Multi-Variable Calculus Assignment
due Friday, April 20, 2001
Terminology (continued)
The term vector field refers to a function y = F(x) where x
and y have the same number of coordinates. A vector field is
essentially the same thing as a transformation of R^{n} except that
when the term transformation is used, both x and y are
regarded as points, but when the term vector field is used,
y is regarded as a vector. Thus, for a vector field one may picture
F(x) as a directed line segment emanating from the point x.
The integral of a vector field F over a piece of curve given
parametrically by
is found by integrating
over the interval of the t-line corresponding to the arc in
question.
The integral of a vector field F over a piece of surface
in R^{3} given parametrically by
is found by integrating
F(s(u, v)) . | ( | {\partial s}/{\partial u} \times {\partial s}/{\partial v} | ) |
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over the appropriate region in the plane of the parameterizing point
(u, v).
Exercises
Let F be the vector field in 3-space that is defined by
F(x, y, z) = (yz, zx, xy) .
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Let C denote the helix in space from the point (2, 0, 0) to
the point (2, 0, 2 pi) that is given by
where the parameter t varies in the interval 0 <= t <=
2 pi. Find the integral of F over C.
Use the parameterization
of the sphere of given radius a to find the integral of F over
the first octant portion of that sphere.
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