Implicit function of two variables
The
function z = z(х; у)
is called implicit if it is given by
unresolved with respect to
.
We will find
and
.
,
,
and
,
.
Tangent plane and surface normal
Definition 3.
The tangent plane ay a regular
point
is a plane which contains the tangent lines at P
of all curves on (S),
passing through P.
the tangent plane of a surface
Let us suppose that P corresponds to
,
.
As shown
above, the tangent
plane is actually the plane which passes through P and
contains the directions
and
.
This
is, its equation is:
The normal
line at a regular point
to (S) is
the perpendicular through P
to
the tangent plane of (S) at
P. Since
the vectors
and
belong to the tangent plane at P
to (S),
it follows that the normal vector of the surface at P
is
.
The unit surface normal at the point P is:
.
4. Partial derivatives and higher order differentials
Definition of second order partial derivatives
We
can apply the partial derivative multiple times on a scalar function
or vector. For example, given a multivariable function,
,
there are four possible second order partial derivatives:
The
last two partial derivatives,
and
are called “mixed derivatives.” An important theorem of
multi-variable calculus is the mixed
derivative theorem. The proof is
beyond the scope of this course and only the results are stated.
Mixed
derivative Theorem: If a function
is continuous and smooth to second order, then the order of operation
of the partial derivatives does not matter. In other words:
for a continuous and smooth (to second order) function
Example:
For the function
,
show
.
Answer Provided:
We
can see that the order of operation of the partial derivative on a
continous and smooth scalar function does not matter.
Second order differential
expression is called second differential or second order differential for function z.
5. Extrema of functions of two variables
When asked to
find the extrema of a function
we are really being asked to find the maximum and/or minimum values
of
if they exist. Once we find these values we have to determine whether
the values are relative or absolute. To do this we need to know the
difference between a relative maximum (or minimum) and an absolute
maximum (or minimum).
Difference between Relative and Absolute Extrema
The distinction between relative and absolute
boils down to the distinction between an open region, disc, or ball,
and a domain. Let’s say that at the point
the value
is greater than or equal to
for all other values of (x,
y) in an
open region, disc, or ball containing
.
Then
has a relative maximum at
.
Suppose instead that
is greater than or equal to
for all other values of (x,
y) in the
entire domain of f.
Then
has an absolute maximum at
.
Note that a point that satisfies both of these conditions is both a
relative maximum and an absolute maximum at the same time. Sometimes
a relative maximum is called a local maximum.
To find the extrema of a function of two variables
we first find a critical point
.
The point
is a critical point if
and
.
Also, if
or
does ot exist, then
is a critical point. Relative extrema occur
only at critical points.
In the above examples we used algebraic means to determine whether critical points gave maximum or minimum values of the function. With more complicated functions we rely on the Second Partials test.
Second Partials Test
1. For the function find a point such that and .
2. Compute
,
and
.
3. Let
.
Then if
d > 0 and
,
has a relative minimum at
;
d > 0 and
,
has
a relative maximum at
;
d < 0 then
is a saddle point;
d = 0 test is inconclusive.
A convenient mnemonic to remember the formula for d is the determinant
,
where
.