※ 引述《sx4152 (呵呵)》之铭言： : 题目: : -> ^ ^ ^ : (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 : 第一小题只要看F的旋度是否是零就可以证了 : 第二小题要怎麽算呢? → (b) 令 ▽G = F δG 则 ----- = (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) + ----- 对照(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) 对照(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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