Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
Proof:
…2†
Z b
a
… r†dl ˆ
ˆ
ˆ
ˆ
Z b
a
Z b
a
Z b
a
Z b
I
a
S
@
@x ^i ‡ @
@y ^j ‡ @
@z ^k …dx^i ‡ dy^j ‡ dz ^k†
@ @ @
dx ‡ dy ‡
@x @y @z dz
@dx
@x dt ‡ @ dy
@y dt ‡ @
dz
dt
@z dt
d
dt ˆ …b†…a†:
dt
@'
@n
ZV
da ˆ r 2 'dV:
Proof: Set ˆ 1 in Eq. (1.87), then @ =@n ˆ 0 ˆr 2
Eq. (1.91).
Z I
…3†
r'dV ˆ '^nda:
V
S
…1:91†
and Eq. (1.87) reduces to
…1:92†
Proof: In Gauss' theorem (1.78), let A ˆ 'C, where C is constant vector. Then
we have
Z
Z
r…'C†dV ˆ 'C ^nda:
Since
V
r…'C† ˆr' C ˆ C r' and 'C ^n ˆ C …'^n†;
S
we have
Taking C outside the integrals,
Z
C
Z
V
Z
C r'dV ˆ
V
S
C …'^n†da:
Z
r'dV ˆ C …'^n†da
S
and since C is an arbitrary constant vector, we have
Z I
r'dV ˆ '^nda:
…4†
Z
V
V
Z
rBdV ˆ
S
S
^n Bda
…1:93†
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