The arrows of the Cartesian vector field $\mathbf{F}\=\mathbf{i}\+2 \mathbf{j}$ must be exactly the same as those of the polar version of the field. This equivalence is explored graphically in the Maple solutions below.

Using the results of Example 9.2.8, namely that the Cartesian field $\mathbf{F}\=a \mathbf{i}\+b \mathbf{j}$ becomes

$\left(a \cos \left(\mathrm{\θ}\right)\+b \sin \left(\mathrm{\θ}\right)\right) {\mathit{e}}_{\mathit{r}}\+\left(b \cos \left(\mathrm{\θ}\right)-a \sin \left(\mathrm{\θ}\right)\right) {\mathit{e}}_{\mathbf{\θ}}$$$

in polar coordinates, the polar form of the field F is then

$\left(\cos \left(\mathrm{\θ}\right)\+2 \sin \left(\mathrm{\θ}\right)\right) {\mathit{e}}_{\mathit{r}}\+\left(2 \cos \left(\mathrm{\θ}\right)-\sin \left(\mathrm{\θ}\right)\right) {\mathit{e}}_{\mathbf{\θ}}$$$$$

Although the vector field is entered as < H >, as soon as the Display button is pressed, the name H will be replaced with the sequence of expressions shown in Figure 9.2.12(a).$$

Define and draw the Cartesian vector field:

The$$  tutor can be used to draw the vector field in polar coordinates. If the launched tutor is carefully examined, it will be seen that the field is entered as a free vector whose components are those of the field in polar coordinates. The coordinates are then set to polar, with coordinate-variable names $r$ and $t$ a necessity. If the code at the bottom of the tutor is inspected, it will be seen that the ambient coordinates have to be set to the polar frame and the FlowLine command invoked to draw the field.

Getting the Vector Field tutor to launch on a vector field defined in polar coordinates requires use of the SetCoordinates command to change the ambient coordinates to polar, and the calling of the VectorFieldTutor command with a second argument, a point from which the FlowLine command can integrate a flow line. As yet, no provision has been made for a simple link via the Context Panel.

The VectorFieldTutor command preceded by comment character (#) to prevent it from launching at this point, should the commands in these cells be executed. The tutor that launches with the$$  button will be in the state that the commands in these cells would provide if they were executed.