推 sx4152 :謝謝 很清楚明瞭! 06/20 19:26
※ 引述《sx4152 (呵呵)》之銘言：
: 題目:
: -> ^ ^ ^
: (a) show that vector F = (y^2 cosx +z^3)i +(2ysinx-4)j+(3xz^2 +2)k is a
: conservative field
: ->
: (b) find the scalar potential function for F
: 第一小題只要看F的旋度是否是零就可以證了
: 第二小題要怎麼算呢?
→
(b) 令 ▽G = F
δG
則 ----- = (y^2)(cosx) + z^3 ------(1)
δx
δG
----- = (2y)(sinx) - 4 ------(2)
δy
δG
----- = (3)(x)(z^2) + 2 ------(3)
δz
由(1)
G(x,y,z) = (y^2)(sinx) + (x)(z^3) + f(y,z)
δG δf
----- = (2y)(sinx) + ----- 對照(2)
δy δy
δf
----- = -4 => f(y,z) = -4y + h(z)
δy
∴ G(x,y,z) = (y^2)(sinx) + (x)(z^3) - 4y + h(z)
δG
=> ----- = (3)(x)(z^2) + h'(z) 對照(3)
δz
(3)(x)(z^2) + h'(z) = (3)(x)(z^2) + 2
=> h'(z) = 2 => h(z) = 2z + c
∴ G(x,y,z) = (y^2)(sinx) + (x)(z^3) - 4y + 2z + c
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