Mathematical Tools for Physics

8—Multivariable Calculus 224

The element of area isR^2 sinθ dθ dφ, so the total charge is

∫

σ dA,

Q=

∫π

0

sinθ dθ R^2

∫ 2 π

0

dφσ 0 sin^2 θcos^2 φ=R^2

∫+1

− 1

dcosθ σ 0

(

1 −cos^2 θ

)∫^2 π

0

dφcos^2 φ

The mean value ofcos^2 is 1 / 2. so theφintegral givesπ. For the rest, it is

σ 0 πR^2

[

cosθ−

1

3

cos^3 θ

]+1

− 1

=

4

3

σ 0 πR^2

8.9 Vectors: Cylindrical, Spherical Bases
When you describe vectors in three dimensions are you restricted to the basisˆx,ˆy,zˆ? In a different coordinate
system you should use basis vectors that are adapted to that system. In rectangular coordinates these vectors have
the convenient property that they point along the direction perpendicular to the plane where the corresponding
coordinate is constant. They also point in the direction in which the other two coordinates are constant.E.g.the
unit vectorxˆpoints perpendicular to the plane of constantx(they-z plane); it also point along the line where
yandzare constant.

x

z

y

ˆx

ˆz

yˆ

θ

r

zˆ

z

ˆr

ˆθ

φ

θ r

ˆr

θˆ

φˆ

Do the same thing for cylindrical coordinates. The unit vectorˆz points perpendicular to thex-yplane.
The unit vectorˆrpoints perpendicular to the cylinderr=constant. The unit vectorθˆpoints perpendicular to
the planeθ=constant and along the direction for whichrandzare constant. The conventional right-hand rule
specifiesˆz=rˆ×ˆθ.
For spherical coordinatesˆrpoints perpendicular to the spherer=constant. Theφˆvector is perpendicular
to the planeφ =constant and points along the direction wherer =constant andθ =constant and toward