WTW258
CALCULUS
STUDY GUIDE 2024 (Semester 1)
ORGANISATIONAL COMPONENT
1. Admittance to the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Lecturers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3. Rules of Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Tutorial classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5. Textbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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6. Pre-knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7. Learning hours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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8. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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STUDY COMPONENT
1. Use of the study guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. General objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3. General learning outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Module structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Theme 1 : Real functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Theme 2 : Multiple integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Theme 3 : Calculus of vector functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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ORGANISATIONAL COMPONENT
1. ADMITTANCE TO THE COURSE
The prerequisites for WTW258 is a pass mark for WTW158 and WTW164.
2. LECTURERS
Module coordinator: Dr DV Moubandjo
Lecturers
Dr MD Mabula
Ms MP Möller
Mrs L Mostert
Dr DV Moubandjo
Dr B Stapelberg
Office
Tel: 012-420Botany 2-5
6818
Mathematics 2-19
2279
Mathematics 1-18
2010
Botany 2-9
5476
Mathematics 2-14
6483
Consulting hours
Hours of consultation with lecturers will be displayed on their office doors and
clickUP. Students may consult lecturers only during the consulting hours as indicated, or by appointment. This policy also holds before tests and examinations. In
other words, lecturers are only available during their normal consulting hours on
the day before a test or examination. This policy aims at encouraging students to
plan their work and to work continuously.
3. RULES OF ASSESSMENT
Evaluation will take the form of regular class tests, two semester tests and a final
examination.
The examination and test instructions of the University of Pretoria must be followed
meticulously.
3.1 Material for semester tests
Material for semester tests will be announced in class and will be published on
clickUP.
3.2 Semester tests
The two semester tests will be written during the following test weeks:
Semester test 1: Saturday 06 April 2024 – Saturday 13 April 2024.
Semester test 2: Saturday 11 May 2024 – Saturday 18 May 2024.
The exact dates are available on your UP portal and will be posted later on clickUP.
3.3 Class tests
Class tests are written on a regular basis in the tutorial classes. They cover both
theory and problems of the relevant sections that are posted weekly on clickUP.
Important: All the class tests that are graded by the lecturers and of which the
marks are entered will contribute to the semester mark.
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3.4 Marked tests
File all your marked class tests and semester tests. They are your only proof that
you have indeed written the tests. Any problems regarding entered marks will only
be considered when you bring all the tests to the lecturer, not just the one(s) that
is(are) queried.
3.5 Arrangements with respect to tests and examinations
• Student cards must be produced at request at tests or examinations.
• No answer books may be removed from the test / examination venue by any
student (this is a serious misconduct).
• All queries concerning the grading of a specific test must be finalized within
3 (three) days after receipt of the graded test. After three days it is assumed
that all marks are final and correct and no further discussion will be entered
into.
3.6 Absence from tests and examinations
In the case of absence from examinations the relevant faculty administration
should be informed.
In the case of absence from tests (tutorial tests and semester tests) the lecturer
concerned must receive the relevant documents within three days from the date
of the test.
Valid original sick notes are accepted if issued by a medical practitioner registered
at the Health Professions Council of South Africa (HPCSA). The only other type of
sick notes that is accepted are those issued by an Advanced Practice Nurse (a registered nurse with a postgraduate qualification) as determined by the South African
Nursing Council who has a BHCF practice number, provided that the diagnosis falls
only within their specific field of specialisation.
An affidavit will only be accepted if supported by substantiating documentation, e.g.
case report or criminal charge with case number obtained from a police station, valid
medical certificate for injuries, a death certificate for a funeral, etc.
Please note that submission of fraudulent sick notes and affidavits is a criminal
offence, which will lead to disciplinary action and may result in dismissal.
In the case of representation, you have to submit a signed, original letter from the
coach or leader of the group, as well as an official notice of the event that includes
the date and location of the event.
In the case of a clash with another test, you have to submit the study guide or a
copy of an email from the module coordinator indicating the date and time of the
test. Evidence for a clash must be presented prior to the test.
Refer also to General Regulation G.2.3
• In the case of illness, follow the procedure below and take note of the requirements for an acceptable medical certificate.
– Hand in a copy of the medical certificate with the lecturer concerned in
person. Your initials, surname and student number must be written clearly
at the back of the copy.
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– The original certificate must accompany the copy and therefore certificates
that are e-mailed to the lecturer will not be accepted.
– A medical certificate stating that a student appeared ill or declared himself
/ herself unfit to write a class test or semester test, will not be accepted.
– The doctor must be consulted on or before the date of the scheduled class
test / semester test.
• Do not slide medical certificates under the door of a lecturer. Certificates that
are received as such, will not be accepted.
• In the case that the three day deadline can not be met (due to unavoidable
circumstances), the student must notify the lecturer concerned/module coordinator of his / her situation (by phoning, e-mail or via a fellow student).
The same rules apply in the case of absence, with a satisfactory proof of the reason
for absence, due to other circumstances.
3.7 Sick test
Once a student has written any test he/she may not write a sick test
(regardless of illness or any other circumstances) to improve the mark.
(The same applies for the examination and re-examination.)
Hence, do not take a test while unwell. Do visit your general practitioner
right away.
There will be a separate sick test for semester test 1 and one for semester test 2.
Writing the sick test is compulsory for a student absent (with a valid proof of
absence) from one or both semester tests. If the sick test is not written (regardless
of the reason for not writing) a mark of 0% will be given for the specific semester
test that has not been written.
Information regarding sick tests will be posted on clickUP during the test week.
