Lecturer:  Andrea Moiola 
Email:  a.moiola@reading.ac.uk 
Phone:  +44 (0) 118 378 4272 
Room:  203 Maths building 
Course webpage:  http://www.personal.reading.ac.uk/~st904897/VC2016/VC2016.html 
Lectures:  Thursday 23 pm (29th September to 8th December, except 3rd November) 
Friday 1011 am (30th September to 9th December, except 4th November)  
Tutorials:  Friday 1112 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.
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 201112,
MA3VC 201112,
MA2VC 201213,
MA3VC 201213,
MA2VC 201314,
MA3VC 201314,
MA2VC 201415,
MA3VC 201415.
MA2VC 201516,
MA3VC 201516.
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.
R.A. Adams, C. Essex, Calculus, a complete course, Pearson Canada, Toronto, 7th ed., 2010. UoR library call number: FOLIO515ADA.
E. Carlen, Multivarible Calculus, Linear Algebra and Differential Equation, lecture notes, 2012. Available on: http://www.math.rutgers.edu/~carlen/291F14/index.html.
M. Corral, Vector Calculus, Schoolcraft College, 2013. Available on: http://www.mecmath.net/.
S. Lang, Calculus of Several Variables, Springer Undergraduate Texts in Mathematics, 3rd ed., 1996. UoR library call number: 515.84LAN.
J.E. Marsden, A. Tromba, Vector calculus, W.H. Freeman, New York, 6th ed., 2012. UoR library call number: 515.63MAR.
M. Spiegel, S. Lipschutz, D. Spellman, Schaum's Outline of Vector Analysis, McGrawHill, 2nd ed., 2009. UoR library call number: FOLIO515.63LIP.
C.J. Smith, Vector Calculus Primer, 2011, Reading. Available here.
On the page of Oliver Knill (Harvard) http://www.math.harvard.edu/~knill/teach/index.html you can find plenty of material on multivariable calculus: lecture notes, exercises and instructive illustrations. Click on one of the Math(s)21a links or on the lecture notes.
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 13 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 
Exercises for tutorials 
1  29.09.2016 Thu 
Introduction to the course. What is vector calculus about? Quick refresh of vectors: notation, canonical basis, components, length, direction, unit vectors.

1.1 

30.09.2016 Fri 
Operations with vectors. Vector addition and scalarvector 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, nonassociativity. Open sets, definition, intuitive description as sets containing spheres centred at each point. 
1.1.1, 1.1.2, 1.1.3 
10.2 (p.577) 10.3 (p.584) 
C.1  
30.09.2016 Fri 
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 twodimensional 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. 
1.2.1, 1.2.2, 1.2.3 
12.1 (p.676) 15.1 (p.848) 
C2,  
2  6.10.2016 Thu 
Recall definitions of curve, path and loop.
Interpretation of a curve as trajectory of a moving point.
Different parametrisations of the same path. Partial derivatives: definition, interpretation, examples. Linearity, product rule and chain rule. Gradient: definition, examples, linearity, product rule and chain rule. 
1.3.1, 1.3.2 
11.3 (p.641) 12.3 (p.687) 12.7 (p.723) 
C.6 
7.10.2016 Fri 
The gradient of f is perpendicular to the level lines of f. 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. 
1.3.3, 1.3.4, 1.3.5 
12.4 (p.692) 16.1 (p.896) 
C.7, 

7.10.2016 Fri (tut) 
Exercises C.2, C.3, C.4. 

3  13.10.2016 Thu 
Curl of a vector field. Definition, example, interpretation as rotation. Vector differential identities: secondorder vector differential operators, vector product rules. Proof of identities (24), (26), (29) and (31). 
1.3.6, 1.4.1, 1.4.2 
16.2 (p.902) 
C.8, C.10, 
14.10.2016 Fri 
Two proofs of Laplacian's product rule (32). Special fields: definition of irrotational, solenoidal and conservative fields, scalar and vector potentials. How to compute potentials? Integration of a differential equation; example: computation of the scalar potential for xi+yj. 
1.5 
15.2 (p.857) 16.2 (p.902) 
C.12 

14.10.2016 Fri (tut) 
Exercises C.5, C.6, C.7. 

4  20.10.2016 Thu 
Example: compute vector potential for yi+xj.
Total derivatives of curves.
Linearity and product rules for derivatives of curves.
Recall chain rules already seen: for composition of real functions and for composition of a real function with a scalar field.

