Partial derivatives
One way to understand a function is to look at
slices. Previously we saw that level curves came from
looking at slices of the form z = c. We could also look
at slices of the from x = x0 or y = y0 . The surface
corresponding to our function when intersected with
such a plane gives a curve in that plane. And we love
curves! In particular, we can find the slope of the tangent lines to these curves. This leads to the idea of
partial derivatives. Given z = f(x, y) we have
∂z
f(x + h, y) − f(x, y)
∂f
(x, y) =
= fx (x, y) = lim
,
h→0
∂x
∂x
h
∂z
f(x, y + h) − f(x, y)
∂f
(x, y) =
= fy (x, y) = lim
.
h→0
∂y
∂y
h
Taking partial derivatives is just like taking derivatives of single variable functions as long as we remember the following rule:
When taking a partial derivative with respect to a
variable, treat all the other variables as constants.
Notationally we use the “∂” symbol (pronounced
“partial”). This acts similarly to “d”, i.e., previously
∂z
we had dy
dx and now we have ∂x . But both notations are asking the same thing, how does one variable change as we perturb the other variable. We will
always use the “d” for functions of a single variable
and always use the “∂” for functions of two or more
variables.
The other notation, fx , is similar to what was previously f 0 , we need the subscript to help specify which
variable we are taking a derivative with respect to.
We can also take higher order partial derivatives,
including mixed partial derivatives. The notation
helps us to keep track of which derivatives we take
and in which order we take them, for example there
are four second order partial derivatives,
∂2 f
∂2 f
∂2 f
∂2 f
=
f
,
=
f
,
=
f
,
= fyy .
xx
xy
yx
∂x2
∂y∂x
∂x∂y
∂y2
Similar notation works for higher order partial
derivatives.
When the function is nice (which will essentially
always be the case in our class) then fxy = fyx . In
other words the order of taking partial derivatives
does not matter. This is true whenever fxy and fyx
are continuous in a neighborhood.
Finally we note that the same notation and ideas
work for functions of three or more variables.
Limits
Often we have to deal with expressions which are
ambiguous, the classic example of this is 0/0. There
is no value for this because any value could work.
So if we cannot know what it is, the we can ask the
question what should it be. This is where limits come
in, namely we look at what is happening to the expression at points nearby and based on what is happening we can indicate what the value should be or
indicate that there is no value that it should be (i.e.,
if the points nearby are giving ambiguous possibilities). Notationally we have
lim
f(x, y) = L
(x,y)→(a,b)
which is read as “the limit as (x, y) approaches (a, b)
of f(x, y) is L”. Intuitively what this means is that as
(x, y) gets close to (a, b) then f(x, y) gets close to L.
When we moved to limits of two variables we
opened up a whole new can of awesome. Previously
we could only talk about approaching a value from
either the left or the right. Now we can approach
from the left, from the top, from a different angle,
along a spiral, along a parabola, along anything we
want. If the limit exists then we will always get the
same answer. However, if we ever get two different answers when we approach in two different ways
then the limit does not exist (DNE for short). As an
example
x2 − y2
= DNE
lim
(x,y)→(0,0) x2 + y2
because if we approach along the x-axis it appears
to approach 1, but if we approach along the y-axis
it appears to approach −1 which are totally not the
same.
A function is continuous when what happens is
what we expected to happen, i.e.,
Continuous ⇐⇒
lim
f(x, y) = f(a, b).
(x,y)→(a,b)
Polynomials are continuous, as are sin, cos, arctan,
e∗ and many others. We can add/subtract/multiply
continuous functions and the result will be a continuous function; we can also divide and the result is
continuous wherever the denominator is not 0. We
can also compose continuous functions (put a function in a function) and get continuous functions.
