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

t ⟼ r(t)

is found by integrating

F (r(t)) · r′(t)

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

(u, v) ⟼ s(u, v)

is found by integrating

F (s(u, v)) · (\frac{∂s}{∂u} × \frac{∂s}{∂v} )

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) .

1. Let C denote the helix in space from the point (2, 0, 0) to the point (2, 0, 2π) that is given by

x = 2 cos t
y = 2 sin t
z = t

where the parameter t varies in the interval 0 ≤ t ≤ 2π. Find the integral of F over C.

2. Use the parameterization

x = a cos u sin v
y = a sin u sin v
z = a cos v

of the sphere of given radius a to find the integral of F over the first octant portion of that sphere.