The original implementation of the VectorCalculus packages was based on the constructs of the free vector and the VectorField. In Cartesian coordinates, the point is identified with the "free" vector whose tail is at the origin and whose head is at the point. This works fine in Cartesian coordinates where the basis vectors are constant. (It does not work well in nonCartesian coordinates where the basis vectors change from point to point.)
Figure 3 uses the SpaceCurve command to plot the (Cartesian) free vector
 (4.1) 

Figure 3 2D curve defined parametrically by a free vector, graphed with the SpaceCurve command



The SpaceCurve command could equally well have been applied to the free vector , thereby producing the graph of a helix. Thus, the distinction between dimensions has been eliminated.
Updates to the VectorCalculus packages introduced the construct of the PositionVector for defining curves (and surfaces). The curve graphed in Figures 1 and 3 would be represented by
 (4.2) 
and would be graphed with the PlotPositionVector command as per Figure 4.

Figure 4 PositionVector graphed by PlotPositionVector command



An option to the PlotPositionVector command plots the arrows of a vector field, and other options provide for plotting the arrows of the tangent, principal normal, and derivative fields for the position vector. This is illustrated in Figure 5 where tangent vectors are shown in black, and normal vectors are shown in green.

Figure 5 Drawing tangent and principal normal vectors with the PlotPositionVector command



The VectorField command defines a vector at each point of . Figure 6 superimposes ten vectors of the vector field
 (4.3) 
on the curve R in Figure 4.

Figure 6 Arrows of the vector field F evaluated along the curve R



The integral of a vector field produces curves called flow lines along which the vectors of the field are tangent. (For example, the flow lines of an electric field are called field lines.) The FlowLine command in the package will draw the flow line emanating from a given point, as illustrated for the field F and Figure 7.

Figure 7 The flow line emanating from for the field F



It is not possible to ask the FlowLine command for more than one flow line. The alternative is to write the differential equations
 (4.4) 
and to integrate them with the DEplot command from the DEtools package. The result for the initial points
 (4.5) 
appears in Figure 8.

Figure 8 Integration of the field F by means of the DEplot command


