Calcule nabla cdot u e nabla times u para os seguintes campos vetoriais:
u(x, y, z)=(x^3+y z^2) imath+(y+x z) jmath+(z+x y) k

Passo 1

O enunciado nos dá o campo vetorial
vec u(x,y,z) = l(x^3+y,z^2r),hatimath + l(y + x,zr),hatjmath + l(z + x,yr),hat k.
O conteúdo aqui é cálculo de divergência e rotacional de um campo em R^3.
Para isso usaremos as definições:
nabla cdot vec u = partial u1partial x + partial u2partial y + partial u3partial z
nabla times vec u = hatimath & hatjmath & hat k [6pt] partialx & partialy & partialz [3pt] u1 & u2 & u3 ]

Passo 2

Vamos calcular a divergência usando
nabla cdot vec u = partialpartial xl(x^3+y,z^2r) + partialpartial yl(y + x,zr) + partialpartial zl(z + x,yr)
Cada termo é
partialpartial x(x^3+y,z^2) = 3,x^2,
partialpartial y(y + x,z) = 1,
partialpartial z(z + x,y) = 1.
Somando, obtemos
nabla cdot vec u = 3,x^2 + 1 + 1 = 3,x^2 + 2

Passo 3

Para o rotacional, calculamos as componentes do determinante:
1. Componente hatimath:
(nabla times vec u)x = partial u3partial y - partial u2partial z = partialpartial y(z + x,y) - partialpartial z(y + x,z) = x - x = 0 2. Componente hatjmath:
(nabla times vec u)y = partial u1partial z - partial u3partial x = partialpartial z(x^3+y,z^2) - partialpartial x(z + x,y) = 2,y,z - y = y,(2,z - 1) 3. Componente hat k:
(nabla times vec u)z = partial u2partial x - partial u1partial y = partialpartial x(y + x,z) - partialpartial y(x^3+y,z^2) = z - z^2
Portanto
nabla times vec u = 0,hatimath + y,(2,z-1),hatjmath + (z - z^2),hat k

Resposta

nabla cdot vec u = 3,x^2 + 2, nabla times vec u = 0,hatimath + y,(2,z - 1),hatjmath + (z - z^2),hat k

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