Лекции по векторному и тензорному анализу. Лосик М.В. - 40 стр.

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СГУ им. Чернышевского | Саратов

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f(
~
j,
~
i) = −f(
~
i,
~
j), f(
~
k,
~
i) = −f(
~
i,
~
k), f(
~
k,
~
j) = −f(
~
j,
~
k).
f
f(
~
j,
~
k) f(
~
k,
~
i) f(
~
i,
~
j)
f
0
(~x, ~y) = (~p, [~x~y])
p
x
, p
y
, p
z
~p
f f
0
f
0
(
~
j,
~
k) = (~p, [
~
j,
~
k]) = (~p,
~
i) = p
x
,
f
0
(
~
k,
~
i) = (~p, [
~
k,
~
i]) = (~p,
~
j) = p
y
,
f
0
(
~
i,
~
j) = (~p, [
~
i,
~
j]) = (~p,
~
k) = p
z
.
f f
0
f(
~
j,
~
k) = p
x
f(
~
k,
~
i) = p
y
f(
~
i,
~
j) = p
z
~p
~p p
x
= f(
~
j,
~
k) p
y
= f(
~
k,
~
i) p
z
=
f(
~
i,
~
j)
~a M
0
M
0
P
M
0
f = f(~x, ~y) ~x ~y
f(~x, ~y) = (P (~x), ~y) − (P (~y), ~x).
f(~y, ~x) = −f (~x, ~y) f(~x, ~y)
~x
1
, ~x
2
, ~y α
1
, α
2
P
f(α
1
~x
1
+ α
2
~x
2
, ~y) = (P (α
1
~x
1
+ α
2
~x
2
), ~y) − (P (~y), α
1
~x
1
+ α
2
~x
2
) =
(α
1
P (~x
1
) + α
2
P (~x
2
), ~y) − α
1
(P (~y), ~x
1
) − α
2
(P (~y), ~x
2
) = α
1
(P (~x
1
), ~y)+
α
2
(P (~x
2
), ~y) − α
1
(P (~y), ~x
1
) − α
2
(P (~y), ~x
2
) = α
1
((P (~x
1
), ~y) − (P (~y), ~x
1
)) +
α
2
((P (~x
2
), ~y) − (P (~y), ~x
2
)) = α
1
f(~x
1
, ~y) + α
2
f(~x
2
, ~y).

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