Find the equation of the plane through that is normal
to the vector .
- Any plane having the stated normal must have an equation of the form
, and the value of the constant is determined
by invoking the stated point. Thus, .
Give a parametric representation for the line through the point
parallel to the line having symmetric equation
- The vector is parallel to the given
line and, therefore, also to the parallel line. Hence, a parametric
representation is give by
When find the following first and second
order partial derivatives:
The position at time of a point moving in is
Find the (scalar) curvature of its path.
- The most convenient formula for the scalar curvature in this case
is
where and .
Evaluate the double integral
where is the triangular region with vertices at the origin, the
point , and the point .
- The triangle lies in the first quadrant, and its sides are the coordinate
axes and the line from to , which has the equation
Thus, the region has the analytic description
This description of permits expression of the double integral as an
iterated integral with evaluation as follows: