`> `
**with(plots):with(linalg):with(plottools):**

**The Polya vector field F of a function f(z)=f(x+iy)=u(x,y)+i*v(x,y) is defined to be the set of vectors**

**F=[F_x,F_y]=[u(x,y),-v(x,y)] in the plane. Here F_x, and F_y, denote the x and y components of**

**the vector field. (They **
__do not__** represent partial derivatives.) In the theory of electricity**

**and magnetism, one can think of the Polya vector field as either **

**an electrostatic field F or a magnetic field B.**

**Below is the Polya vector field corresponding to f(z)=1/z. Note the behavior of the field at the**

**singularity z=0. In physics, the singularity at z=0 represents a point charge.**

`> `
**f:=z->z^2;**

`> `
**u:=unapply(evalc(Re(f(x+I*y))),(x,y));**

`> `
**v:=unapply(evalc(Im(f(x+I*y))),(x,y));**

`> `
**F:=[u(x,y),-v(x,y)];**

`> `
**g1:=fieldplot(F,x=-1..1,y=-1..1,arrows=thick,color=red,grid=[5,5]):display(g1);**

`> `
**g := transform((x,y) -> [2*x,y]):
display({g(g1)});**

**1. Where are the arrows largest in length? Where are they smallest in length. Through how many **

**revolutions do the arrows rotate as one travels a path around the origin?**

**2. Repeat the above question using f(z)=z^2 and then f(z)=z^3.**

**The DEL operator is defined to be the partial derivative operator**

** + **
** + **
**. **

**The **
__divergence__** of a vector field F = [F_x, F_y] at a point is defined to be the **

__scalar__** quantity obtained by taking the **
__dot product__** of DEL with the value of the **

**vector F at that point. In other words, **

**div(F) = **
** + **
** **

**Intuitively, the divergence measures net inflow/outflow of **

**the vector field at that point. **

**3. Show that if f is differentiable at a point z=z0, then the divergence of the corresponding **

**Polya vector field is **
**zero there.**
** (Hint: Use the dot product, together with the Cauchy-Riemann equations.)**

**Below Maple calculates the divergence for the Polya field above.**

`> `
**diverge([op(F),0], [x,y,z]);**

**The **
__curl__** of a vector field F = [F_x, F_y] at a point is defined to be the**

__vector__** quantity obtained by taking **
__cross product__** of the DEL operator**

**with the vector F_z*i + F_y*j + F_z*k at that point.**

**Intuitively, the curl measures rotational tendency of the vector field at that**

**point. Think of placing a paddle wheel placed at a point with its blade perpendicular**

**to the plane. If the paddle wheel rotates, the curl is different from zero.**

**4. Show that if f is differentiable at a point z=z0, then the curl of the corresponding **

**Polya vector field is **
**zero there.**
** (Hint: Use the cross product, together with the Cauchy-Riemann equations.)**

**Below Maple calculates the curl for the Polya field above.**

`> `
**curl([op(F),0], [x,y,z]);**

**The above results have a significant connection to physics, as the curl and divergence are **

**essential for formulating the differential form of Maxwell's equations, the four primary **

**equations that govern the theory electricity and magnetism. **