Geology Reference
In-Depth Information
Appendix A
Elementary results from vector analysis
In this appendix we tabulate some of the elementary results, assumed as back-
ground, from vector analysis.
A.1 Vector identities
For arbitrary vectors
a
,
b
,
c
,
d
, we have a number of identities frequently used in
elementary analysis. The
scalar triple product
obeys the cyclic identity
a
·
(
b
×
c
)
=
b
·
(
c
×
a
)
=
c
·
(
a
×
b
).
(A.1)
The
vector triple product
can be expanded as
a
×
(
b
×
c
)
=
(
a
·
c
)
b
−
(
a
·
b
)
c
.
(A.2)
The
scalar quadruple product
can be successively developed as
·
b
d
)
(
a
×
b
)
·
(
c
×
d
)
=
a
×
(
c
×
·
(
b
c
)
d
=
a
·
d
)
c
−
(
b
·
=
(
a
·
c
)(
b
·
d
)
−
(
a
·
d
)(
b
·
c
).
(A.3)
Finally, the
vector quadruple product
obeys the identity
=
(
a
d
c
−
(
a
c
d
.
(
a
×
b
)
×
(
c
×
d
)
×
b
)
·
×
b
)
·
(A.4)
A.2 Vector calculus identities
There are a variety of identities from vector calculus involving the vector operator
del
or
nabla
, written
. These involve the arbitrary scalars φ and ψ as well as the
arbitrary vectors
a
and
b
. We first have the gradient of the sum of scalars,
∇
∇
(φ
+
ψ)
=∇
φ
+∇
ψ,
(A.5)
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