When dealing with limits we first check to see if it
is continuous and plug in the point to see what we
get; if we get a value then we are done and if we get
6=0
0 then the limit does not exist. If we get 0/0 then
we start looking at different ways to approach the
limiting point; if we get two different values we are
done. If we are still not done then we need to try to
rewrite the function or perhaps bound one part; this
is nontrivial and there is a whole class dedicated to
teaching this (we will mostly avoid this situation!).
For future reference we will need some notation.
Given a set S, an interior point x is a point where we
can put a small ball centered at x completely inside
of S; a boundary point y is a point where every small
ball centered around y contains points both inside
and outside of S. An open set is a set where every
point is an interior point; a closed set is a point which
includes all the boundary points. A bounded set is a
set which can be placed inside of a single large ball.
Differentiability
A function is differentiable if locally the function
is flat, i.e., zoom in close enough and we see what
is approximately a plane. We have the following, a
function is locally linear if
so that the gradient corresponds to “scaling” this vector by f. We note that the gradient vector satisfies
many basic properties including
• ∇(f + g) = ∇f + ∇g, showing ∇ is linear
• ∇(cf) = c∇f, for some constant c
• ∇(fg) = g∇f + f∇g, the product rule.
Finally, we note that if a function is differentiable at
a point then it must also be continuous at that point.
Review problems
1. Find fx (0, 1) and fy (0, 1) for the function
f(x, y) =
∂f
∂f
(x0 , y0 )(x − x0 ) +
(x0 , y0 )(y − y0 )
f(x0 , y0 ) +
∂x
∂y
|
{z
}
f(x, y) = sin x + y2 cos x + y4 arctan(x(y2 − 1))
+ ln(2esin x − 1) sec(xy) tan(y − 1).
= tangent plane / linear approximation
+ g1 (x, y)(x − x0 ) + g2 (x, y)(y − y0 ) .
|
{z
}
= error
where g1 , g2 → 0 as (x, y) → (x0 , y0 ), which is to say
that for points (x, y) near (x0 , y0 ) the error is very
small and becomes vanishingly small as we move
close.
The choice of the partial derivatives is driven by
the need to match up with the cross sections with
the planes x = x0 and y = y0 . It is useful to note
that this notation is consistent with what was done
in single variable calculus.
In general to have a function which is differentiable
we need to have the partial derivatives exist. To guarantee differentiable it suffices to have that the partial
derivatives exist and are continuous.
Since the partial derivatives will play an important
rule in what is to follows we have a special vector
which consists of the partial derivatives, this is the
gradient vector. Given f we denote the gradient of f
by ∇f.
∂f ∂f
If z = f(x, y) then ∇f =
,
∂x ∂y
∂g ∂g ∂g
If w = g(x, y, z) then ∇g =
,
,
∂x ∂y ∂z
The gradient will take over many of the roles that
was previously done by the derivative. For example
the above formulation of differentiability becomes
f(x, y) ≈ f(x0 , y0 ) + ∇f(x0 , y0 ) · hx − x0 , y − y0 i.
In a later chapter we will see it is useful to think of
“∇” as a vector of partial derivatives, i.e.,
∂ ∂ ∂
,
,
”,
“∇ =
∂x ∂y ∂x
2. Find all the second partial derivatives for the
function g(x, y) = xexy .
3. Show that u(x, t) = sin(x + sin t) satisfies the differential equation ut uxx = ux utx .
4. Show h(x, y) = x3 y − xy3 satisfies hxx + hyy = 0.
x sin x
.
x2 + y4
p
x |y| sin x
6. Determine
lim
.
(x,y)→(1,1) x2 + y4
5. Determine
7. Determine
lim
(x,y)→(0,0)
x4 − y4
.
(x,y)→(0,0) x2 + y2
lim
8. Given f(x, y, z) = x2 yz + y2 z3 find ∇f.
9. Find all points (x, y) so that ∇f(x, y) = 0 where
f(x, y) = x2 − 6x + 2y2 − 10y + 2xy + 137.
10. Show that ∇(fn ) = nfn−1 ∇f.