Se \varphi=x y+y z+z x e \mathbf{A}=x^{2} y \hat{\mathbf{x}}+y^{2} z \hat{\mathbf{y}}+z^{2} x \hat{\mathbf{z}}, ache (a) \mathbf{A} \cdot \nabla \varphi e (b) \varphi \nabla \cdot \mathbf{A} no ponto (3,-1,2). (Valor 1,0 ponto)

Passo 1

O enunciado define o campo escalar \varphi = x y + y z + z x
e o campo vetorial \mathbf{A} = x^{2}y\,\hat{\mathbf{x}} + y^{2}z\,\hat{\mathbf{y}} + z^{2}x\,\hat{\mathbf{z}},
e pede:
(a) \mathbf{A}\cdot\nabla\varphi
(b) \varphi\,\nabla\cdot\mathbf{A}
no ponto (3,-1,2).
Para isso vamos usar definição de gradiente, produto escalar e divergência.

Passo 2

Calcular o gradiente de \varphi. Por definição,
\nabla\varphi = \frac{\partial\varphi}{\partial x}\,\hat{\mathbf{x}} + \frac{\partial\varphi}{\partial y}\,\hat{\mathbf{y}} + \frac{\partial\varphi}{\partial z}\,\hat{\mathbf{z}}.
Como
\frac{\partial\varphi}{\partial x} = y + z,\quad \frac{\partial\varphi}{\partial y} = x + z,\quad \frac{\partial\varphi}{\partial z} = x + y,
segue
\nabla\varphi = (y+z)\,\hat{\mathbf{x}} + (x+z)\,\hat{\mathbf{y}} + (x+y)\,\hat{\mathbf{z}}.

Passo 3

Escrever \mathbf{A}\cdot\nabla\varphi de forma geral:
\mathbf{A}\cdot\nabla\varphi = x^{2}y\,(y+z) \;+\; y^{2}z\,(x+z)\;+\; z^{2}x\,(x+y).

Passo 4

Calcular a divergência de \mathbf{A}.
\nabla\cdot\mathbf{A} = \frac{\partial}{\partial x}(x^{2}y) + \frac{\partial}{\partial y}(y^{2}z) + \frac{\partial}{\partial z}(z^{2}x).
Como
\frac{\partial}{\partial x}(x^{2}y)=2xy,\quad \frac{\partial}{\partial y}(y^{2}z)=2yz,\quad \frac{\partial}{\partial z}(z^{2}x)=2zx,
segue
\displaystyle \nabla\cdot\mathbf{A} = 2xy + 2yz + 2zx = 2\varphi.

Passo 5

Logo
\varphi\,\nabla\cdot\mathbf{A} = \varphi\cdot(2\varphi) = 2\varphi^{2}.

Passo 6

Avaliar em (x,y,z)=(3,-1,2):
  1. Primeiro \varphi:
\varphi = 3\cdot(-1) + (-1)\cdot2 + 2\cdot3 = -3 -2 +6 = 1.
  1. Depois \nabla\varphi:
\nabla\varphi = (y+z,\,x+z,\,x+y) = (-1+2,\,3+2,\,3-1) = (1,5,2).
  1. E \mathbf{A}:
\mathbf{A} = (x^{2}y,\,y^{2}z,\,z^{2}x) = (9\cdot(-1),\,1\cdot2,\,4\cdot3) = (-9,\,2,\,12).
  1. Então
\mathbf{A}\cdot\nabla\varphi = -9\cdot1 + 2\cdot5 + 12\cdot2 = -9 +10 +24 = 25.
  1. E
\varphi\,\nabla\cdot\mathbf{A} = 2\cdot(1)^{2} = 2.

Resposta

(a) 25
(b) 2

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