Differentiability of a function of three variables
All of the preceding results for differentiability of functions of two variables can be generalized to functions of three variables. First, the definition:
Definition
A function
is differentiable at a point
if for all points
in a
disk around
we can write
where the error term
E satisfies
If a function of three variables is differentiable at a point
then it is continuous there. Furthermore, continuity of first partial derivatives at that point guarantees differentiability.
Key concepts
The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
Tangent planes can be used to approximate values of functions near known values.
A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
The total differential can be used to approximate the change in a function
at the point
for given values of
and
Key equations
Tangent plane
Linear approximation
Total differential
Differentiability (two variables)
where the error term
satisfies
Differentiability (three variables)
where the error term
satisfies
For the following exercises, find a unit normal vector to the surface at the indicated point.
For the following exercises, find parametric equations for the normal line to the surface at the indicated point. (Recall that to find the equation of a line in space, you need a point on the line,
and a vector
that is parallel to the line. Then the equation of the line is
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