Line Integrals of Vector Fields

Using Maple and the vec_calc Package

This worksheet shows how to compute line integrals of vector fields using Maple and the vec_calc package. As examples we compute

* The Work Done by an Electric Field

* The Circulation of a Fluid

To start the vec_calc package, execute the following commands:

> restart;

> libname:=libname,"C:/mylib/vector";

> with(vec_calc): vc_aliases:

Warning, the protected names norm and trace have been redefined and unprotected

Warning, the name changecoords has been redefined

Package:   vec_calc   Version 4.3

For all HELP, execute: ?vec_calc

To use aliases, execute:   vc_aliases;

>

The Work Done by an Electric Field

The electric field of a line of charge with charge density is where is a constant. A particle of charge moves through this field along the line segment from to . We want to find the work done by this electric field on the charged particle.

We input the field as the function:

> E:=MF([x,y,z],[2*k*lambda*x/sqrt(x^2+y^2),2*k*lambda*y/sqrt(x^2+y^2),0]);

We input the line segment as

> r:=MF(t, evall([1,2,3]+t*([3,2,1]-[1,2,3])));

where . So the velocity is

> v:=D(r);

On the line segment, the electric field is

> Er:=E(op(r(t)));

The work is defined as = . Consequently,

> qEv:=q*Er &. v(t);

> Work:=Int(qEv,t=0..1); Work:=value(%);

Another way to compute this line integral is to use the Line_int_vector command (or its alias Liv ) from the vec_calc package which works directly with the parametrized curve and the vector field:

> Liv(E,r,t=0..1); Work:=q*value(%);

There is a second way to compute this work. The electric field is conservative because it is curl-free:

> CURL(E);

So the electric field has a scalar potential, .

> POT(E,'phi');

> phi(x,y,z);

By the Fundamental Theorem of Calculus, the work is the change in the potential energy:

> Work:=q*phi(3,2,1)-q*phi(1,2,3);

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The Circulation of a Fluid

The velocity of the water in a sink going down the drain is . We want to find the circulation of the fluid counterclockwise around the circle with .

We enter the fluid velocity as a function:

> V:=MF([x,y,z],[(-y-z)/(x^2+y^2), (x-z)/(x^2+y^2), z^2*(-x-y)/((x^2+y^2)^2)]);

The circle may be parametrized as

> r:=MF(t,[2*cos(t),2*sin(t),3]);

Its tangent vector (velocity) is:

> v:=D(r);

Caution: Don't confuse the velocity (tangent vector) of the curve with the velocity of the fluid .

On the curve, the fluid velocity becomes:

> V(op(r(t))); Vr:=simplify(%);

The circulation is defined as = . Consequently,

> Vr_v:=Vr &. v(t);

> Circ:=Int(Vr_v,t=0..2*Pi); Circ:=value(%);

Another way to compute this line integral is to use the Line_int_vector command (or its alias Liv ) from the vec_calc package which works directly with the parametrized curve and the vector field:

> Liv(V,r,t=0..2*Pi); Circ:=value(%);

>