Лекции по векторному и тензорному анализу. Лосик М.В. - 50 стр.

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СГУ им. Чернышевского | Саратов

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Рубрика: 

~a
~
b
div [~a,
~
b] = (
~
b, rot~a) − (~a, rot
~
b)
rot [~a,
~
b] = ~a div
~
b −
~
b div ~a + (
~
b, ∇)~a −(~a, ∇)
~
b
grad (~a,
~
b) = (~a, rot
~
b) + (
~
b, rot ~a) + (~a, ∇)
~
b + (
~
b, ∇)~a
∇
div [~a,
~
b] = (∇, [~a,
~
b]) = (∇, ~a,
~
b) + (∇, ~a,
~
b
c
) + (∇, ~a
c
,
~
b) = (
~
b, ∇, ~a)−
(~a, ∇,
~
b) = (
~
b, [∇, ~a]) − (~a, [∇,
~
b]) = (
~
b, rot ~a) − (~a, rot
~
b).
∇
rot [~a,
~
b] = [∇, [~a,
~
b]] = [∇, [~a,
~
b
c
]] + [∇, [~a,
~
b
c
]] = (
~
b, ∇)~a −
~
b(∇,~a)+
~a(∇,
~
b) − (~a, ∇)
~
b = ~a div
~
b −
~
b div ~a + (
~
b, ∇)~a − (~a, ∇)
~
b.
~p (~a,
~
b) = [
~
b, [~p,~a]] + (
~
b, ~p)~a.
∇
grad (~a,
~
b) = ∇(~a,
~
b) = ∇(~a,
~
b
c
) + ∇(~a
c
,
~
b) = [
~
b, [∇,~a]] + (
~
b, ∇)~a+
[~a, [∇,
~
b]] + (~a, ∇)
~
b = (~a, rot
~
b) + (
~
b, rot ~a) + (~a, ∇)
~
b + (
~
b, ∇)~a.
(~a, ∇)
~
b
~a = |~a|~a
0
~a
0
~a
∇
(~a, ∇)
~
b = |~a|(~a
0
, ∇)
~
b = |~a|
∂
~
b
∂~a
0
.

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