Contents
What do you need to know about line integrals?
For a scalar line integral, we let C be a smooth curve in a plane or in space and let be a function with a domain that includes C. We chop the curve into small pieces.
When does a line integral on a smooth curve always exist?
If is a continuous function on a smooth curve C, then always exists. Since is defined as a limit of Riemann sums, the continuity of is enough to guarantee the existence of the limit, just as the integral exists if g is continuous over Before looking at how to compute a line integral, we need to examine the geometry captured by these integrals.
Why is it difficult to calculate a scalar line integral?
Note that in a scalar line integral, the integration is done with respect to arc length s, which can make a scalar line integral difficult to calculate. To make the calculations easier, we can translate to an integral with a variable of integration that is t.
Which is the second kind of line integral?
Line Integrals – Part II – In this section we will continue looking at line integrals and define the second kind of line integral we’ll be looking at : line integrals with respect to x x, y y, and/or z z. We also introduce an alternate form of notation for this kind of line integral that will be useful on occasion.
How to write a description of a scalar line integral?
For a formal description of a scalar line integral, let be a smooth curve in space given by the parameterization Let be a function with a domain that includes curve To define the line integral of the function over we begin as most definitions of an integral begin: we chop the curve into small pieces.
Is the line integral independent of the parameterization?
Changing the parameterization did not change the value of the line integral. Scalar line integrals are independent of parameterization, as long as the curve is traversed exactly once by the parameterization.