Vector Calculus (1016-410-01)

Plan for the day - Tuesday 12/4

1

Functions of several variables

A function of several variables, say n variables, is a mapping that takes a
set of values x1 , x2 , ..., xn (an n-tuple), and returns a real number
w = f (x1 , x2 , ..., xn )
The set of n-tuples for which the function is defined is called the domain.
The values that w or equivalently f takes is called the range. In this
notation, w is the dependent variable, since it depends on the values of
x1 , x2 , ..., xn , the independent variables.
In general, the domain is all of the points except those where the function
doesnt work no logarithms of non-poitive numbers, roots of negative
numbers, or arcsin or arccos of values outside the range [1, 1]. The range
is sometimes all reals, and sometimes requires more care.
Functions of two variables: We denote these in the form z = f (x, y)
typically. Points in the domain can be in the interior of the domain or
the boundary, depending on whether every close point is also a member
of the domain. How close is close? Infinitesimally so. Points not in the
domain are classified as being in the exterior.
An open region is one containing all interior points but not the boundary.
A closed region contains every boundary point.
A domain that fits within a disk of finite radius is called bounded.
A level curve for a function is the set of points where we find f (x, y) = C,
for some constant value C. Imagine a 3-d graph of a function, and we are
looking for isocontours - curves of constant height z.
Interiors, boundaries, and domains work fine for functions of three variables as well, though we talk about balls around a point rather than discs.
Level surfaces are defined by the condition f (x, y, z) = C for some constant C.

Limits and continuity

As with the case of scalar and vector-valued functions of one variable,
we can
p define limits and continuity for functions of many variables. Let
r = (x x0 )2 + (y y0 )2 + ... be the distance from a point (x, y, ...) to

a specified point (x0 , y0 , ...). The function f (x, y) has a limiting value L
at (x0 , y0 ), i.e.,
lim

f (x, y) = L

(x,y)(x0 ,y0 )

If for every value  > 0 there exists a value > 0 such that |f (x, y, ...)
L| <  whenever 0 < r < .
The same definition extends to functions of three variables, four variables,
etc.
All the standard limit laws work: constant multiples, addition, subtraction, products, quotients, powers, roots...
If you get different limiting values on different paths toward the point in
question, the limit does not exist. Consider
f (x, y) =

y
x + 2y
=1+
x+y
x+y

What is lim(x,y)(0,0) f (x, y)?

Along the path y = 0, we find limx0 1 + x0 = 1. Along the path x = 0,
we find limy0 1 + yy = 2. Since the limits differ along different paths, the
limit does not exist.
Continuity works like always. A function is continuous at a point if the
limit exists at a point, the function exists there, and the function equals
the limiting value.
We can compose functions and assume continuity in some cases. If f (x, y)
is continuous at (x0 , y0 ) and g(x) is continuous at the value of f (x0 , y0 ),
then g(f (x0 , y0 ) is continuous.at (x0 , y0 ).

Partial derivatives
For a function of two or more variables f (x, y, ...), we can reduce down to
a function of one variable, say g(x) or h(y) by holding the rest constant.
Define
g(x) = f (x, y0 , ...);

h(y) = f (x0 , y, ....)

The derivative of g(x) or h(y) is known as a partial derivative, since

we only capture part of the variation of the whole function the part
changing in a given direction.

f 
)f (x0 ,y0 )
d
dx
g(x) = limxx0 f (x,y0xx
= limh0 f (x0 +h,y0h)f (x0 ,y0 )
0
x (x0 ,y0 )

f 
(x0 ,y0 )
(x0 ,y0 )
d
h(y) = limyy0 f (x0 ,y)f
= limh0 f (x0 ,y0 +h)f
dx
yy0
h
y 
(x0 ,y0 )

 Alternate notion is

f
x

fx and

f
y

fy .