Sick test 1 may be written in the week directly after the first test week or at the
same time as sick test 2 directly after the second test week.
It is the responsibility of the student to get the information regarding
the sick test.
There is no sick test for absence from any of the class tests. Absence from a class
test with a valid reason will be taken in account when calculating the final class test
mark.
3.8 Calculations of marks
Semester mark
Semester test 1
30%
Semester test 2
40%
Class tests
30%
Final mark
Semester mark
50%
Examination mark 50%
3.9 Examination: Admittance and pass requirements
• To obtain admittance to the examination a semester mark of at least 40% is
required.
• To pass the course a final mark of at least 50% is required and a subminimum
of 40% for the examination.
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3.10 Supplementary examination
A student qualifies for a supplementary examination if he/she complies with one of
the following criteria:
• the final mark is between 45% and 49%,
• the final mark is between 40% and 44% and either the examination mark or
the semester mark is at least 50%.
The final mark for a supplementary examination is the average of the semester
mark and the supplementary examination mark.
To pass the course a final mark of 50% is required and a subminimum of 40% for
the supplementary examination.
The final mark awarded may not be more than 50%.
3.11 Winter school
Refer also to General Regulation G11.2(2.1) and G.12.6(6.5)
(i) To gain entry into Winter school a student must have obtained exam entry for
the module in the previous semester and received a final mark of at least 40%.
(ii) If a student fails the Winter School presented, the module needs to be repeated.
(iii) If a student qualifies to do more than one Winter school and there is an overlap
between the Winter Schools, the student needs to select one between the two. You
cannot do both.
No exceptions are made. Contact the EBIT Administration (ENG I Level
6) for further enquiries.
4. TUTORIAL CLASSES
Attendance of all classes and tutorial classes is compulsory.
The tutorial slot of 3 (three) hours, in your timetable, is divided into 90 (ninety)
minutes for WTW258 and 90 (ninety) minutes for WTW256.
Students who are taking both modules will therefore have a tutorial session of 180
minutes in total on the same day.
A detailed tutorial allocation, with specific time and venue for each group, will be
posted on clickUP (according to study program and surname).
You may not attend another tutorial class than the one that you are
allocated to. If you do attend another tutorial class the test will not be
taken into consideration and you will be regarded as absent.
All the problems as indicated in the study guide must be done. The assignments
for the tutorial classes will be posted weekly on clickUP. You are expected to
• prepare the theoretical part thoroughly before the tutorial class and
• do all the exercises for the tutorial class beforehand. The idea of the tutorial
class is to sort out the problems that you had while preparing for the tutorial
class and not to start doing the exercises in the tutorial class.
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5. TEXTBOOK
Authors: J. Stewart, D. Clegg, S. Watson
Title: CALCULUS Early Transcendentals (Ninth Edition, Metric Version)
That is the textbook used in WTW158/WTW164
6. PRE-KNOWLEDGE
You are advised to revise certain sections of first year Calculus. It is your responsibility to do the necessary revision in time.
For Theme 1: All the work in Lecture units 1.1 to 1.3 as well as maximum and
minimum values of functions of one variable.
For Theme 2: Techniques of integration : Standard integrals, integration by parts
and the integration of powers of sin and cos. Polar coordinates/equations: Convert
polar coordinates to Cartesian coordinates and vice versa.
7. LEARNING HOURS
This module carries a weighting of 8 credits, indicating that on average a student
should spend about 80 hours to master the required skills (including time for preparation for tests and examinations). A student must devote on average 6 hours
of study time per week to this module. The scheduled contact time is approximately 3 hours per week which means that another 3 hours per week of own study
time should be devoted to the module. The actual time required to complete the
module successfully, depends on the abilities and circumstances of each student.
8. GENERAL
8.1 Announcements:
The study guide does not necessarily contain all the information. Important announcements may be made during lectures and will be posted on clickUP.
8.2 ClickUP
All important information will appear on clickUP.
8.3 Pigeon holes:
If you are in the Mathematics building with room 1-14 to your right and the notice
boards to your left, you look at the pigeon holes for the course. All unclaimed class
tests will be put in the pigeon holes.
8.4 Calculators
Calculators may not be used in Semester tests/examinations and most tutorial
tests.
Only the prescribed calculators, that is Sharp EL 531, Casio FX 82 or Casio FX 82
Plus, may be used in tutorial tests when a calculator is allowed.
8.5 Communication via email
When you send an email to a lecturer, you have to use a respectful tone and include
all the following aspect:
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• A clear and explanatory subject line which include the course code (e.e. ”WTW258
submission sick note- P Mduli”);
• Your full name and surname at the end of the mail;
• Your student number; and
• Short and clear message.
8.6 Compliments and complaints
You are more than welcome to express your appreciation to your lecturer or tutor
and supply feedback about aspects of the course that you enjoy and find valuable.
If you have a query or complaint, you have to submit it in writing with specifics
of the issues or the nature of the complaint. It is imperative that you follow the
procedure outlined below in order to resolve the issues:
1. Consult the class representative or lecturer concerned about your
complaint/concerns. If the matter has not yet been resolved,
2. consult the course coordinator. If the matter has not yet been resolved,
3. consult the Head of Department(Prof Banda). If the matter has not yet
been resolved,
4. consult the Dean of the Faculty.
8.7 Previous semester tests and examination papers
Enquiries with regard to semester tests and examination papers of previous years will
only be answered after the student provided proof that all the prescribed problems
of the relevant tutorial classes have already been done. We strongly advise students
not to put too much emphasis on previous papers when preparing for the tests and
examinations.
