Mathematical Tools for Physics

8—Multivariable Calculus 219

The symbol∇is commonly used for the gradient operator. This vector operator will appear in several other
places, the curl of a vector field will be the one you see most often.

∇=xˆ

∂

∂x

+yˆ

∂

∂y

+ˆz

∂

∂z

(11)

From Eq. ( 9 ) you have

gradf=∇f (12)

8.7 Plane Polar Coordinates

When doing integrals in the plane there are many coordinate systems to choose from, but rectangular and polar
coordinates are the most common. You can find the element of area with a simple sketch: The lines (or curves)
of constant coordinate enclose an area that is, for small enough increments in the coordinates, a rectangle. Then
you just multiply the sides. In one case∆x.∆yand in the other case∆r.r∆θ.

Vibrating Drumhead
A circular drumhead can vibrate in many complicated ways. The simplest and lowest frequency mode is approxi-
mately

z(r,θ,t) =z 0

(

1 −r^2 /R^2

)

cosωt (13)

whereRis the radius of the drum andωis the frequency of oscillation. (The shape is more accurately described by
Eq. (4.17) but this approximation is pretty good for a start.) The kinetic energy density of the moving drumhead

isu=^12 σ

(

∂z/∂t

) 2

. That is, in a small area∆A, the kinetic energy is∆K=u∆Aand the limit as∆A→ 0 of
∆K/∆Ais the area-energy-density. In the same way,σis the area mass density,dm/dA.