8—Multivariable Calculus 188Still another way is from the Stallone-Schwarzenegger brute force school of computing. Put
everything in rectangular coordinates and do the partial derivatives using Eqs. (8.15) and (8.6).
(
∂(1/r)
∂x
)
y,z=
(
∂(1/r)
∂r
)
θ,φ(
∂r
∂x
)
y,z=−
1
r^2
∂
∂x
√
x^2 +y^2 +z^2 =−
1
r^2
x
√
x^2 +y^2 +z^2
Repeat this foryandzwith similar results and assemble the output.
−gradkq
r
=
kq
r^2
xˆx+yyˆ+zˆz
√
x^2 +y^2 +z^2
=
kq
r^2
~r
r
=
kq
r^2
ˆr
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
+zˆ
∂
∂z
(8.17)
From Eq. (8.15) you have
gradf=∇f (8.18)
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∆φ.
x x+dx
y
y+dy
r r+dr
φ
φ+dφ
Vibrating Drumhead
A circular drumhead can vibrate in many complicated ways. The simplest and lowest frequency mode
is approximately
z(r,φ,t) =z 0
(
1 −r^2 /R^2
)
cosωt (8.19)
whereRis the radius of the drum andωis the frequency of oscillation. (The shape is more accurately
described by Eq. (4.22) 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.
What is the total kinetic energy because of this oscillation? It is