Taking a partial derivative introduces no new calculus! You just need to

remember to hold all variables not being differentiated against constant.
If youd like, replace the undifferentiated variables by some other language,
and only differentiate in Roman characters. Let f (x, y, z) = sin(x + 2y) +
f
f
xyz + zex + yz . What are f
x , y , and z ?

y
f (, y, ) = sin( + 2y) + y + e +

f (, , z) = sin( + 2) + z + ze +
z
f (x, , ) = sin(x + 2) + x + ex +

:
:
:

f
= cos(x + 2) + + ex = cos(x + 2y) + yz + zex
x
f
1
1
= 2 cos( + 2y) + + = 2 cos(x + 2y) + xz +
y

z
f

y
= + e 2 = xy + ex 2
z
z
z

or just capitalize the variables you are holding constant:

Y
Z
y
f (X, y, Z) = sin(X + 2y) + XyZ + ZeX +
Z
Y
f (X, Y, z) = sin(X + 2Y ) + XY z + zeX +
z
f (x, Y, Z) = sin(x + 2Y ) + xY Z + Zex +

:
:
:

f
= cos(x + 2Y ) + Y Z + Zex
x
f
1
= 2 cos(X + 2y) + XZ +
y
Z
f
Y
= XY + eX 2
z
z

We can differentiate implicitly. If z + xy sin z = 0, then we find z/x

by taking derivatives of x and z with respect to x, but holding all other
variables (in this case, only y)
z
z
y sin z
z
+ xy cos z
+ y sin z = 0
=
x
x
x
1 + xy cos z
We define second partial derivatives as the partial derivative of a partial
derivative. Thus, we have unmixed second derivatives, e.g.,
 
2f
f
=
x2
x x
and mixed second derivatives, e.g.
f
xy

=
=

 
f
?
y x
 
f
?
x y

Luckily, we need not worry about the order in which we take mixed second
partial derivatives, because we get the same answer each way. From one

of the examples above, with f (x, y, z) = sin(x + 2y) + xyz + zex + yz , we

find
 
f
f
=
= 2 sin(x + 2y) + z
xy
y x
 
f
=
= 2 sin(x + 2y) + z
x y
This is known as the Mixed derivative theorem, of Clairauts theorem.
A function of many variables is said to be differentiable if each of the
partial derivatives is continuous throughout a region. Differentiable functions are continuous as well.

The chain rule

The chain rule explains how to differentiate compositions of functions.
d
df dg
[f (g(x))] = f 0 (g(x))g 0 (x) =
dx
dg dx
Quick (sort-of-)proof:
d
[f (g(x))]
dx

=
=

f (g(x + h)) f (g(x)) g(x + h) g(x)

f (g(x + h)) f (g(x))
=
h0
h
g(x + h) g(x)
h
f 0 (g(x))g 0 (x)
lim

What if f is a function of two variables, say x and y, both functions of

some other variable, say t:
d
f (x(t), y(t))
dt

f (x(t + h), y(t + h)) f (x(t), y(t))

h
f (x(t + h), y(t + h)) f (x(t), y(t + h)) f (x(t), y(t + h)) f (x(t), y(t))
= lim
+
h0
h
h
f (x(t + h), y(t + h)) f (x(t), y(t + h)) x(t + h) x(t)
= lim
h0
x(t + h) x(t)
h
f (x(t), y(t + h)) f (x(t), y(t)) y(t + h) y(t)
+
y(t + h) y(t)
h
f dx f dy
=
+
x dt
y dt

lim

h0

In general, to take the derivative of a function f of multiple variables, that

are themselves possibly functions of many variables, etc., some of which
in the end are functions of some variable q, we need to
Trace every composition of functions back to q.
4

 For each path, write the chain rule, with partial derivatives for functions of multiple variables and standard derivatives for functions of
single variables.
Sum all the terms.
We can draw a diagram to summarize the idea:
f (x, y)
f /x
x
@
dx/dt @
R

@ f /y
@
R y
@
t

dy/dt

We can expand to three variables as well, with an example: f (x, y, z) =

x + y 2 + z 3 + xyz, where x(t) = t, y(t) = sin t, z(t) = et . As a picture, we
find:

which results in the derivative expression

df
f dx f dy f dz
=
+
+
dt
x dt
y dt
z dt
For our particular example, we find
df
dt

(1 + yz)(1) + (2y + xz)(cos t) + (3z 2 + xy)(et )

(1 + sin tet ) + (2 sin t + tet ) cos t + (3e2t + t sin t)(et )

1 + 2 sin t cos t + et (sin t + t cos t + t sin t + 3e2t )

A quick check shows that f (t) = t + sin2 t + e3t + t sin tet , with derivative
df
= 1 + 2 sin t cos t + 3e3t + et (sin t + t cos t + t sin t)
dt
which is equivalent.