8.8 Application for extra time during semester tests and examinations
Students who need extra time for semester tests and examinations must get a valid
and applicable document (a letter on a letter heading from the Faculty of Engineering) from student administration (floor 6, Engineering 1). No other letter will be
accepted.
A copy of this letter must be handed in at the module coordinator not later than a
week before the first semester test. The original letter must also be shown.
8.9 Disciplinary cases
The policy of the Department of Mathematics and Applied Mathematics is without
exception to refer all cases where a suspicion of irregularity exists to the disciplinary
committee of the university.
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STUDY COMPONENT
1. USE OF THE STUDY GUIDE
1.1 The course is divided into a number of THEMES. Each theme is subdivided
into LECTURE UNITS, each with its own LEARNING OUTCOMES, in order
to provide you with an overview of the structure of the course. It also tells you
exactly what is expected from you.
1.2 The material you have to master is indicated clearly in the learning outcomes
and under the heading SOURCE. Unless indicated otherwise, you must comprehend and know everything in full. Please note that amongst other reasons,
the text book is prescribed to accustom you with the book in order that you
will be able to do further reading about topics not covered in the course when
you need more information on such topics.
1.3 The LEARNING OUTCOMES are basic guidelines. It does not mean that
examination questions will consist only of theory and the type of problems spelt
out in the outcomes. It may sometimes be necessary to combine your knowledge
of different themes to solve a problem. The first step however remains to check
after each lecture unit that you have indeed reached the set learning outcomes.
1.4 The tutorial problems test whether you have reached the learning outcomes.
Solving problems also ensures that you get the necessary training in the application of your knowledge. It is of utmost importance that these problems are
done as soon as possible after the completion of a lecture unit. In this way you
ensure that you do not lapse behind.
2. GENERAL OBJECTIVES
To introduce the theory and applications of the differential and integral Calculus of
multi-variable functions.
3. GENERAL LEARNING OUTCOMES
After completion of this module you should be able to
3.1 sketch and recognise two-variable functions as surfaces;
3.2 find, interpret and apply tangent planes and normal vectors to surfaces;
3.3 find, interpret and apply directional derivatives of multi-variable functions;
3.4 find, interpret and apply extreme values of multi-variable functions;
3.5 find, interpret and apply double, triple, line and surface integrals of multivariable functions;
3.6 interpret and apply the basic integral theorems of multi-variable functions.
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4. MODULE STRUCTURE
The subject matter for the course is divided into three themes:
THEME 1 : REAL FUNCTIONS OF SEVERAL VARIABLES
(6 lectures)
1.1 Functions of several variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.2 Partial derivatives (SELF STUDY) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.3 Tangent planes and linear approximations (SELF STUDY) . . . . . . . . . . . . .
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1.4 The chain rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5 Directional derivatives and the gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.6 Maximum and minimum values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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THEME 2 : MULTIPLE INTEGRALS
(7 lectures)
2.1 Double integrals and Iterated integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.2 Double integrals in Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2.3 Triple integrals for functions of three variables . . . . . . . . . . . . . . . . . . . . . . . . .
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2.4 Triple integrals in Cylindrical and in Spherical coordinates . . . . . . . . . . . . .
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2.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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THEME 3 : CALCULUS OF VECTOR FUNCTIONS
(11 lectures)
3.1 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.2 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.3 The Fundamental Theorem for line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.4 Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.5 Rotation and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.6 Parametric surfaces and their areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.7 The Surface integral and Flux integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.8 Stokes’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3.9 The Divergence Theorem of Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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STUDY THEME 1 : REAL FUNCTIONS OF SEVERAL VARIABLES
We consider functions with domain in IR2 or IR3 and real function values. An example
of such a function is one denoting the temperature at a given position on a heated plate.
The aim of this theme is the study of the differential calculus for functions of several
variables. This theme is divided into 6 lecture units:
1.1 Functions of several variables
1 lecture
1.2 Partial derivatives
Self study
1.3 Tangent planes and linear approximations
1.4 The chain rule
1 lecture
1.5 Directional derivatives and the gradient
2 lectures
1.6 Maximum and minimum values
2 lectures
1.1 FUNCTIONS OF SEVERAL VARIABLES
NUMBER OF LECTURE PERIODS : 1
(REVISION)
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. explain what is meant by a function of several variables.
2. determine the domain and range of a function of several variables.
3. sketch circles, ellipses, parabolas, hyperbolas and straight lines in the plane
from their Cartesian equations.
4. represent the well-known surfaces such as planes, paraboloids, cones, spheres,
ellipsoids and cylinders geometrically in space by determining their curves of
intersection with the coordinate planes.
5. derive curves of intersection between the above mentioned surfaces and sketch
their projections on the three coordinate planes.
6. identify and sketch contour diagrams of a given function.
7. use functions that are defined numerically.
Remark
You need not to be able to sketch a hyperboloid or a hyperbolic paraboloid. (Thus
you may leave Examples 5 and 6 on p 877 – 878.)
SOURCE
Calculus : Section 12.6,
p 875 – 881.
Calculus : Section 14.1,
p 933 – 946.
