Questão 9 - Determine o rotacional do campo vetorial F(x, y, z)=(x e^y z, y e^x z, z e^x y).

Passo 1

O enunciado pede o rotacional de F(x,y,z)=l(xe^yz,,ye^xz,,ze^xyr). Vamos usar a definição
nablatimesF =l(partial Rpartial y-partial Qpartial z,;partial Ppartial z-partial Rpartial x,;partial Qpartial x-partial Ppartial yr),
com P=xe^yz, Q=ye^xz e R=ze^xy.

Passo 2

Calculamos a primeira componente:
partial Rpartial y =partialpartial yl(ze^xyr) =xz,e^xy,
partial Qpartial z =partialpartial zl(ye^xzr) =xy,e^xz
Logo
(nablatimesF)1 = xz,e^xy-xy,e^xz

Passo 3

Calculamos a segunda componente:
partial Ppartial z =partialpartial zl(xe^yzr) =xy,e^yz,
partial Rpartial x =partialpartial xl(ze^xyr) =yz,e^xy
Logo
(nablatimesF)2 = xy,e^yz-yz,e^xy

Passo 4

Calculamos a terceira componente:
partial Qpartial x =partialpartial xl(ye^xzr) =yz,e^xz,
partial Ppartial y =partialpartial yl(xe^yzr) =xz,e^yz
Logo
(nablatimesF)3 = yz,e^xz-xz,e^yz

Resposta

nablatimesF =l(xz,e^xy-xy,e^xz,;xy,e^yz-yz,e^xy,;yz,e^xz-xz,e^yzr)

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