 The picture is more simple if we have f (x), involving a single variable

x(r, s) which is dependent on two variables r and s:

f
df x f
df x
=
;
=
r
dx r s
dx s
There is a subtlety here. The standard, or total derivative df /dt represents
the total change in f as t changes, which allows for the fact that the
other parameters may change with t as well.The partial derivative, f /t,
represents the change in f as t changes, assuming x, y, z, i.e. all of the
other parameters, are all held constant.
Example: consider f (x, y, z) = x3 + xy 2 + xz 2 . We find by holding y and
z constant that
f
= 3x2 + y 2 + z 2
x
p
Defining r = x2 + y 2 + z 2 , we find f (x, r) = xr2 and
f
= r2
x
In the first example, we held y and z constant, whereas in the latter case
we held r constant and treated it like one of the fundamental variables,
which is not the same thing. To confirm that math does really work, note
that if r = r(x, y, z), we find
df
dx

=
=

f
f r
+
x
r x
r2 + 2rx(x/r) = r2 + 2x2 = 3x2 + y 2 + z 2

where we use the derivative r/x = x/r.

Implicit differentiation: Consider a function F (x, y) = 0 that defines y
implicitly in terms of x, so that we can view it as F (x, y(x)) = 0. Partial
differentiating with respect to x, we find
dF
dx
F dy

y dx
dy
dx
0=

F
F dy
+
x
y dx
F
=
x
F/x
=
F/y
=

For three or more variables, consider F (x, y, z) = 0. If we want to take

a partial derivative, say z/x, then we are assuming that we implicitly
have z(x), and thus F (x, y, z(x)). Taking a derivative, we find
dF
dx
F z

z x
z
x
0=

F
F z
+
x
z x
F
=
x
F/x
=
F/z
=

and this formula generalize to all other examples.

Directional derivatives and gradient vectors.

Consider a surface z = f (x, y). A curve drawn along the surface in time
will have the form (x(t), y(t), f (x(t), y(t))). The height will change in time
according to the chain rule:
dz
dt

f dx f dy
+
x dt
y dt
 


f
dx
dy
f
+

+

=
x
y
dt
dt


f
f
=
+
~v
x
y
=

Motivated by this discussion, we can define the gradient vector f ,

given in two or three dimensions by

 

f
f
f f
f =
+
=
,
x
y
x y

 

f
f f f
f
f
=
+
+
k =
,
,
x
y
z
x y z
The gradient vector takes a scalar function f and defines a vector by
defining the component in a given coordinate direction as the derivative
with respect to that coordinate.
Above, we derived the rule for changes in the values of a function along a
curve:
df
= (f ) ~v
dt
If you remember that the rate of change of arclength involves the velocity,
ds
v |, we see that
dt = |~
df
df /dt
1
~v
=
=
[(f ) ~v ] = f
= f v
ds
ds/dt
|~v |
|~v |
7

 The rate of change of a function in the direction of a unit vector ~u is known

as a directional derivative, and is denoted D~u f . When evaluated at a
point P0 , we write it D~u f |P0 .
Since |df /ds| = |f | cos , we can interpret its meaning, the gradient vector points in the direction in which the function increases most rapidly at
a point. It decreases at the same rate in the direction opposite.
At an angle of 90 to the gradient vector, we find df /ds, so that the value of
the function is instantaneously constant. This is the direction of the level
curves, or isocontours. Level curves are perpendicular to the direction of
maximum increase/decrease. Remember that these point in two, exactly
opposite, directions!
Gradients are derivatives, and follow some familiar rules.
(f g) = f g

(cf ) = cf
 
gf f g
f
=

g
g2

(f g) = f g + gf

In three dimensions, the gradient is still the direction of maximum increase, and perpendicular directions still give no instantaneous change.
Since we find a plane perpendicular to a given vector, not a line, the
perpendicular directions define the level surface at the point.