PROBLEMS
1. Find the domain and range and sketch the following functions:
(1) f (x, y) = p
6 − 2x − 3y
(2) f (x, y) = 4 − p
x2 − y 2
(3) f (x, y) = 4 − x2 − y 2
(4) f (x, y) = 4 − x2 + y 2
(5) f (x, y) = 4 − x
(6) f (x, y) = 4 − y 2
(7) f (x, y) = 4
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2. Sketch the surface:
(1) x2 + y 2 = 9
(3) y = x2 + 4z 2
(5) y 2 + z 2 = 4
(2) 4x2 + y 2 + 2z 2 = 8
(4) y 2 = x2 + 4z 2
3. Sketch:
p
(1) f (x, y) = x2 + y 2, g(x, y) = 2 − f (x, y) and h(x, y) = f (x, y − 2)
(2) x2 + y 2 + z 2 = 4y
(3) x2 + (y − 2)2 = 4, z ∈ IR
(4) the closedp
region bounded by the surfaces f (x, y) = 6 − x2 − y 2 and
g(x, y) = x2 + y 2 . Also find the curve of intersection of these surfaces.
4. Sketch a few level curves of the following functions:
y
(1) f (x, y) = x2 + 9y 2
(2) f (x, y) = 2
(3) f (x, y) = yex
x + y2
5. Describe the level surfaces of the function:
(1) f (x, y, z) = x + y + z
(2) f (x, y, z) = x2 + y 2 + z 2
(3) f (x, y, z) = x2 + y 2 − z
6. Find and sketch the domain of the following functions:
p
(1) f (x, y) = x2 + y 2 − 1 + ln(9 − x2 − y 2 )
p
(2) f (x, y, z) = 4 − x2 − y 2 − 4z 2 (give also the range)
7. Exercises 14.1 p 946 nos 4; 6; 12; 15; 21; 35; 37
8. Exercises 12.6 p 882 nos 37; 45; 46
1.2 PARTIAL DERIVATIVES
SELF STUDY
(REVISION)
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. use the definition to determine partial derivatives of functions of two variables.
2. find partial derivatives of functions of two or three variables.
3. interpret partial derivatives of functions of two variables geometrically.
4. estimate partial derivatives from contour diagrams.
5. estimate partial derivatives from numerical data.
6. calculate higher order partial derivatives.
7. use the different notations for higher order partial derivatives.
8. discuss when the second order mixed partial derivatives of a function of two
variables will be equal.
9. use implicit differentiation to find partial derivatives.
SOURCE
Calculus : Section 14.3,
p 961 – 968.
PROBLEMS
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1. Exercises 14.3 p 969 nos 6; 7; 11; 23; 32; 44; 55; 60; 63; 73; 74(a)-(b); 94
∂u ∂u
∂u
x+y
, find
,
and
.
2. If u =
y+z
∂x ∂y
∂z
3. Use the definition to find fx (0, 0) and fy (0, 0) if
 3
 x + 2y 3
, (x, y) 6= (0, 0)
f (x, y) =
x2 + y 2

0
, (x, y) = (0, 0)
1.3 TANGENT PLANES AND LINEAR APPROXIMATIONS
SELF STUDY
(REVISION)
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. find the equation of the tangent plane to the surface z = f (x, y) at a point.
2. find a linear approximation (tangent plane approximation) for a function of
two variables at a point and use it to estimate function values.
3. describe which conditions the partial derivatives must satisfy for a function of
two variables to be differentiable at a point.
4. explain what it means geometrically if a function is differentiable at a point.
Remark
You may leave the definition of differentiability (that is Definition 7 on p 977).
SOURCE
Calculus : Section 14.4,
p 974 – 978.
PROBLEMS
1. Exercises 14.4 p 981 nos 4; 9; 17; 25; 26
2. Determine whether f is differentiable at (0, 0) if
i. f (x, y) = x2 + 4y 2
p
ii. f (x, y) = x2 + 4y 2
1.4 THE CHAIN RULE
NUMBER OF LECTURE PERIODS : 1
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. use the chain rule for composite functions.
2. find higher order derivatives by means of the chain rule.
3. find partial derivatives for functions that are defined implicitly.
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SOURCE
Calculus : Section 14.5,
p 985 – 991.
PROBLEMS
1. Exercises 14.5 p 991 nos 5; 7; 13; 15; 18; 19; 29; 37; 38; 43; 47; 49; 51; 53
y z ∂u
∂u
∂u
show that x
,
+y
+z
= 3u
2. If u = x3 f
x x
∂x
∂y
∂z
1.5 DIRECTIONAL DERIVATIVES AND THE GRADIENT
NUMBER OF LECTURE PERIODS : 2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. explain the geometrical interpretation of a directional derivative of a function
of two variables.
2. estimate directional derivatives from contour diagrams or tables.
3. find the gradient of a function at a point and use it.
4. write the directional derivative in terms of the gradient and be able to use it
to determine directional derivatives.
5. derive the theorem on the maximum value of the directional derivative at a
point and the direction in which it is reached and be able to use it.
6. apply the connection between the direction in which a function increases most
rapidly and the direction of the gradient.
7. apply the fact that the gradient is perpendicular to the level curves of functions
of two variables.
8. apply the fact that the gradient is orthogonal on the level surfaces of a function
of three variables.
9. determine the equations of normal lines and tangent planes using the gradient.
SOURCE
Calculus : Section 14.6,
p 994 – 1004.
PROBLEMS
1. Exercises 14.6 p 1005 nos 1; 5; 9; 12; 15; 18; 23; 26; 27; 31; 33(b);
38; 42; 44; 47; 50; 56; 60; 61
2. Find the point on the paraboloid z = 9 − 4x2 − y 2 at which the tangent plane
is parallel to the plane z = 4y.
3. Find the point on the surface z = 2x2 −y 2 +3y where the normal line is parallel
to the line that joins the points A(1, 1, 2) and B(9, −4, 3).
