作者LeoRen (Eugene Hecht)
看板trans_math
标题Re: [考古] 94成大
时间Wed Jun 25 21:59:13 2008
※ 引述《jbsh (乡民)》之铭言：
: Let F(x1,x2)be a real-valued function defined on R^2 with partial derivatives
: Fi=dF/dxi for i=1,2 .Suppose that variable v is implicitly defined as a
: function of u by the equation F(u-v,u+v)=0 .find dv/du in terms if F1 and
: F2
依题意: F为x1,x2之函数且 x1=u-v, x2=u+v, v是u的隐函数
故F最终为u的函数! 图解示意..
F
/ \
x1 x2
/ \ / \
u v u v
| |
u u
以下 "@" 表偏微，原po题目打错，应为 Fi=@F/@xi for i=1,2
x1=u-v, x2=u+v
F(u-v,u+v)=0 => dF/du =0
dF/du = (@F/@x1)(@x1/@u) + (@F/@x1)(@x1/@v)(dv/du)
+ (@F/@x2)(@x2/@u) + (@F/@x2)(@x2/@v)(dv/du)
= F1*1 + F1*(-1)*(dv/du) + F2*1 + F2*1*(dv/du)
= (F1+F2) + (dv/du)*(F2-F1)
= 0
<=> dv/du = (F1+F2)/(F1-F2)
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