PARTIAL
DIFFERENTIATION, TANGENT PLANES etc
For a
function $y = f(x)$
of one variable the derivative $f'(a)$ was
defined as the limit
$\displaystyle{\lim_{h\,\to\,0}
\frac{f(a+h) - f(a)}{h}}$
of the Newtonian Quotient. The value of $f'(a)$ was
the
rate of change
of $f(x)$
at $x = a$.
It was interpreted, for instance, as the limit of the slope of
secant lines passing through the point $(a, f(a))$ as
shown to
the right in the
case of a parabola and $a
= -1$.
Knowing a few derivatives, we computed many more using the
Product,
Quotient, and Chain Rules. Via the Point Slope formula the
equation of the tangent line at $(a, f(a))$
became
$y = f(a) +
f'(a)(x-a).$
This provided a Linearization,
$L(x) = f(a)
+ f'(a)(x-a)$,
of $f$ that was
useful
in various estimates. In addition, first and second order derivatives
turned out to
be very helpful with determing graphs and with optimization. |
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So how can we deal with functions $z = f(x, y)$
of two variables? Given all that we've done with surfaces, slicing and
vectors, it should come as no surprise that we employ these ideas to
the full, just one variable at a time. Partial
derivatives, $f_x,
f_y, ... $ are the rates of change of $f$ with
respect to each variable separately:
$\displaystyle{f_x(a, b) =
\frac{\partial f}{\partial x}\Big|_{(a, b)} = \lim_{h \, \to \, 0}
\frac{f(a+h, b) - f(a, b)}{h}, \quad f_y(a, b) =
\frac{\partial f}{\partial y}\Big|_{(a, b)} = \lim_{k \, \to\, 0}
\frac{f(a, b+k) - f(a, b)}{k}. }$
In other words, we differentiate with respect to one variable exactly
as in the one variable case, holding all the other variables fixed. All
the standard techniques of single variable calculus thus apply. And
once we've understood what it all means for functions $f(x, y)$ of
two variables, then functions $f(x,
y, z, ... )$ of more than two variables can be dealt with
in exactly the same way - just don't expect to look at 'pictures' of
things for functions of three or more variables!
Let's see
what it means geometrically:
take vertical slices in
the $x$
and $y$-directions
passing through $P$.
Then the
trace of the surface on these slices are the graphs of the respective
vector
functions
r${}_1(x) = \big\langle x,
b, f(x, b)\big \rangle, $r${}_2(y) = \big\langle a,
y, f(a, y)\big \rangle$,
both shown in orange on the surface. The vector
derivatives
T${}_1 = $r$'_1(a) = \big \langle 1,
0,
f_x(a, b) \big \rangle = $ i $+ f_x(a,
b)$ k,
T${}_2
= $r$'_2(b) = \big \langle 0,
1,
f_y(a, b) \big \rangle = $ j $+ f_y(a, b)$ k,
are tangent vectors at $P$
to these space curves, and the tangent plane at $P$ to the
surface is the one containing these two tangent vectors. After a bit of
calculation using i $\times$ k $= -$ j
etc
we then get
n = T${}_1 \times$ T${}_2 = -f_x(a,b)
$i
$-
f_y(a, b)$ j $+$
k
for the normal to the
surface at $P$.
|
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Ah,
now I see! The tangent line in 2-space
becomes the tangent plane in 3-space and
this is where the normal vector n
comes in: at the point $P
= (a, b, f(a, b))$ on the surface an equation for the
tangent plane is
$\big \langle x - a, y - b,
z - f(a, b) \big \rangle\cdot $n
$ = 0$. i.e.,
$z
= f(a, b) + (x-a)f_x(a, b) + (y-a)f_y(a, b)$.
So near $P$ the Linearization
of $f$ will be
$L(x, y) =
f(a, b) + (x-a) f_x(a, b) + (y-a) f_y(a, b)$,
while the Change in $f$
will be
$\Delta f =
f(x, y) - f(a, b) \approx L(x, y) - f(a,
b) = (x-a) \Delta x + (y-b) \Delta y$.
The pattern is just the same as for one variable, making allowances for
the appearance of the second variable. And the corresponding formulas
for functions of more than two variables will follow exactly this
pattern. Thus we are set to exploit differential calculus in several
variables in a completely analogous way to the one variable case.
What about the Chain Rule? In one variable
it refers to the derivative of the composition of two functions of one
variable and says that if $y
= f(x)$ and $x
= x(t)$, then $y
= f(x(t))$ is a function of $t$ such that
$\displaystyle{\frac{d y
}{d t} = \frac{d f}{d x} \frac{d x}{d t}}$.
