861)
Ex. 1-9

15.4 (p.869)
Ex. 1-7



Recall line integrals of scalar and vector fields and their interpretation.
Other kind of line integrals: int_Gamma f dx.

Recall fundamental theorem of calculus (for real functions).
Fundamental theorem of vector calculus; example.
Path independence of the line integral of conservative fields.
Scalar potentials as primitives for line integrals.

Theorem (only stated): given a vector field F, three conditions are equivalent: (i) F is conservative, (ii) loop integrals of F are zero, (iii) line integrals of F are path-independent (for paths with the same endpoints).

Definition of star-shaped domain.
Theorem: given vector field F defined on a star-shaped domain, F is conservative if and only if it is irrotational.


Fri (tut)

Justification of formula for the line integral for scalar fields using Riemann sums (on equispaced points).

Exercises C.13 and C.14.



7 10.11.2016
Thu (tut)

Exercises C.15 and C.16.


Double integrals: iterated integral on y-simple regions, geometric interpretation as volume between graph and xy-plane, comments on more general regions. Example: integral of xy on rectangle (0,2)x(0,1) and on the triangle with vertices 0, 2i and 2i+j.

Change of variables for double integrals: motivation from integration by substitution in 1D, new coordinates xi and eta, vector field T, xi-eta-plane, unit vectors, Jacobian and Jacobian determinant, change of variables formula.



14.1 (p.796)
Ex. 13-22

14.2 (p.802)
Ex. 1-28

14.4 (p.817)
Ex. 32-34



Double integral example: integral of (x+y)¹² over the square with vertices i, j, -i, -j.

Triple integrals, z-simple domains, volume of a domain, change of variables formula.

Comparison: path integrals extend scalar integrals from "flat" to curvilinear (1D) domains of integration.
Similarly, extending double integrals from "flat" to curvilinear (2D) domains we obtain surface integrals.
Surfaces as two-dimensional curved objects in R³. Definition of parametric surfaces


14.5 (p.823)
Ex. 1-20

8 17.11.2016

Recall definition of parametric surfaces.
Example: parametrisations of paraboloid {z=x²+y²} and cylinder {x²+y²=1}.
Graph surfaces as parametric surfaces.
Tangent and orthogonal vectors on a surface.

Formula for the integral of a scalar field over a parametric surface.
Independence of parametrisation.
Special case: integral of a scalar field over a graph surface.


14.7 (p.838)
Ex. 1-10

15.5 (p.880)
Ex. 3, 4, 7-10, 13-16


Example: integral of f=√(1+4z) over the paraboloid defined by X=ui+wj+(u²+w²)k over R=(-1,1)².

Flux of a fluid flowing through a surface.
Unit normal vector fields, orientations and oriented surfaces. Non-orientable surfaces (Moebius strip).
Parametric surfaces (including graphs), boundaries and level sets are orientable and have a preferred choice of orientation.

Flux as surface integral of normal component of vector field; analogy with line integrals.
Formula for fluxes on a parametric surface and on a graph surface.
Example: flux of F=-yi+xj+zk through the paraboloid defined by X=ui+wj+(u²+w²)k over R=(-1,1)².

Path orientation induced by an oriented surface on its boundary.


15.6 (p.886)


Fri (tut)

Exercises C.18, C.19, C.20

9 24.11.2016

Special coordinate systems are standard changes of variables used in presence of some geometric symmetries.

Polar coordinates r and ϑ. Definition, Jacobian determinant and area element, area of a star-shaped region, disc, orthogonal unit vector fields r and ϑ.

Cylindrical coordinates r, ϑ and z. Definition, volume element, volume of a solid of revolution.

Spherical coordinates ρ, Φ and ϑ. Definition, volume element.




8.5 (p.488)
Ex. 1-28

8.6 (p.492)
Ex. 1-11

10.6 (p.600)
Ex. 1-14

14.4 (p.817)
Ex. 1-30

14.6 (p.830)
Ex. 1-16

15.2 (p.857)
Ex. 19-22



Introduction to new part: three theorems on integration of derivatives.
Recall fundamental theorem of calculus and fundamental theorem of vector calculus.
Possible generalisations of the different objects appearing in these theorems.

Green's theorem: assumptions, statement, comments, derivation from Lemma 3.3.
Proof of part 1 of Lemma 3.3.


16.3 (p.906)
Ex. 1-7

Fri (tut)

Table of comparison of the different integrals considered so far.

Exercise C.21.

10 1.12.2016

Completion of the proof (parts 2 and 3) of Lemma 3.3.
Use of Green's theorem to compute areas.

Exercise: compute circulation of F = (x-y)i + xj around a circle or a triangle in the xy-plane.
Exercise: for a smooth real function g, prove that the circulation of a vector field F = g(x+y)(i + j) around the boundary of any planar region is zero.

Divergence theorem: assumptions, statement, interpretation.



Recall divergence theorem. Proof using Lemma 3.13.
Proof of Lemma 3.13.

Divergence of a vector field as average flux through infinitesimal spherical surface.


16.4 (p.912)
Ex. 1-14


Fri (tut)

Comments on errors found in the assignments.
Exercise C.22.

11 8.12.2016

Integration by parts in three dimensions, derivation of formula in 3.26.
Derivation of first and second Green's identities (Corollary 3.22).

Stokes' theorem.
Statement and simple consequences.

"Physical" definitions of divergence as (averaged) flux through an infinitesimal sphere and of curl as (averaged) circulation around an infinitesimal disc (Remark 3.33).

Recall relation between conservative fields and path independence of line integrals (box (50)); similar relation between fields admitting a vector potential and "surface independence" of fluxes (Remark 3.34).

Example of application of the divergence theorem: radial fields that are solenoidal outside a ball decay as 1/r²; for example, gravitational and electrical fields.


16.5 (p.916)
Ex. 1-12


Fri (tut)

Example of application of the divergence theorem to physics: derivation of continuity equation for a fluid in absence of sources and sinks.

Exercises C.27, C.24, C.26.

Fri (tut)

Exercises C.28 and C.30.

R 18.4.2017
JJT-DB, 5pm

Exercises 1, 2 and 4 of the 2016 MA2VC exam.