John E. Gilbert, Heather Van Ligten, and Benni Goetz
Calculus for functions
z
=
f
(
x, y
)
of two (or more) variables relies heavily on what you already know about
the calculus of functions
y
=
f
(
y
)
of one variable. But a few preliminary ideas about vectors and various
coordinate systems in two and three dimensions need to be developed before those single variable ideas can be
exploited. Once that’s done, then we can get back to calculus!
Let’s start with vectors in the plane  you may have met them already, and you’ll certainly make good use of
them in a number of your other courses!
What is a vector:
A quantity, be it geometric, scientiﬁc or whatever, is a
vector
so long as it has both a
magnitude
(or
length
)
and a
direction
. For instance,
velocity
can be described by a vector because it has a magnitude, namely
speed
,
as well as a direction: the wind blows at a speed of 5 mph from the northwest, Joe heads due north at 75
mph in his car, and so on.
Displacements
provide a diﬀerent type of example:
let’s look at where Bob lives in relation to Alice. His
house is at point
B
which is
223
ft.,
18
◦
ENE, from
Alice’s house at point
A
. If we represent this as an
arrow from
A
to
B
, it determines a
displacement
vector
→
AB
with
magnitude
the distance from
A
to
B
,
and
direction
the direction from
A
to
B
. It’s natural to
represent this vector by an arrow with
A
the
tail
and
B
the
head
.
A
B
N
E
100
ft
In general, we’ll usually label vectors by single boldfaced letters like
a
,
v
, ... , and so on. Beware: physicists
and engineers sometimes use diﬀerent notation. The
length
of a vector
v
is denoted by
∥
v
∥
or by

v

; this
length is a positive number except for the
zero
vector
0
which has length
0
. Of course, not all quantities can
be represented as vectors: for instance, mass, temperature and distance have magnitude, but no direction.
Such directionless quantities are real numbers