Calcule o rotacional do campo vetorial F: U subset R^3 R^3
F(x, y, z)=(y cos (x y),-x sin (x y z), z ln (x y z))

Passo 1

O enunciado nos dá o campo vetorial
F:UsubsetR^3toR^3 definido por
F(x,y,z)=(ycos(xy),;-xsin(xyz),;zln(xyz))
e pede o rotacional de F. Vamos usar a definição de rotacional em termos de derivadas parciais.

Passo 2

Seja
;F1(x,y,z)=ycos(xy), F2(x,y,z)=-xsin(xyz), F3(x,y,z)=zln(xyz)
Por definição,
nablatimesF =l(partial F3partial y-partial F2partial z,; partial F1partial z-partial F3partial x,; partial F2partial x-partial F1partial yr)

Passo 3

Calculamos a primeira componente:
partial F3partial y =zcdotpartialpartial yl(ln(xyz)r) =zcdotxzxyz =zy,
partial F2partial z =-xcdotpartialpartial zl(sin(xyz)r) =-xcos(xyz)cdot(xy) =-x^2ycos(xyz)
Logo,
(nablatimesF)1 =zy-l(-x^2ycos(xyz)r) =zy+x^2ycos(xyz)

Passo 4

Calculamos a segunda componente:
partial F1partial z =partialpartial zl(ycos(xy)r) =0,
partial F3partial x =zcdotpartialpartial xl(ln(xyz)r) =zcdotyzxyz =zx
Logo,
(nablatimesF)2 =0-zx =-zx

Passo 5

Calculamos a terceira componente:
partial F2partial x =partialpartial xl(-xsin(xyz)r) =-sin(xyz)-xcos(xyz)cdot(yz) =-sin(xyz)-xyzcos(xyz),
partial F1partial y =partialpartial yl(ycos(xy)r) =cos(xy)-xysin(xy)
Logo,
(nablatimesF)3 =l(-sin(xyz)-xyzcos(xyz)r) -l(cos(xy)-xysin(xy)r) =-sin(xyz)-xyzcos(xyz)-cos(xy)+xysin(xy)

Passo 6

Reunindo as componentes, obtemos
nablatimesF(x,y,z) =l(zy+x^2ycos(xyz),; -zx,; -sin(xyz)-xyzcos(xyz)-cos(xy)+xysin(xy)r)

Resposta

nablatimesF(x,y,z) =l(zy+x^2ycos(xyz),; -zx,; -sin(xyz)-xyzcos(xyz)-cos(xy)+xysin(xy)r)

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