1.6, 1.7 
11.1 (p.627) 
C.13, 
21.10.2016 Fri 
Chain rule for the partial derivatives of the composition of a scalar field with a vector field.
Summary of the programme covered so far, structure of chapter one of the notes.
General introduction to vector integration. Line integrals of scalar fields: formula for the length of a path and for the line integral of a scalar field. 
2.1.1 
12.5 (p.702) 
C.16 

21.10.2016 Fri (tut) 
Exercises C.8, C.9, C.10, C.12. 

5  27.10.2016 Thu 
Recall line integrals of scalar fields and their interpretation. 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. 
2.1.2 
11.3 (p.641) 15.3 (p.861) 15.4 (p.869) 
C.17 
28.10.2016 Fri 
Recall line integrals of scalar and vector fields and their interpretation. Recall fundamental theorem of calculus (for real functions). 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 pathindependent (for paths with the same endpoints). Definition of starshaped domain. 
2.1.3 

28.10.2016 Fri (tut) 
Justification of formula for the line integral for scalar fields using Riemann sums (on equispaced points). Exercises C.13 and C.14. 

6  ...  
7  10.11.2016 Thu (tut) 
Exercises C.15 and C.16. 

11.11.2016 Fri 
Double integrals: iterated integral on ysimple regions, geometric interpretation as volume between graph and xyplane, 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, xietaplane, unit vectors, Jacobian and Jacobian determinant, change of variables formula. 
2.2.1, 2.2.2 
14.1 (p.796) 14.2 (p.802) 14.4 (p.817) 
C.18, 

11.11.2016 Fri 
Double integral example: integral of (x+y)¹² over the square with vertices i, j, i, j. Triple integrals, zsimple domains, volume of a domain, change of variables formula. Comparison: path integrals extend scalar integrals from "flat" to curvilinear (1D) domains of integration. 
2.2.3, 
14.5 (p.823) 

8  17.11.2016 Thu 
Recall definition of parametric surfaces.
Formula for the integral of a scalar field over a parametric surface. 
2.2.4 
14.7 (p.838) 15.5 (p.880) 

18.11.2016 Fri 
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. Flux as surface integral of normal component of vector field; analogy with line integrals. Path orientation induced by an oriented surface on its boundary. 
2.2.5 
15.6 (p.886) 
C.22 

18.11.2016 Fri (tut) 
Exercises C.18, C.19, C.20 

9  24.11.2016 Thu 
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 starshaped 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. 
2.3.1, 2.3.2, 2.3.3 
8.5 (p.488) 8.6 (p.492) 10.6 (p.600) 14.4 (p.817) 14.6 (p.830) 15.2 (p.857) 
C.23, 
25.11.2016 Fri 
Introduction to new part: three theorems on integration of derivatives.
Green's theorem: assumptions, statement, comments, derivation from Lemma 3.3. 
3.1 
16.3 (p.906) Ex. 17 

25.11.2016 Fri (tut) 
Table of comparison of the different integrals considered so far. Exercise C.21. 

10  1.12.2016 Thu 
Completion of the proof (parts 2 and 3) of Lemma 3.3.
Exercise: compute circulation of F = (xy)i + xj around a circle or a triangle in the xyplane.
Divergence theorem: assumptions, statement, interpretation. 
C.26 

2.12.2016 Fri 
Recall divergence theorem. Proof using Lemma 3.13.
Divergence of a vector field as average flux through infinitesimal spherical surface. 
3.2 
16.4 (p.912) 
C.27, 

2.12.2016 Fri (tut) 
Comments on errors found in the assignments. 

11  8.12.2016 Thu 
Integration by parts in three dimensions, derivation of formula in 3.26.
Stokes' theorem. "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. 
3.3 
16.5 (p.916) 
C.30, 
9.12.2016 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. 

9.12.2016 Fri (tut) 
Exercises C.28 and C.30. 

R  18.4.2017 Revision JJTDB, 5pm 
Exercises 1, 2 and 4 of the 2016 MA2VC exam. 