Determine o rotacional e o divergente do campo vetorial dado.
F(x, y, z)=x y z i-x^2 y j;

Passo 1

Resumindo: temos o campo vetorial
F(x,y,z)=xyz,i-x^2y,j+0,k
e queremos calcular o divergente nablacdot F e o rotacional nablatimes F.
Para isso vamos usar as definições
nablacdot F=partial Fxpartial x+partial Fypartial y+partial Fzpartial z,
nablatimes F=l(partial Fzpartial y-partial Fypartial zr)i +l(partial Fxpartial z-partial Fzpartial xr)j +l(partial Fypartial x-partial Fxpartial yr)k

Passo 2

Calculemos primeiro o divergente:
Fx=xyz, Fy=-x^2y, Fz=0
Logo
partial Fxpartial x=partial (xyz)partial x=yz, partial Fypartial y=partial (-x^2y)partial y=-x^2, partial Fzpartial z=partial 0partial z=0
Então
nablacdot F=yz - x^2 + 0 = yz - x^2

Passo 3

Agora o rotacional, usando cada componente:
(nablatimes F)x =partial Fzpartial y-partial Fypartial z =0-partial(-x^2y)partial z=0,
(nablatimes F)y =partial Fxpartial z-partial Fzpartial x =partial(xyz)partial z-0=xy,
(nablatimes F)z =partial Fypartial x-partial Fxpartial y =partial(-x^2y)partial x-partial(xyz)partial y =-2xy - xz
Portanto
nablatimes F =0,i + xy,j +(-2xy - xz),k

Resposta

Divergente:
nablacdot F = yz - x^2
Rotacional:
nablatimes F = xy,j - l(2xy + xzr),k

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