5
Vector and scalar fields
5.1
scalar fields
A “scalar field” is a fancy name for a function of space, i.e.
it associates a real
number with every position in some space, e.g.
in 3D
φ
=
φ
(
x, y, z
).
We’ve
already encountered examples without calling them scalar fields, e.g. the temper
ature
T
(
x, y
) in a metal plate, or the electrostatic potential
φ
=
φ
(
x, y, z
).
The
gravitational potential is another, and it’s frequently convenient to think about
potential “landscapes”, imagining that a set of hills is a kind of paradigm for a
varying potential, since the height in this case scales with the potential
mgh
(
x, y
)
itself.
Formally, scalar is a word used to distinguish the field from a vector field. We
can do this because a scalar field is
invariant
under the rotation of the coordinate
system:
φ
0
(
x
0
, y
0
, z
0
) =
φ
(
x, y, z
)
.
(1)
In other words, I may label the point on top of one of the hills by a different set of
coordinates, but this doesn’t change the height I assign to it. This is in contrast to
a vector field, where the values of the components do change in the new coordinate
system, as we have discussed.
5.1.1
gradients of scalar fields
If you’re standing on the hill somewhere, say not on the top, there’s one direction
in
xy
space which gives you the direction of the fastest way down. This vector is
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 Fall '07
 HIRSHFELD
 Electron, General Relativity, Pauli exclusion principle, Condensed matter physics, Fermi

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