16.1 Introduction
Often, topics in engineering mathematics can be presented most clearly by means of
diagrams rather than symbols. This topic is perhaps the opposite to this – the pictures used
in illustrating calculus of more than one variable are sometimes not easy to visualise – but
don’t worry, in this case the symbols are much easier to handle.
Previously we have considered mainly functions of asinglevariable, e.g.
y=f(x)In practice, in engineering and science, we usually have to deal with functions ofmany
variables.
For example the pressurePof a perfect gas depends on its volume and temperature:
P=RT
V=P(V,T)Rbeing thegas constant. We want to consider such questions as, how doesPchange if
VandTvary at given rates?
In general we will restrict consideration to functions oftwo variables, because these can
be portrayed graphically. A function of two variables can be represented by a surface in
3-dimensional space. However, most of the ideas we cover are easily extended to functions
of more than two variables, even if the pictures are not.
Exercises on 16.1
- In the relation betweenP,V,andT, how doesP vary with (i)T, (ii)V? How does
Vvary with (iii)T,(iv)P? - IfVincreases by 10% whileTremains constant, by what percentage, approximately,
doesPchange?
Answers
- (i) In proportion; (ii) In inverse proportion; (iii) In proportion; (iv) In inverse
proportion. - Decreases by approximately 10%.
16.2 Function of two variables
Letz=f(x,y)be a function of two variables. If we take a 3-dimensional Cartesian
coordinate system as described in Section 11.4 then any point in 3-dimensional space
can be represented by a point referred to this system,P(x,y,z). For each point(x, y)
of thexyplane, we can calculate the value ofzfromz=f(x,y)and plot this at an
appropriatez-level fixed by thez-axis. In this way, the functionf(x,y)gives asurface
in 3 dimensions, as shown in Figure 16.1.