Moving swiftly along from our discussion of vector fields, we shall go straight into the method of finding the lines of force for a given vector field.
In a physics perspective, suppose we know the strength of a point in a magnetic field. We then want to deduce from this information the lines of force which are generated by the magnetic field.
VECTOR INTEGRAL CALCULUS >
FINDING LINES OF FORCE.1
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FINDING LINES OF FORCE.2
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VIDEO LECTURE
part 1
MAIN CONCEPTS
To find the lines of force of a given vector field, we use the equation  Get the Printer Friendly VersionCOMMENTS
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CONCEPT
The idea behind this method is that in a given vector field
 in three-space, we construct a curve by the knowledge we have of its tangents at each point.
We start by defining the curve C with the parametric equations
parametrized in terms of arc length along the curve. Do not confuse this with the vector field. This parametric equations define the curve C we are to find. We can then define the position vector to be
We do this so we can proceed with our analysis of vector differentiation.
We know from differential calculus that
and by definition
We let
These equations, while may be useful, is a little messy. We can put them in a more simple form by rearranging the
assuming A point to note, All information presentated, less questions and exercises, is original content of Donny, with slight references to various books.
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is a unit vector to the curve. But our vector field 
or by our parametric equations, 
is a tangent to the curve at 
and thus parallel to 
. So, for any given s, there is a scalar 
such that 
and by equating the i, j, and k, componenets, we have 
on one side to give 
are non-zero. This is our final form of the differential equation we will use to find the lines of force. 
need NOT be only in terms of 
respectively. For example, although the i componenet of the vector field is labelled as 
, it can be a function of 
.
