Mathematical Tools for Physics - Department of Physics - University

8—Multivariable Calculus 192

when you have to change the order of integration, the new limits may not be obvious. Are there any
special techniques or tricks to doing this? Yes, there is one, perhaps obscure, method that you may
not be accustomed to.

Draw Pictures.
If you have an integral such as the first one, you have to draw a picture of the integration domain
to switch limits.

∫ 1

0

dy

∫√ 2 −y 2

y

dxf(x,y)

x

y

[∫

1

0

dx

∫x

0

dy+

∫√ 2

1

dx

∫√ 2 −x 2

0

dy

]

f(x,y) (8.22)

Of course, once you’ve drawn the picture you may realize that simply interchanging the order of
integration won’t help, but that polar coordinates may.
∫√ 2

0

rdr

∫π/ 4

0

dφ

8.9 Vectors: Cylindrical, Spherical Bases

When you describe vectors in three dimensions are you restricted to the basisxˆ,yˆ,zˆ? In a different

coordinate system you should use basis vectors that are adapted to that system. In rectangular coordi-
nates 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 constant

x(they-zplane); it also point along the line whereyandzare constant.

x

z

y

xˆ

ˆz

yˆ

φ

r

zˆ

z

rˆ

φˆ

φ

θ r

ˆr

θˆ

φˆ

Do the same thing for cylindrical coordinates. The unit vectorzˆpoints perpendicular to thex-y

plane. The unit vectorrˆpoints perpendicular to the cylinderr=constant. The unit vectorφˆpoints

perpendicular to the planeφ=constant and along the direction for whichrandzare constant. The

conventional right-hand rule specifieszˆ=ˆr×φˆ.

For spherical coordinates ˆrpoints perpendicular to the sphere r =constant. Theφˆvector

is perpendicular to the planeφ=constant and points along the direction wherer =constant and

θ=constant and toward increasing coordinateφ. Finallyθˆis perpendicular to the coneθ=constant

and again, points toward increasingθ. Thenφˆ=rˆ×θˆ, and on the Earth, these vectorsrˆ,θˆ, andφˆ

areup,ˆ South, andˆ East.ˆ

Solenoid
A standard solenoid is cylindrical coil of wire, so that when the wire carries a current it produces a
magnetic field. To describe this field, it seems that cylindrical coordinates are advised. Until you know
something about the field the most general thing that you can write is

B~(r,φ,z) =rBˆ r(r,φ,z) +φBˆ φ(r,φ,z) +ˆzBz(r,φ,z)