MATH 2019: ENGINEERING MATHEMATICS 2E

SEMESTER 1, 2015

Course information:Lecture Stream 1 Instructor: Dr. Shane Keating

Red Centre 2081, Email: s.keating@unsw.edu.au

Lecture Stream 2 Instructor: Prof. John RobertsRed Centre 3065, Email: jag.roberts@unsw.edu.au

In this course you will learn and master some of the mathematical tools you will need inyour scientific and engineering careers. As with any new tool, practice makes perfect,so we will work though a lot of examples and practice problems during lectures. Inaddition, you will participate in small-group tutorials, where you will work through theproblems in the MATH2019 tutorial problem set.

We will be meeting for 5 lectures per week and you will also have 1 tutorial per week.Lectures run from Weeks 1-12 and the tutorials from Week 2-13. There are no tutorials inWeek 1. The tutorial problems are available for download on the Moodle page.

Every now and then the lecture will be a problem class where we will go through aselection of problems from past exam papers.

You need to hold a Math1231 pass or better to enrol in Math2019.

The lecture notes are available for download from Moodle. You must bring a print-out of the lecture material to each of the lectures. On Moodle you will also findthe course outline as well as links to extra learning materials.

You do not need to purchase a text book for this course. Your printed lecture notes willbe sufficient. Additional course notes are available on the Moodle course page.

There is no MAPLE in Math2019.

Course Assessment: There will be two short tests (valued at 20% each) in the Week6 and 10 tutorials. The final exam is worth 60%. Details are in the course outline.

1

LECTURE 1

PARTIAL DIFFERENTIATION

The definition of partial differentiation:Suppose z = f(x, y). Define the partial derivatives of f with respect to x and y as

f

x= lim

x0f(x+ x, y) f(x, y)

x,

f

y= lim

y0f(x, y + y) f(x, y)

y

Notation

f

x= fx = zx,

f

y= fy = zy

In your previous studies the focus was on functions of a single variable y = f(x) and their

rates of changedy

dx. It is however quite rare for a quantity of interest to depend on only

one variable and in complicated physical systems it may be the case that the variable youare concerned with may depend upon dozens of other variables. Partial differentiation isthe extension of our usual calculus to functions of several variables.Given a function of two variables z = f(x, y) we denote the rates of change in the x

and y directions asz

xand

z

yor simply as zx and zy. The formal definitions of these

derivatives are presented above however in reality we only need to remember a few thingsto differentiate partially:

Simple rules for differentiation

y y

xn nxn1

eax aeax

sin(ax) a cos(ax)cos(ax) a sin(ax)ln(ax)

1

xsinh(ax) a cosh(ax)cosh(ax) a sinh(ax)

General rules for differentiation

(uv) = uv + vu Product Rule(uv

)=vu uv

v2Quotient Rule

The only new issue that needs to be kept in mind is that the partial derivative treats allother variables exactly as if they were constant.

2

Example 1 Findz

xand

z

yif z = x2 + y5 + 7.

F zx

= 2x,z

y= 5y4 F

Example 2 Suppose that z = f(x, y) = x3y5 + 3x 8y + 2. Find the function valueand the rate of change of f in the x direction at the point (1, 2).

F f(1, 2) = 21, zx

(1, 2) = 99 F

3

Example 3 Findw

uand

w

vif w = u3v4 + sinh(v9) .

F wu

= 3u2v4,w

v= 4u3v3 + 9v8 cosh(v9) F

Example 4 Findz

xand

z

yif z =

e7y

x3 + 1.

F zx

=3e7yx2(x3 + 1)2

,z

y=

7e7y

x3 + 1F

4

Plotting in Space

Before examining partial derivatives from a geometrical point of view let us consider theissue of sketching in higher dimensions.

Example 5 Plot the points

124

and 124

in R3.

You will observe that plotting in R3 is somewhat problematic as you are trying to squeezethree dimensions onto a two dimensional page. It gets worse!

Example 6 Plot the point

1247

in R4.

5

You will recall that the graph of y = f(x) is generally a curve in R2......lines, parabolas,hyperbolas etc. The graph of z = f(x, y) is always a surface in R3.

Example 7 Sketch each of the following surfaces in space:

a) 3x+ 4y + 6z = 12

b) x2 + y2 + z2 = 25

c) x2 + y2 = 9

d) z = 3x2 + y2

e) z = x2 + y2

6

Question How about x = y2 + z2?

SUMMARY

a) ax+ by + cz = d is a plane.

b) x2 + y2 + z2 = r2 is a sphere centred on the origin with radius r.

c) If a variable is absent then the function is independent of this variable. Extrude thetwo dimensional curve into the missing direction.

d) z = x2 + y2 is a cone with semi-vertical angle tan1(

1

).

e) z = (x2 + y2) is a paraboloid of revolution.

7

GEOMETRICAL INTERPRETATION OF THE PARTIAL DERIVATIVES

8

There is of course no reason why we must restrict ourselves to two independent variables!!

Example 8 If f(x1, x2, x3, x4, x5, x6) = x31x

43 +

x5sin(x6)

+ ln(x4) sinh(x2)ex1

findf

x4.

F 1x4

F

9

As in single variable calculus we make extensive use of second and higher order deriva-tives. However with partial differentiation we have many more options!

Example 9 If z = f(x, y) = x2 sin(y) + x3y + y5 find

z

x

z

y

2z

xy

2z

yx

2z

x2

2z

y2

You will observe in the above example that2z

xy=2z

yxThis is true for most reasonably

well behaved functions. Note however that in general2z

x26=

2z

y2.

Note also that2z

xyis most definitely not equal to (

z

x)(z

y).

1You can now do Qs 1 to 4

10