Calcule div F e rot F sendo:
a) F(x, y, z)=(2 z-3 y, 3 x-z, y-2 x)

Passo 1

O enunciado dá o campo vetorial
vec F(x,y,z) = l(P,Q,Rr) = l(2z - 3y,;3x - z,;y - 2xr)
e pede (a) o divergente divvec F e (b) o rotacional rot,vec F. Para isso vamos usar as definições em coordenadas cartesianas, calculando derivadas parciais de P, Q e R.

Passo 2

Aplicando a definição
divvec F = partial Ppartial x + partial Qpartial y + partial Rpartial z,
temos
partial Ppartial x = partial(2z-3y)partial x = 0,
partial Qpartial y = partial(3x - z)partial y = 0,
partial Rpartial z = partial(y - 2x)partial z = 0
Logo
divvec F = 0 + 0 + 0 = 0

Passo 3

O rotacional é dado por
rot,vec F = l(partial Rpartial y-partial Qpartial z,;partial Ppartial z-partial Rpartial x,;partial Qpartial x-partial Ppartial yr)
Calculamos as derivadas parciais necessárias:
partial Rpartial y = 1, partial Qpartial z = -1,
partial Ppartial z = 2, partial Rpartial x = -2,
partial Qpartial x = 3, partial Ppartial y = -3

Passo 4

Substituindo em cada componente do rotacional:
partial Rpartial y-partial Qpartial z = 1 - (-1) = 2,
partial Ppartial z-partial Rpartial x = 2 - (-2) = 4,
partial Qpartial x-partial Ppartial y = 3 - (-3) = 6
Portanto
rot,vec F = (2,,4,,6)

Resposta

divvec F = 0,
rot,vec F = (2,,4,,6)

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