Second order partial derivatives are connected with
'concavity', but the relationship is more subtle than in the one
variable case. If the tangent plane at a point $P(a,\,b,\, f(a,\,b))$
on the graph of $z = f(x,\, y)$ gives the best Linear
Approximation
$$L(x,\,y) \ = \ f(a,\,b) + f_x\Bigl|_{(a,\,b)}
(x-a) + f_y\Bigl|_{(a,\,b)}(y-b) $$ to $f$ near $P$, then we
can expect that some Quadric surface will give the best Quadratic
Approximation to $f$ near $P$. Just as the best Linear
Approximation is the degree 1 Taylor polynomial centered at $(a,\, b)$
for $f$, so this best Quadratic Approximation is the degree 2 Taylor
polynomial. For brevity we'll speak of these degree one and degree two
Taylor polynomials as simply the respective Linear and Quadratic
Approximations to $z = f(x,\,y)$ at $(a,\,b)$.
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The Quadratic Approximation to a function $z = f(x,\,y)$ at $(a,\, b)$ is given by
$$Q(x,\,y) \ = \ f(a,\,b) + f_x\Bigl|_{(a,\,b)}
(x-a) + f_y\Bigl|_{(a,\,b)}(y-b) \hskip1.0in $$
$$ \hskip1.0in + \frac{1}{2} f_{xx}\Bigl|_{(a,\,b)} (x-a)^2+ f_{xy}\Bigl|_{(a,\,b)} (x-a)(y-b) + \frac{1}{2} f_{yy}\Bigl|_{(a,\,b)} (y-b)^2 \,.$$
At $(x,y)=(a,b)$, the value, first derivatives and
second derivatives of $Q(x,y)$ equal the value,
first derivatives and second derivatives of $f(x,y)$. In fact,
$Q(x,y)$ is the unique quadratic polynomial with this property.
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Example: Find the linear and quadratic approximations to $f(x,\,y)
=\displaystyle{\frac{2}{xy}}$ near $(1,2)$.
Solution: We first compute derivatives at $(1,2)$:
$$f(1,\,2) = 1, \quad f_x(1,\,2) = -1, \quad f_y(1,\,2) = -
\frac{1}{2},$$ $$f_{xx}(1,\,2) = 2, \quad f_{xy}(1,\,2) =
\frac{1}{2}, \quad f_{yy}(1,\,2)= \frac{1}{2}\,,$$
so the Linear Approximation to $\displaystyle{ f (x,\,y) = \frac{2}{xy}}$ near
$(1,\,2)$ is given by
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\begin{eqnarray*}\frac{2}{xy} \ \approx \ L(x,\,y)
& = & 1 - (x-1) - \frac{1}{2}(y-2) \cr &= & 3 - x -\frac{1}{2}y\,,
\end{eqnarray*}
while the Quadratic Approximation is
\begin{eqnarray*}\frac{2}{xy} \ \approx \ Q(x,\,y)
& = & 1 - (x-1) - \frac{1}{2}(y-2) \cr && +(x-1)^2 + \frac{1}{2}(x-1)(y-2)
+ \frac{1}{4}(y-2)^2\,,\end{eqnarray*}
which after some algebra becomes
$$ \frac{2}{xy}\ \approx \ 6 - 4x - 2y
+x^2 + \frac{1}{2}xy + \frac{1}{4}y^2\,.$$
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But what does all this mean graphically for a function $z =
f(x,\,y)$? Well, the graph of a linear equation $Ax + by+Cz=D$ is a
plane, while the graph of a quadratic equation is a quadric
surface. So near a point $(a,\, b)$ the Linear Approximation $L(x,\,
y)$ at $(a,\, b)$ approximates the graph of $z = f(x,\,y)$ by a
plane - the Tangent Plane - while the Quadratic Approximation
$Q(x,\, y)$ at $(a,\,b)$ approximates the graph of $z = f(x,\,y)$ by
a quadric surface such as a paraboloid, hyperbolic paraboloid, or a
hyperboloid.
To illustrate these ideas, let's compute some more
Quadratic Approximations. Set
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$$z \ = \ f(x,\, y) \ = \ \sin x \sin y, \quad -\pi \le x,\, y \le \pi;$$
the graph of $f$ is shown to the right in interactive form - try grabbing and moving it!
Then
$$\frac{\partial f}{\partial x} = \cos x \sin y,\quad \frac{\partial f}{\partial y} = \sin x \cos y\,,$$
while
$$\frac{\partial^2 f}{\partial x^2} =-\sin x \sin y\,, \ \ \frac{\partial^2 f}{\partial y^2} =-\sin x \sin y\,.$$
and
$$\frac{\partial^2 f}{\partial x\partial y} \,=\,\cos x \cos y\,, $$
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Thus: when $f(x,\, y) = \sin x \sin y$,
near
$(0,\,0): \quad \displaystyle{ f(x,\, y) \ \ \approx \ \ xy\,, \quad }$ a hyperbolic paraboloid;
- near $\displaystyle{\Big(\frac{\pi}{2},\ \frac{\pi}{2}\Big): \quad
f(x,\, y) \ \ \approx \ \ 1 - \frac{1}{2} \Big(x - \frac{\pi}{2}\Big)^2
- \frac{1}{2} \Big(y - \frac{\pi}{2}\Big)^2\,,\quad }$ a paraboloid
opening downwards;
- near $\displaystyle{\Big(\frac{\pi}{2},\ -\frac{\pi}{2}\Big): \quad
f(x,\, y) \ \ \approx \ \ -1 + \frac{1}{2} \Big(x - \frac{\pi}{2}\Big)^2
+ \frac{1}{2} \Big(y +\frac{\pi}{2}\Big)^2\,,\quad }$ a paraboloid
opening upwards.
Do these seem reasonable given the graph of $f$ shown above? For a
function $y = f(x)$ of one variable, the degree two approximating
polynomial in $x$ would be a parabola, opening up or down, but in two
variables things are more subtle because there are several possible
approximating quadric surfaces.
Question: all the definitions and calculations so far have been for functions $z = f(x,\,y)$ of two variables. Is it clear how to extend the definitions to a function $w = f(x,\, y,\, z)$ of three variables?
Just as in one dimension, we can use higher derivatives to get an
even more accurate approximation, and to express functions as power series.
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Taylor Series in Two Variables: If $f(x,y)$ is an analytic function
of two variables, then
$$f(x,y) = \sum_{n,m=0}^\infty c_{n,m} (x-a)^n (y-b)^m,$$ where
$$c_{n,m}= \frac{1}{n!m!}\frac{\partial^{n+m}f}{\partial x^n\partial
y^m}(a,b).$$
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