13
1.6 MAXIMUM AND MINIMUM VALUES
NUMBER OF LECTURE PERIODS : 2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. explain the definition of an absolute (global) and a local extreme of a twovariable function.
2. determine the critical points of a two-variable function.
3. use the second derivative test to classify the extreme points of a two-variable
function.
4. explain what is meant by a saddle point of a two-variable function.
5. find the extreme values and/or saddle points of a two-variable function.
6. use these techniques in application problems.
7. find the absolute maximum and minimum of a continuous function on a closed
bounded set.
SOURCE
Calculus : Section 14.7,
p 1008 – 1015.
PROBLEMS
1. Exercises 14.7 p 1016 nos 1; 3; 7; 19; 20; 21; 35; 36; 43; 45; 51; 55
14
STUDY THEME 2 : MULTIPLE INTEGRALS
In this theme the definition of the integral of a real-valued function of a real variable is
extended to the integral of a real-valued function of more than one variable.
This theme is divided into 5 lecture units:
2.1 Double integrals and Iterated integrals
2 lectures
2.2 Double integrals in Polar coordinates
1 12 lecture
1
2.3 Triple integrals for functions of three variables
lecture
2
2.4 Triple integrals in Cylindrical and Spherical coordinates 2 lectures
2.5 Transformations
1 lecture
2.1 DOUBLE INTEGRALS AND ITERATED INTEGRALS
NUMBER OF LECTURE PERIODS : 2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. discuss and use the definition of a double integral of a function of two variables
in terms of a Riemann sum.
2. explain that a double integral of a positive function over a bounded region
gives a volume.
3. explain what is meant by a double integral and an iterated integral.
4. apply the theorem stating how double integrals may be calculated using iterated integrals (Fubini’s Theorem).
5. write areas that are described in words or that are given as a sketch, in the
form
{(x, y)| a < x < b, g1 (x) < y < g2 (x)}
(type I) or
{(x, y)| h1 (y) < x < h2 (y), c < y < d}
(type II).
6. sketch regions that are given in one of the above forms.
7. evaluate double integrals on a region by writing the double integral as an
iterated integral (or the sum of iterated integrals).
8. reverse the order of integration of iterated integrals.
9. use the properties of double integrals.
10. use a double integral to calculate volumes and areas.
11. evaluate the average value of a function that is defined on a region.
SOURCE
Calculus : Section 15.1,
p 1037 – 1048.
Calculus : Section 15.2,
p 1051 – 1059.
15
PROBLEMS
1. Exercises 15.1 p 1049 nos 4; 7; 10; 20; 21; 22; 26; 29; 36; 45*; 53
2. Exercises 15.2 p 1059 nos 6; 7(a); 10(a); 20; 23; 26*; 28*; 30(a); 31*;
38*; 39*; 47; 58; 59; 61; 64; 66; 71; 74
* Give only an iterated integral that may be used for the evaluation. (Do not
evaluate.)
Z 2 Z 9−x2
(x2 + y 2 ) dydx
4. Reverse the order of integration in
−3
x+3
5. In 5.1 – 5.3 give an iterated integral that may be used to evaluate the volume
of the solid region:
5.1 The region below the plane z = 4 + x + 2y, inside the cylinder x2 + y 2 = 1 and
above the xy-plane.
5.2 The p
region below the paraboloid z = 6 − x2 − y 2 and above the cone
z = x2 + y 2 .
5.3 The region above the paraboloid z = x2 + y 2 and below the plane z = 2y.
2.2 DOUBLE INTEGRALS IN POLAR COORDINATES
NUMBER OF LECTURE PERIODS : 1 21
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. discuss and use the connection between the Cartesian coordinates and polar
coordinates of a point.
2. describe regions in terms of polar coordinates.
3. explain the justification for the formula for the calculation of a double integral
by means of polar coordinates.
4. evaluate double integrals by means of polar coordinates.
5. evaluate the mass of a lamina with given density.
Remark
Leave moment and the center of mass of a lamina. Thus we only look at the
calculation of the mass in Examples 2 and 3 (p 1071 and 1072).
SOURCE
Calculus : Section 10.3
p 684 – 688.
Calculus : Section 15.3
p 1062 – 1066.
Calculus : Section 15.4
p 1069 – 1072.
16
PROBLEMS
1. Exercises 10.3 p 692 nos 9; 11; 15; 16; 17; 18; 22; 23; 26
2. Exercises 15.3 p 1067 nos 2; 4; 6; 8; 9; 11*; 13*; 30*; 35*; 36*; 39*; 41*; 47*;
49*
3. Exercises 15.4 p 1078 nos 8; 13*; 14* (Find only the mass.)
* Give only an iterated integral in polar coordinates that will be used for the
evaluation. (Do not evaluate.)
4. Describe the following regions R in terms of polar coordinates:
√
i. R is the region bounded by the curves y = |x| and y = 4 − x2 .
ii. R is the region bounded by the circle x2 + y 2 = 4x.
iii. R is the region bounded by the circle x2 + y 2 = 2y.
iv. R is√the region in the first quadrant bounded by the lines y = 0 and
y = 3x and the circle x2 + y 2 = 4.
v. R is the region bounded by the lines y = x, y = 3 and x = 0.
5. Find the mass of the plate bounded by the curves y 2 = 2x and y = x, if the
density at any point on the plate is equal to the distance from the point to the
y axis.
6. In 6.1 and 6.2 give an iterated integral in polar coordinates that may be used
to find the volume of the solid region:
p
6.1 The region above the cone z = x2 + y 2 and below the plane z = 3.
2
2
6.2 The region
pabove the xy-plane, inside the cylinder x + y = 4x and below the
cone z = x2 + y 2 .
