Vector Fields with no Flow and no Flux

[Graphics:Images/noflownoflux_gr_1.gif]

Example 1:  [Graphics:Images/noflownoflux_gr_2.gif]

[Graphics:Images/noflownoflux_gr_3.gif]
[Graphics:Images/noflownoflux_gr_4.gif]

Here is a picture of F (the arrows are scaled, to make the picture clearer).

[Graphics:Images/noflownoflux_gr_5.gif]

[Graphics:Images/noflownoflux_gr_6.gif]

[Graphics:Images/noflownoflux_gr_7.gif]

[Graphics:Images/noflownoflux_gr_8.gif]
[Graphics:Images/noflownoflux_gr_9.gif]

[Graphics:Images/noflownoflux_gr_10.gif]

Now look at the level curves of the two components of F1. These components are  P1=[Graphics:Images/noflownoflux_gr_11.gif],  Q1=-2xy.
Their level curves are supposed to be perpendicular to each other. We first show P1, then Q1, then both together.

[Graphics:Images/noflownoflux_gr_12.gif]
[Graphics:Images/noflownoflux_gr_13.gif]
[Graphics:Images/noflownoflux_gr_14.gif]

[Graphics:Images/noflownoflux_gr_15.gif]

[Graphics:Images/noflownoflux_gr_16.gif]

[Graphics:Images/noflownoflux_gr_17.gif]

You can see the curves are perpendicular.

[Graphics:Images/noflownoflux_gr_18.gif]

[Graphics:Images/noflownoflux_gr_19.gif]
[Graphics:Images/noflownoflux_gr_20.gif]

[Graphics:Images/noflownoflux_gr_21.gif]

[Graphics:Images/noflownoflux_gr_22.gif]

[Graphics:Images/noflownoflux_gr_23.gif]

Example 2:   Start with f(x)=Cos[x]. It turns out that  Cos[x-iy]=Cos[x]Cosh[y]+iSin[x]Sinh[y],
so       P3=Cos[x]Cosh[y],      Q3=Sin[x]Sinh[y], and   F3=(Cos[x]Cosh[y],Sin[x]Sinh[y]).
Here is a picture of F3:

[Graphics:Images/noflownoflux_gr_24.gif]
[Graphics:Images/noflownoflux_gr_25.gif]

[Graphics:Images/noflownoflux_gr_26.gif]

Now F3=∇(Sin[x]Cosh[y]), and  the  functions
P4=Sin[x]Cosh[y]    Q4=-Cos[x]Sinh[y]
satisfy the Cauchy-Riemann equations, so, as with F1, the level curves of Q4 are the flow lines of F3.

[Graphics:Images/noflownoflux_gr_27.gif]
[Graphics:Images/noflownoflux_gr_28.gif]
[Graphics:Images/noflownoflux_gr_29.gif]

[Graphics:Images/noflownoflux_gr_30.gif]

[Graphics:Images/noflownoflux_gr_31.gif]

[Graphics:Images/noflownoflux_gr_32.gif]


Converted by Mathematica      April 4, 2000