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)) . 
({\partial s}/{\partial u} \times {\partial s}/{\partial 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, 2 pi) that is given by

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

    where the parameter t varies in the interval 0 <= t <= 2 pi. 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.

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