Calcule o divergente e o rotacional do seguinte campo:
F(x, y, z)=y e^z i+x e^z j+x y e^z k

Passo 1

O enunciado pede para calcular o divergente e o rotacional do campo vetorial
F(x,y,z)=y e^zi+x e^zj+x y e^zk.
Vamos usar as definições de cálculo vetorial:
div,F=partial Ppartial x+partial Qpartial y+partial Rpartial z
e
nablatimesF=l(tfracpartial Rpartial y-tfracpartial Qpartial z,;tfracpartial Ppartial z-tfracpartial Rpartial x,;tfracpartial Qpartial x-tfracpartial Ppartial yr),
onde P=y e^z, Q=x e^z, R=x y e^z.

Passo 2

Para o divergente, calculamos as derivadas parciais:
partial Ppartial x=0,
partial Qpartial y=0,
partial Rpartial z=x y e^z.
Logo,
div,F =0+0+x y e^z =x y e^z

Passo 3

Para o rotacional, aplicamos
nablatimesF =l(partial Rpartial y-partial Qpartial z,; partial Ppartial z-partial Rpartial x,; partial Qpartial x-partial Ppartial yr)

Passo 4

Calculamos cada componente:
partial Rpartial y=x e^z,
partial Qpartial z=x e^z ;Rightarrow;1ª componente =0.
partial Ppartial z=y e^z,
partial Rpartial x=y e^z ;Rightarrow;2ª componente =0.
partial Qpartial x=e^z,
partial Ppartial y=e^z ;Rightarrow;3ª componente =0.
Portanto,
nablatimesF=0

Resposta

div,F=x y e^z,nablatimesF=0

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