Determine (a) o rotacional e (b) o divergente do campo vetorial.
F(x, y, z)=lang<=arctg(x y), arctg(y z), arctg(z x)rang<=

Passo 1

O enunciado dá o campo vetorial
F(x,y,z)=llang<= arctg(xy),,arctg(yz),,arctg(zx)rrang<=
e pede (a) o rotacional nablatimes F e (b) o divergente nablacdot F. Para isso vamos usar as definições clássicas de cálculo vetorial e a regra da cadeia para derivadas parciais.

Passo 2

Para calcular o rotacional, lembramos que, se F=lang<= P,Q,Rrang<=, então
nablatimes F =l(partial Rpartial y-partial Qpartial z,; partial Ppartial z-partial Rpartial x,; partial Qpartial x-partial Ppartial yr)
Aqui temos
P=arctg(xy),
Q=arctg(yz) e
R=arctg(zx).

Passo 3

Agora aplicamos a regra da cadeia em cada derivada:
1. partial Rpartial y =partialpartial yl[arctg(zx)r]=0 2. partial Qpartial z =11+(yz)^2cdot y =y1+y^2z^2 3. partial Ppartial z =partialpartial zl[arctg(xy)r]=0 4. partial Rpartial x =11+(zx)^2cdot z =z1+z^2x^2 5. partial Qpartial x =partialpartial xl[arctg(yz)r]=0 6. partial Ppartial y =11+(xy)^2cdot x =x1+x^2y^2
Substituindo em nablatimes F, obtemos
nablatimes F =l(0-y1+y^2z^2,;0-z1+z^2x^2,;0-x1+x^2y^2r) =l(-y1+y^2z^2,,-z1+z^2x^2,,-x1+x^2y^2r)

Passo 4

Para o divergente, usamos a fórmula
nablacdot F =partial Ppartial x+partial Qpartial y+partial Rpartial z

Passo 5

Calculemos cada termo:
1. partial Ppartial x =11+(xy)^2cdot y =y1+x^2y^2 2. partial Qpartial y =11+(yz)^2cdot z =z1+y^2z^2 3. partial Rpartial z =11+(zx)^2cdot x =x1+z^2x^2
Somando,
nablacdot F =y1+x^2y^2+z1+y^2z^2+x1+z^2x^2

Resposta

(a) nabla times F = l(-y1+y^2z^2,,-z1+z^2x^2,,-x1+x^2y^2r)
(b) nabla cdot F = y1+x^2y^2 + z1+y^2z^2 + x1+z^2x^2

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