This article explains how to use Mathcad to compute line integrals of vector fields.
A vector field is a function that assigns a vector to each point in n-dimensional space. For example, the following is a 2-dimensional vector field.
One of the most important applications of vector fields is to represent forces, such as electromagnetism or gravitation, that act at every point in space.
You can use Mathcad to visualize a 2 or 3-dimensional vector field
Here's how to create a vector field plot for the vector field F defined above:
Endpoints of the x and y ranges
Number of points in each range
x-coordinates of the array.
Evaluate F on the array.
x-coordinate of F(i, j)
y-coordinate of F(i, j)
The vector field plot of F is shown below.
A line integral of a vector field is an integral defined over a curve. Usually, the curve is parameterized by a function from an interval of real numbers into the space of vectors. For example, the following function r(t) parameterizes an ellipse in the plane.
The plot below shows the ellipse.
The curve traverses the ellipse in the counter-clockwise direction. Note that the initial point, r(0), and the final point, r(2π), of the curve are the same.
A curve with this property is called a closed curve.
The line integral of a vector field F along a parameterized curve r(t), from r(a) to r(b), is defined to be
If F represents a force field, the line integral is the work done when an object moves along the curve from r(a) to r(b).
For a 2-dimensional vector field of the form
you can write the line integral as
The line integral of the vector field F
If you prefer to use the first form of the line integral, you must rewrite F as a function of a vector v, using subscripted variables v0 and v1 in place of the variables x and y.
To enter the subscripts of v, press the [ key.
Then evaluate the line integral with the symbolic equal sign.
A vector field is conservative if it is the gradient of a scalar function
The following is an example of a conservative vector field.
The coordinate functions of G are
To find the potential function
You can check the gradient of
Reset the symbolic values of x and y.
The line integral of a conservative vector field over any curve from r(a) to r(b) is given by the following formula.
The result depends only on the values of
If you evaluate a line integral of a conservative vector field around a closed curve, the result must be 0, because the initial and final points are the same. If you evaluate G around the ellipse from the previous section, the result is 0.
On the other hand, the vector field F described in the previous section cannot be conservative, because the integral around the ellipse is not 0.
Green's theorem relates the line integral of a 2-dimensional vector field around a closed curve C to a double integral over the region D bounded by C.
A common notation for the line integral of a 2-dimensional vector field of the form
around a curve C is
With this notation, Green's theorem says the following:
The theorem was first discovered around 1828 by George Green. Green, a mainly self-taught mathematician, did his early research while working in his father's grain mill in Nottingham, England.
Green's theorem provides an alternate way to evaluate line integrals over closed curves. For example, you can compute the line integral of the vector field F over the ellipse C, described in the section Curves and Line Integrals , by evaluating the right-hand side of the equation above as follows.
The equation for the ellipse is
So you can evaluate the double integral on the right-hand side of Green's Theorem as follows.
The result is the same as for the line integral.
Since the integrand of the above integral is
the value of the double integral is twice the area of the ellipse, so the ellipse has area 6π. More generally, for any closed curve C, the value of the line integral
is twice the area of the region bounded by C. This is the principle underlying the planimeter, a mechanical device for finding the areas of regions with curved boundaries. When you trace the boundary of the region with a planimeter, the device computes a line integral like the one above.
The physicist James Clerk Maxwell, whose pioneering work on electricity and magnetism relied heavily on Green's work on vector calculus, designed an early version of the planimeter.