Line Integral

The line integral report is represented in the simulation tree by a node. Right-click this node to access the pop-up menu of the line integral report.

As a statistical report, the line integral largely has the same properties as the others of this type. However its Field Function property selects a scalar or vector field function value upon which the report is based.

A line integral refers to an integral of a function, whether it is a scalar or a vector field, along a curve. One example of using a line integral is calculating the amount of work done on a particle by a force field as it moves along an arbitrary curve C in that vector field.

To compute a line integral, a parameterization of the curve C is used, that is, C is expressed by a vector function r(t), with atb. Here, r(a) and r(b) are the initial and end point of the curve, respectively.

The line integral of a scalar field f along a parametric curve C is:

Figure 1. EQUATION_DISPLAY
Cfds=abf(r(t))|r(t)|tif(r(ti))Δsi
(387)

where Δsi is the distance between points on the curve.

The line integral of a vector field F(r)is:

Figure 2. EQUATION_DISPLAY
CF(r)dr=abF(r(t))r(t)tiF(r(ti))Δsi
(388)

where Δsi is the distance vector between points on the curve.

Reversing the orientation of curve C does not change the value of the line integral of a scalar field but reverses the sign of the line integral of a vector field.

The following limitations exist when you work with vector quantities:

  • The start and end points for the integration cannot be user-specified.
  • The display does not indicate the direction of integration of the line.