Next: Mass of a wire, Up: Line integrals. Previous: Line integrals.

## Definition and properties.

Consider the curve defined by the parametrization

If the curve passes through the domain of a function w=f(x,y,z), the values of f on the curve are given by w=f(x(t),y(t),z(t)). If we integrate this function of the variable t with respect to the arc length for t in the interval [a,b], we get the line integral of f on ; it is denoted

We suppose that the curve is smooth, i.e. for every value of t, the functions x, y and z are differentiable and nowhere all equal to 0. Denote .

Example 8.2   Let f(x,y,z)=xy+yz2+x and let be the segment joining the origin to the point A(1,2,1).

A parametrization of is given by . Thus and .

We have:

Example 8.3   We define a function by f(x,y,z)=z+y-z and consider the curve , dispalyed on Fig.  1, with the following parametrization:

Then:

and

Remark 8.4   After replacing ds by , the integral becomes a regular integral of a function of one real variable on an interval.

Proposition 8.5
1.
.
2.
.
3.
If , where each is a smooth curve and the endpoint of coincides with the startpoint of , then

Example 8.6   Let f(x,y,z)=x+y+2z; we compute the integral of f on the broken line displayed on Fig.  2. This curve is the union of two line segments:
• The segment OA, with parametrization: .
• The segment AB, with parametrization: .

We have:

• On OA, , hence . Therefore:

• On AB, , hence . Therefore:

Finally, we have:

Next: Mass of a wire, Up: Line integrals. Previous: Line integrals.
Noah Dana-Picard
2001-05-30
ar