Determine o rotacional e o divergente do campo vetorial dado.
F(x, y, z)=1raizx^2+y^2+z^2(x i+y e^z j+k k);

Passo 1

O campo vetorial é
F(x,y,z)=1rl(x,,y e^z,,1r), r=raizx^2+y^2+z^2
A tarefa é calcular o divergente ( nablacdot F) e o rotacional ( nablatimes F). Vamos usar as fórmulas:
– Divergente: nablacdot F=partialxF1+partialyF2+partialzF3.
– Rotacional:

Passo 2

Para o divergente, calculamos cada derivada parcial:
F1=xr, partialxF1=1r+xl(-1r^2r)xr =1r-x^2r^3=y^2+z^2r^3,
F2=y e^zr, partialyF2=e^zr-y e^zyr^3= race^z(r^2-y^2)r^3= race^z(x^2+z^2)r^3,
F3=1r, partialzF3=-1r^2zr=-zr^3
Somando:
nablacdot F=y^2+z^2r^3+e^z(x^2+z^2)r^3-zr^3 =y^2+z^2+e^z(x^2+z^2)-zr^3

Passo 3

Para o rotacional, usamos
(nablatimes F)1=partialyF3-partialzF2,
partialyF3=-yr^3, partialzF2=y e^zr-y e^zzr^3= racy e^z(r^2-z)r^3,
(nablatimes F)1=-yr^3-y e^z(r^2-z)r^3=-yigl[1+e^z(r^2-z)r]r^3 
(nablatimes F)2=partialzF1-partialxF3 =-xzr^3-l(-xr^3r)=x(1-z)r^3
(nablatimes F)3=partialxF2-partialyF1 =-x y e^zr^3-l(-x yr^3r)=xy(1-e^z)r^3
Portanto:
nablatimes F=l(-y[1+e^z(r^2-z)]r^3,,x(1-z)r^3,,xy(1-e^z)r^3r)

Resposta


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