Multivariable Calculus (Math 214) Assignment

due November 3, 1999

1. Definitions and Terminology

If C : r(t) , a <= t <= b , is a parameterized path, there are two kinds of integrals that may be taken ``over'' C.

Integral of a scalar-valued function.

If f is a function that is continuous at every point of C, then the integral of f over C is INT[_{C} {f}] = INT[_{a}^{b} f(r(t)) ||r'(t)|| d t] .

If f is regarded as a 1-dimensional mass density along C, then INT[_{C} {f}] is the mass of C. If f is the constant 1, then INT[_{C} {1}] is the arc length of C.

Integral of a vector field.

(This use of the term vector field refers to the case where a function y = F(x) is characterized by x and y both having the same number n of coordinates. One may think of such an F as either a ``motion'' of R^{n} or as the attachment of an ``arrow'' to each point of R^{n}.) If F is continuous at every point of C, then the integral of F over C (a scalar) is INT[_{C} {F}] = INT[_{a}^{b} F(r(t)) . r'(t) d t] .

If F is a force field, then INT[_{C} {F}] is the work contributed by F to a point particle whose motion along C is described by r.

2. Reading Assignment

Read the first two sections of Chapter 13 in the text as if they were not going to be explained carefully in class.

3. Exercises

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.

1. Find the arc length of C.

2. Let F be the vector field in 3-space that is defined by

   F(x, y, z) = (yz, zx, xy) .

   Find the integral of F over C.

3. Let f be the function defined by f(x, y, z) = z. Find the integral of f over C.