In two variables
there is a similar formula, but there are more possibilities. The General Version of the Chain Rule starts with a
function $z = f(x,
y)$ of two variables and says that if $x = x(s, t)$
and $y = y(s, t)$
are themselves functions of two variables $s, t$, then
the composition $z
= f(x(s, t), y(s, t))$ is a function of $s, t$ and
$\displaystyle{\frac{\partial
z}{\partial s} = \frac{\partial f}{\partial x}
\frac{\partial x}{\partial s} + \frac{\partial f}{\partial
y}\frac{\partial y}{\partial s},}
$\displaystyle{\frac{\partial
z}{\partial t} = \frac{\partial f}{\partial
x}\frac{\partial x}{\partial t} + \frac{\partial f}{\partial
y}\frac{\partial y}{\partial t}.}$
The
pattern should be clear. For instance, if $w = f(x, y, z)$
and $x = x(r, s,
t), y = y(r, s, t), z = z(r, s, t)$,
then
$w = f(x(r, s,
t), y(r, s, t), z(r, s, t))$
is a function of $r,
s, t$ while
$\displaystyle{\frac{\partial
w}{\partial r} = \frac{\partial f}{\partial
x}\frac{\partial r}{\partial r} + \frac{\partial f}{\partial
y}\frac{\partial y}{\partial r} + \frac{\partial f}{\partial z}
\frac{\partial z}{\partial r},}$
and so on; it looks like dot products really are involved in some way!
Why might we be interested in such
compositions? Well, earlier we looked at the idea of using both
Cartesian and Polar coordinates in the plane. In this case $x = x(r, \theta) = r \cos
\theta$ and $y
= y(r, \theta) = r \sin \theta$, so under
the change of variable from Cartesian to Polar coordinates, a function $z = f(x, y)$
becomes a function $z
= f(x(r, \theta), y(r, \theta))$ of polar
variables and
$\displaystyle{\frac{\partial
z}{\partial r} =
\frac{\partial f}{\partial x}\frac{\partial x}{\partial r}+
\frac{\partial f}{\partial y} \frac{\partial y}{\partial r}
= \cos \theta \frac{\partial
f}{\partial x} + \sin \theta \frac{\partial f}{ \partial y}, }$
$\displaystyle{\frac{\partial
z}{\partial \theta} =
\frac{\partial f}{\partial x} \frac{\partial x}{\partial \theta} +
\frac{\partial f}{\partial y}\frac{\partial y}{\partial \theta}
= -r \sin \theta
\frac{\partial f}{\partial x} + r \cos \theta \frac{\partial
f}{\partial y}}$.
Conversely, if we start with a function $z = f(r, \theta)$
and change to Cartesian coodinates, what will be the corresponding
formulas for ${\partial
z}\big/{\partial x}$ and ${\partial z}\big/{\partial
y}$? In 3-space
also there is often the need to change from Cartesian to other sets of
coordinates (or the other way round), so here too there will be a need
for the Chain Rule, this time for functions of three variables.
A 'Less General Version' of the Chain Rule would be the case when $ z = f(x, y)$ is a function of two variables, but $x = x(t), y = y(t)$ are just functions of a single variable, say $t$. Then $z = f(x(t), y(t))$ is simply a function of $t$, so that we can ask for $z'(t)$. The Chain Rule is basically the same:
$\displaystyle{z'(t)
= \frac{d z }{dt} =
\frac{\partial f}{\partial x} \frac{d x}{d t} + \frac{\partial
f}{\partial y} \frac{dy}{{dt}} = x'(t) \frac{\partial
f}{\partial x} + y'(t) \frac{\partial f}{\partial y}$.
Again
this looks like a dot product, and the next few sections will show
exactly why it is and why it too occurs very naturally.
An 'Even Less General Version' of the Chain Rule would be a case that really occured in single variable calculus except that now we can be 'cleverer' by regarding an equation $f(x, y) = 0$ as defining $y$ implicitly as a function of $x$. In all previous examples in single variable calculus there was an explicit equation like $x^2 + y^2 = 1$ relating $x$ and $y$, and the single variable Chain Rule was then used to compute $dy \big/ dx$. But now if the 'Less General Version' of the Chain Rule is used with $t = x$, then $f(x, y(x)) = 0$ and
$\displaystyle{\frac{d
f}{dx} = \frac{\partial f}{\partial x}\frac{d x}{ d x} +
\frac{\partial f}{\partial y} \frac{d y }{dx} =
\frac{\partial f}{\partial x} + \frac{\partial f}{\partial
y}\frac{d y}{d x} = f_x + f_y \frac{d y}{d x} =
0 }$, i.e., $\displaystyle{\frac{d y}{d x} = - \frac{f_x}{f_y}}$.