z=f(x,y)=g(r,Θ)利用隐函数证明 In parts(a)-(e),suppose that the equation z=f(x,y) is expressed in the polar from z=g(r,Θ) by making the sub-stitution x=rcosΘ and y=rsinΘ. (a)View r and Θ as functions of x and y and use implicit differentiation to show that dr dΘ sinΘ ____ =cosΘ and ___ = - _____ dx dx r (b)View r and Θ as functions of x and y and use implicit gifferentiation to show that dr dΘ cosΘ ____ =sinΘ and ___ = _____ dy dy r (c)Use the results in parts (a) and (b) to show that dz dz 1 dz ____ = ____cosΘ- ___ ____sinΘ dx dr r dΘ dz dz 1 dz ____ = ____sinΘ+ ___ ____cosