全微分z=(x^2+y^2)e^[(x^2+y^2)/xy] 最后有过程,可以的话偏导数是多少也说一下?
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全微分z=(x^2+y^2)e^[(x^2+y^2)/xy] 最后有过程,可以的话偏导数是多少也说一下?
记:z = f(x,y) = (x^2+y^2)e^[(x^2+y^2)/xy] = u(x,y) e^[u(x,y)/v(x,y)]
其中:u(x,y) = (x^2+y^2);
v(x,y) = xy;
全微分:dz = df(x,y) = [∂f(x,y)/∂x] dx + [∂f(x,y)/∂y] dy
∂f(x,y)/∂x = ∂u(x,y)/∂x e^(u/v) + u e^(u/v) ∂(u/v)/∂x
∂f(x,y)/∂y = ∂u(x,y)/∂y e^(u/v) + u e^(u/v) ∂(u/v)/∂y
∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^2+y^2) (2x-(x^2+y^2))/(x^2y)]
∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (x^2+y^2) (2y-(x^2+y^2))/(xy^2)]
这是 z对x,y的两个偏导数.
z的全微分:
dz = df(x,y) = e^[(x^2+y^2)/xy] {[2x + (x^2+y^2) (2x-(x^2+y^2))/(x^2y)] dx +
+ [2y + (x^2+y^2) (2y-(x^2+y^2))/(xy^2)] dy }
再问: ∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^2+y^2) (2x-(x^2+y^2))/(x^2y)] ∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (x^2+y^2) (2y-(x^2+y^2))/(xy^2)] 不应该是 ∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^2+y^2) (2x^2-(x^2+y^2))/(x^2y)] 吗? ∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (x^2+y^2) (2y^2-(x^2+y^2))/(xy^2)]
再答: 我检查了一下,您说的对,应该是: ∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^4 - y^4)/(x^2y)] ∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (y^4 - x^4)/(xy^2)] dz = e^[(x^2+y^2)/xy] {[2x + (x^4 - y^4)/(x^2y)] dx + [2y + (y^4 - x^4)/(xy^2)] dy }
其中:u(x,y) = (x^2+y^2);
v(x,y) = xy;
全微分:dz = df(x,y) = [∂f(x,y)/∂x] dx + [∂f(x,y)/∂y] dy
∂f(x,y)/∂x = ∂u(x,y)/∂x e^(u/v) + u e^(u/v) ∂(u/v)/∂x
∂f(x,y)/∂y = ∂u(x,y)/∂y e^(u/v) + u e^(u/v) ∂(u/v)/∂y
∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^2+y^2) (2x-(x^2+y^2))/(x^2y)]
∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (x^2+y^2) (2y-(x^2+y^2))/(xy^2)]
这是 z对x,y的两个偏导数.
z的全微分:
dz = df(x,y) = e^[(x^2+y^2)/xy] {[2x + (x^2+y^2) (2x-(x^2+y^2))/(x^2y)] dx +
+ [2y + (x^2+y^2) (2y-(x^2+y^2))/(xy^2)] dy }
再问: ∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^2+y^2) (2x-(x^2+y^2))/(x^2y)] ∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (x^2+y^2) (2y-(x^2+y^2))/(xy^2)] 不应该是 ∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^2+y^2) (2x^2-(x^2+y^2))/(x^2y)] 吗? ∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (x^2+y^2) (2y^2-(x^2+y^2))/(xy^2)]
再答: 我检查了一下,您说的对,应该是: ∂f(x,y)/∂x = e^[(x^2+y^2)/xy] [2x + (x^4 - y^4)/(x^2y)] ∂f(x,y)/∂y = e^[(x^2+y^2)/xy] [2y + (y^4 - x^4)/(xy^2)] dz = e^[(x^2+y^2)/xy] {[2x + (x^4 - y^4)/(x^2y)] dx + [2y + (y^4 - x^4)/(xy^2)] dy }
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