Multidimensional Calculus: Differential Theory
Chapter & Page: 9–9
Final Comments on Christoffel Symbols and Acceleration
Using the above, we can find formulas for acceleration (and Christoffel symbols) in Euclidean
spaces and in nonEuclidean subpaces contained in Euclidean spaces (e.g., curves and spheres
in Euclidean three-space), at least when the coordinates on the subspace are a subset of the
coordinates of the coordinates of the larger Euclidean space. If we had time to develop the metric
tensor, we would also be able to obtain another formula for the Christoffel symbols that would
allow us to compute these quantities without referring back to some Cartesian system.
This
formula also applies in nonEuclidean spaces, and allows us to find formulas for acceleration in
those spaces, again, without recourse to any Cartesian system.
9.2
Scalar and Vector Fields
Basic Definitions and Concepts
In the following, all points/positions refer to points in some
N
-dimensional space
S
.
A
scalar field
Ψ
(on
S
) is a scalar-valued function of position,
Ψ(
x
)
=
some scalar value corresponding to position
x
in
S
.
Some examples:
T
(
x
)
=
temperature at position
x
,
Φ(
x
)
=
the gravitational potential at
x
due to the surrounding masses
,
x
2
(
x
)
=
the second coordinate of
x
with respect to some given coordinate system
and
r
(
x
)
=
distance between
x
and some fixed point
O
(i.e.,
r
(
x
)
=
dist
(
x
,
O
)
)
.
The
coordinate formula
for a scalar field
Ψ
with respect to a given coordinate system
{
(
x
1
,
x
2
,...,
x
N
)
}
is just the formula
ψ(
x
1
,
x
2
,...,
x
N
)
for computing the value of
Ψ(
x
)
from the coordinates for
x
. So
Ψ(
x
)
=
ψ(
x
1
,
x
2
,...,
x
N
)
with
x
∼
(
x
1
,
x
2
,...,
x
N
)
.
For example, if our space is a plane and we are using Cartesian coordinates
{
(
x
,
y
)
}
with
O
as
the origin, then
the coordinate formula for
r
(
x
)
=
dist
(
x
,
O
)
is
radicalbig
x
2
+
y
2
.
If, instead, we are using polar coordinates
{
(ρ,φ)
}
with
O
as the origin, then
the coordinate formula for
r
(
x
)
=
dist
(
x
,
O
)
is
ρ
.
A
vector field
(on
S
) is a vector-valued function of position,
V
(
x
)
=
some vector corresponding to position
x
in
S
.