wcwcwc asked in 科學數學 · 1 decade ago

advanced calculus

determine whether each of the following vector fields is the gradient of a function f, and if so, find f

a). F(x,y,z) = (yz-ysin(xy))i+(xz-xsin(xy)+zcos(yz))j+(xy+ycos(yz))k

b). F(x,y,z) = (y-z)i+(x-z)j+(x-y)k

1 Answer

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  • 1 decade ago
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    版大問題中之函數,均為二階可導函數,

    故只需檢查 curl(F)是否為向量0i+0j+0z即可

    a)curl(F)=0i+0j+0k

    故F(x,y,z)=grad(f) , for some real valued function f(x,y,z), 即

    ∂f/∂x= yz-ysin(xy) => f(x,y,z)=xyz+cos(xy)+C1(y,z)

    ∂f/∂y= xz-xsin(xy)+zcos(yz)=>f(x,y,z)=xyz+cos(xy)+sin(yz)+C2(z,x)

    ∂f/∂z= xy+ycos(yz)=>f(x,y,z)=xyz+sin(yz)+C3(x,y)

    故 f(x,y,z)= xyz+cos(xy)+sin(yz)+C (for any constant number C)

    b) curl(F)= -2j ≠0i+0j+0k

    故F(x,y,z)≠grad(f)

    Source(s): me
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