Determine (a) o rotacional e (b) o divergente do campo vetorial.
\mathbf{F}(x, y, z)=x e^{y} \mathbf{j}+y e^{z} \mathbf{k}

Passo 1

O enunciado pede o rotacional e o divergente do campo vetorial F(x,y,z)=x e^yj+y e^zk.
O conteúdo aqui é cálculo vetorial: vamos usar as definições de nablatimesF e nablacdotF.

Passo 2

Primeiro identificamos as componentes de F:
Fx=0
Fy=x e^y
Fz=y e^z

Passo 3

Para calcular o rotacional usamos:
nablatimesF =(tfracpartial Fzpartial y-tfracpartial Fypartial z),i -(tfracpartial Fzpartial x-tfracpartial Fxpartial z),j +(tfracpartial Fypartial x-tfracpartial Fxpartial y),k
Calculando as derivadas:
tfracpartial Fzpartial y=e^z,
tfracpartial Fypartial z=0,
tfracpartial Fzpartial x=0,
tfracpartial Fxpartial z=0,
tfracpartial Fypartial x=e^y,
tfracpartial Fxpartial y=0.
Logo:
nablatimesF =e^z,i+e^y,k

Passo 4

Para calcular o divergente usamos:
nablacdotF =tfracpartial Fxpartial x+tfracpartial Fypartial y+tfracpartial Fzpartial z
e temos:
tfracpartial Fxpartial x=0,
tfracpartial Fypartial y=x e^y,
tfracpartial Fzpartial z=y e^z.
Portanto:
nablacdotF =x e^y+y e^z

Resposta

(a) nablatimesF=e^z,i+e^y,k
(b) nablacdotF=x e^y+y e^z

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