Z 1 Z √1−x2
1 dy dx as an iterated integral in polar coordinates.
7. Write
√
0
x−x2
8. Use a double integral to find the area of the region inside the circle r = 4 sin θ
but outside the circle r = 2.
2.3 TRIPLE INTEGRALS FOR FUNCTIONS OF THREE VARIABLES
NUMBER OF LECTURE PERIODS :
1
2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. calculate triple integrals by means of iterated integrals.
2. evaluate of the volume and the mass of a solid using triple integrals.
SOURCE
Calculus : Section 15.6
p 1082 – 1092.
17
PROBLEMS
1. Exercises 15.6 p 1092 nos 7; 17*; 21; 22*; 23*; 33; 35; 51(a); 57*
* Give only an iterated integral. (Do not evaluate.)
2. E is the solid region bounded by the coordinate planes and the plane
2x + y + z = 1. Give an iterated integral that may be used to find the mass of
E if the density at any point of E is equal to
i. the distance from the point to the xy-plane.
ii. the distance from the point to the z-axis.
2.4 TRIPLE INTEGRALS IN CYLINDRICAL AND SPHERICAL
COORDINATES
NUMBER OF LECTURE PERIODS : 2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. discuss and use the connection between the Cartesian coordinates and cylindrical coordinates of a point.
2. discuss and use the connection between the Cartesian coordinates and spherical
coordinates of a point.
3. find the Cartesian-, cylindrical- or spherical coordinate equation of a given
surface.
4. evaluate triple integrals by means of cylindrical coordinates.
5. evaluate triple integrals by means of spherical coordinates.
SOURCE
Calculus : Section 15.7
p 1095 – 1099.
Calculus : Section 15.8
p 1102 – 1105.
PROBLEMS
1. Exercises 15.7 p 1100 nos 5; 6; 7; 8; 9; 12; 17; 23; 27(a)*; 30*; 32*
2. Exercises 15.8 p 1106 nos 5; 7; 8: 9(a); 10(a); 12; 13; 15; 17; 18; 19;
20; 25*; 26*; 32*; 34(a)*; 36*; 43*; 45*
* Give only an iterated integral. (Do not evaluate.)
3. In 3.1 – 3.4 give an iterated integral in either cylindrical or spherical coordinates
that may be used to find the volume of the solid region:
3.1 The p
region inside the sphere x2 + y 2 + z 2 = 4 and above the cone
z = 3(x2 + y 2).
p
3.2 The region below the cone z = 8 − x2 + y 2, inside the cylinder x2 + y 2 = 2x
and above the xy-plane.
18
3.3 The region below the plane z = y and above the paraboloid z = x2 + y 2.
3.4 The region between the spheres x2 + y 2 + z 2 = 1 and x2 + y 2 + z 2 = 4 and
below the top half of the double cone 3z 2 = x2 + y 2.
4. In 4.1 – 4.3 give an iterated integral in either cylindrical or spherical coordinates
that may be used to evaluate the mass of the solid region:
4.1 The region between the spheres x2 + y 2 + z 2 = 4 and x2 + y 2 + z 2 = 16 if the
density at (x, y, z) is equal to the reciprocal of the distance from (x, y, z) to
the origin.
4.2 The region bounded by the cylinder x2 + y 2 = 9 and the planes z = 0 and
z = 5 if the density at any point is equal to the distance from the point to the
axis of the cylinder.
p
4.3 The region bounded by the cone z = 9x2 + 9y 2 and the plane z = 9 if the
density at any point is equal to the distance from the point to the plane z = 9.
Z 2 Z √4−x2 Z 4
5. Write
f (x, y, z) dz dy dx as an iterated integral in both cylin√
− 4−x2
−2
x2 +y 2
drical and spherical coordinates.
2.5 TRANSFORMATIONS
NUMBER OF LECTURE PERIODS : 1
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. explain what is meant by a transformation.
2. calculate the Jacobian of a transformation.
3. use the result of the transformation theorem for double integrals.
SOURCE
Calculus : Section 15.9
p 1109 – 1115.
PROBLEMS
1. Exercises 15.9 p 1116 nos 5; 13; 15; 17; 21; 23(a); 25; 27; 28
ZZ
2. Write
x dA as an iterated integral in polar coordinates if R is the region
R
bounded by the circle x2 + y 2 + 4x − 6y + 4 = 0.
ZZ
3. Evaluate
e(2x−y)/(x+y) dA with R the region bounded by the lines x = 2y,
R
y = 2x, x + y = 1 and x + y = 2.
19
STUDY THEME 3 : CALCULUS OF VECTOR FUNCTIONS
In the first year we integrated real-valued functions on intervals in IR. We now extend this
concept to the line integral of a real-valued or a vector function on a portion of a curve
in IR2 or IR3 . The double integral is also extended to an integral on a surface in IR3 .
This theme is divided into 9 lecture units:
1
3.1 Vector fields
lecture
2
1
3.2 Line integrals
2 2 lectures
3.3 The Fundamental Theorem for Line integrals 1 lecture
3.4 Green’s Theorem
1 lecture
1
3.5 Rotation and Divergence
lecture
2
3.6 Parametric surfaces and their areas
1 12 lectures
3.7 The Surface integral and Flux integral
2 lectures
3.8 Stokes’s Theorem
1 12 lectures
1
lecture
3.9 The Divergence Theorem of Gauss
2
3.1 VECTOR FIELDS
NUMBER OF LECTURE PERIODS :
1
2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. explain what is meant by a vector field.
