Calcule div \mathbf{F} e (\operatorname{rot} \mathbf{F}).
\mathbf{F}(x, y, z)=7 y^{3} z^{2} \mathbf{i}-8 x^{2} z^{5} \mathbf{j}-3 x y^{4} \mathbf{k}

Passo 1

O enunciado dá o campo vetorial
F(x,y,z) = 7y^3z^2,i - 8x^2z^5,j - 3xy^4,k
e pede para calcular a divergência e o rotacional. O conteúdo é cálculo vetorial, usando as definições
div,F =partial Ppartial x+partial Qpartial y+partial Rpartial z,
rot,F =nablatimesF =l(tfracpartial Rpartial y-tfracpartial Qpartial z,;tfracpartial Ppartial z-tfracpartial Rpartial x,;tfracpartial Qpartial x-tfracpartial Ppartial yr),
onde P=7y^3z^2, Q=-8x^2z^5 e R=-3xy^4.

Passo 2

Calcular a divergência:
partial Ppartial x =partialpartial x(7y^3z^2)=0,
partial Qpartial y =partialpartial y(-8x^2z^5)=0,
partial Rpartial z =partialpartial z(-3xy^4)=0
Somando,
div,F=0+0+0=0

Passo 3

Calcular o rotacional usando cada componente:
partial Rpartial y =partialpartial y(-3xy^4) =-12xy^3, partial Qpartial z =partialpartial z(-8x^2z^5) =-40x^2z^4,
partial Ppartial z =partialpartial z(7y^3z^2) =14y^3z, partial Rpartial x =partialpartial x(-3xy^4) =-3y^4,
partial Qpartial x =partialpartial x(-8x^2z^5) =-16xz^5, partial Ppartial y =partialpartial y(7y^3z^2) =21y^2z^2
Logo,
rot,F =l((-12xy^3)-(-40x^2z^4),;(14y^3z-(-3y^4)), ;(-16xz^5-21y^2z^2)r),
rot,F =l(40x^2z^4-12xy^3,;14y^3z+3y^4,;-16xz^5-21y^2z^2r)

Resposta

div,F=0, rot,F =l(40x^2z^4-12xy^3,;14y^3z+3y^4,;-16xz^5-21y^2z^2r)

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