2. sketch simple vector fields.
3. find the gradient vector field of a function.
4. describe in your own words what is meant by a conservative vector field and a
potential function.
SOURCE
Calculus : Section 16.1
p 1124 – 1129.
PROBLEMS
1. Exercises 16.1 p 1129 nos 4; 7; 13; 19; 20; 26; 28
3.2 LINE INTEGRALS
NUMBER OF LECTURE PERIODS : 2 21
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. parametrize a given curve.
2. reverse the orientation of a curve by changing the parametrization.
3. use the various notations for the line integral along a curve.
4. calculate the line integral of a real-valued function along a curve.
20
5. calculate the line integral of a vector field along a curve.
6. discuss the connection between a line integral along the curve C and along the
curve −C.
R
7. determine the sign of C F· dr from the sketch of a vector field F and a curve
C.
SOURCE
Calculus : Section 16.2
p 1131 – 1141.
ADDITIONAL NOTES
Standard Parametrizations
1. The line segment from point A(a1 , a2 , a3 ) to B(b1 , b2 , b3 ) in IR2 or IR3
r(t) =< a1 , a2 , a3 > +t < b1 − a1 , b2 − a2 , b3 − a3 >,
t ∈ [0, 1]
= [a1 + t(b1 − a1 )]i + [a2 + t(b2 − a2 )]j + [a3 + t(b3 − a3 )]k,
t ∈ [0, 1]
B
A
The direction of parametrization is from A to B.
2. The circle with center (a, b), radius c and oriented counter-clockwise in IR2
r(t) =< a, b > +c < cos t, sin t >,
= (a + c cos t)i + (b + c sin t)j,
t ∈ [0, 2π]
c
t ∈ [0, 2π]
(a, b)
The direction of parametrization is
counter-clockwise with
initial point = (a + c, b) = terminal point.
3. The function y = f(x),
x ∈ [a, b] in IR2
Let x = t. Then
r(t) =< t, f (t) >,
t ∈ [a, b]
= ti + f (t)j,
t ∈ [a, b]
y = f (x)
The direction of parametrization is
from (a, f (a)) to (b, f (b)).
a
b
Similarly, for the function x = g(y), y ∈ [c, d] in IR2 , let y = t. Then
r(t) =< g(t), t) >= g(t)i + tj,
t ∈ [c, d].
The direction of parametrization is from (g(c), c) to (g(d), d).
21
4. Reversal of Parametrization.
If a curve C has a given orientation, then −C is the curve with the same points as
C but with the opposite orientation.
If C is the curve with parametrization
r(t) =< x(t), y(t), z(t) >,
t ∈ [a, b]
then −C is the curve with parametrization
r1 (t) = r(−t) =< x(−t), y(−t), z(−t) >,
t ∈ [−b, −a].
PROBLEMS
1. Parametrize the following directed curves C and give a parametrization for
−C:
i. The line segment from the point (−1, 2, 3) to the point (2, 1, −1).
ii. The arc of a circle centered at origin with radius 2 from (0, −2) to (−2, 0)
oriented counterclockwise.
iii. The arc of a circle centered at the origin with radius 2 from (0, −2) to
(−2, 0) oriented clockwise.
iv. The part of the curve y = x2 − 4 from (3, 5) to (0, −4).
v. The part of the curve x = y 2 from (1, 1) to (4, −2).
2. Exercises 16.2 p 1141 nos 3; 5; 7; 10; 17; 19; 23; 34(a); 36*; 38*; 42
* Find only the mass.
3. Find the mass of a wire in the form of a helix that is parameterized by
r(t) = sin ti − cos tj + 4tk, π ≤ t ≤ 2π if the density at any point on the wire
is equal to
i. the square of the distance from the point to the x-axis.
ii. the square of the distance from the point to the origin.
3.3 THE FUNDAMENTAL THEOREM FOR LINE INTEGRALS
NUMBER OF LECTURE PERIODS : 1
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. use the Fundamental Theorem of line integrals.
R
2. explain what is meant by “ C F·dr is independent of the path”.
R
3. describe and apply two equivalent conditions for “ C F · dr is independent of
the path”.
4. describe under which conditions a vector field will be conservative (that is a
gradient function).
5. find a potential function for a conservative vector field.
SOURCE
Calculus : Section 16.3
p 1144 – 1151.
22
PROBLEMS
1. Exercises 16.3 p 1151 nos 1; 2; 5; 7; 9; 11; 22; 24; 25; 28; 30; 31; 36
3.4 GREEN’S THEOREM
NUMBER OF LECTURE PERIODS : 1
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. apply Green’s Theorem.
2. illustrate the result of Green’s Theorem by means of an example.
3. find areas using Green’s Theorem.
SOURCE
Calculus : Section 16.4
p 1154 – 1159.
PROBLEMS
1. Exercises 16.4 p 1159 nos 2; 8; 9; 11; 17; 22; 23*; 24; 25; 32
* t ∈ [0, 2π] for no 23.
R
2. Evaluate C F·dr where F(x, y) = hy 2 − x2 y, xy 2i and C consists of the circle
√ √
√ √
x2 + y 2 = 4 from (2, 0) to ( 2, 2) and the line segments from ( 2, 2) to
(0, 0) and from (0, 0) to (2, 0). The orientation of C is counter-clockwise.
3.5 ROTATION AND DIVERGENCE
NUMBER OF LECTURE PERIODS :
1
2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. use the definition of rotation in a vector field (curl F).
2. determine whether a vector field F is conservative using curl F.
3. use the definition of divergence in a vector field (div F).
4. discuss the ”value” of curl(∇f ) and div(curl F).
SOURCE
Calculus : Section 16.5
pp 1161 – 1166.
23
PROBLEMS
1. Exercises 16.5 p 1168 nos 2; 3; 5; 8; 15; 16; 20; 23*; 24*
* F is irrotational if curl F = 0 and incompressible if div F=0.
3.6 PARAMETRIC SURFACES AND THEIR AREAS
NUMBER OF LECTURE PERIODS : 2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. find a parametric representation for a given surface.
2. find a equation of the tangent plane to a parametric surface at a point.
3. find the area of a given surface.
Remarks
1. Leave surfaces of revolution on p 1115.
2. If the surface S is given by r(φ, θ) = ha sin φ cos θ, a sin φ sin θ, a cos φi, that
is if S is parameterized using spherical coordinates, then you may assume in
problems that | rφ × rθ | = a2 sin φ. (See Example 10 on p1178.)
SOURCE
Calculus : Section 16.6
p 1170 – 1180.
Calculus : Section 15.5
p 1079 – 1081.
PROBLEMS
1. Exercises 16.6 p 1180 nos 3; 4; 5; 20; 23; 24; 25; 33; 36; 39; 40; 44; 47; 49
2. Exercises 15.5 p 1081 nos 1; 4; 7; 14
3. In 3.1 and 3.2 find the area of the surface:
3.1 The part of the sphere x2 +y 2 +z 2 = 9 that is inside the paraboloid x2 +y 2 = 8z.
3.2 r(u, v) = hu cos v, u sin v, ui,
1 ≤ u ≤ 2,
0 ≤ v ≤ π.
3.7 THE SURFACE INTEGRAL AND FLUX INTEGRAL
NUMBER OF LECTURE PERIODS : 1 21
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. evaluate surface integrals.
2. evaluate the mass of a surface.
24
3. explain what is meant by the orientation of a surface.
4. explain what is meant by a flux integral.
5. evaluate flux integrals.
SOURCE
Calculus : Section 16.7
p 1182 – 1192.
PROBLEMS
1. Exercises 16.7 p 1192 nos 5; 6; 7; 9; 10; 17; 21; 23; 27; 28; 40
ZZ p
2. Evaluate
4x2 + 4y 2 + 1 dS with S the part of the paraboloid z = x2 + y 2
S
below the plane z = y.
3. F(x, y, z) = xi − j + 2x2 k and S is the part of the paraboloid z = x2 + y 2 above
the region in the xy-plane bounded by the parabolas x = 1 −y 2 and x = y 2 −1.
(Normal vectors n are
Z Zdirected downward.) Give an iterated integral that may
be used to evaluate
F·dS.
S
4. Evaluate
ZZ
S
x−y
√
dS with S the surface represented by
2z + 1
r(u, v) = hu + v, u − v, u2 + v 2 i, 0 ≤ u ≤ 1,
ZZ
5. In 5.1 and 5.2 evaluate
F·dS for
0 ≤ v ≤ 2.
S
p
5.1 F(x, y, z) = hx, y, z i and S the part of the cone z = x2 + y 2 beneath the
plane z = 1 with downward orientation.
p
5.2 F(x, y, z) = h−y, x, 3zi and S the hemisphere z = 16 − x2 − y 2 , y ≥ 0
with upward orientation.
4
3.8 STOKES’S THEOREM
NUMBER OF LECTURE PERIODS : 1 21
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. apply the result of Stokes’s rotation Theorem.
2. explain how this theorem is a three-dimensional extension of Green’s Theorem.
SOURCE
Calculus : Section 16.8
p 1195 – 1199.
PROBLEMS
1. Exercises 16.8 p 1199 nos 1; 3; 5; 9; 10; 11; 14; 15(a); 18; 23
25
2. Use Stokes’s theorem to evaluate
ZZ
(curl F)·dS with
S
F(x, y, z) = xz 2 i+x3 j+cos xzk and S the part of the ellipsoid x2 +y 2 +3z 2 = 1
below the xy-plane with outward orientation.
3. Let F(x, y, z) = (x2 + z)i + (y 2 + x)j + (z 2 + y)kpand let C be the intersection of
the sphere xZ2 + y 2 + z 2 = 1 and the cone z = x2 + y 2 . Use Stokes’s theorem
to evaluate
above.
F·dr where C has a counterclockwise orientation as seen from
C
4. Let C be the intersection of the paraboloid z = x2 + y 2 and the plane z = y
and suppose C is oriented
counterclockwise as seen from above. Use Stokes’s
Z
xy dx + x2 dy + z 2 dz.
theorem to evaluate
C
3.9 THE DIVERGENCE THEOREM OF GAUSS
NUMBER OF LECTURE PERIODS :
1
2
LEARNING OUTCOMES
After completion of this lecture unit you should be able to
1. apply the divergence theorem of Gauss.
2. determine from the sketch of a vector field whether a point is a source or a sink
(that is to determine the sign of div F).
SOURCE
Calculus : Section 16.9
p 1201 – 1205.
PROBLEMS
1. Exercises 16.9 p 1206 nos 2; 5; 7; 11; 12; 19; 22; 28; 29
ZZ
2. Use the divergence theorem to evaluate
F·dS with
S
F(x, y, z) = x2 i+ yj−2z 2 k and S the surface of the solid region bounded below
by the xy-plane, above by the plane z = x and on the sides by the parabolic
cylinder y 2 = 2 − x (outward orientation).
26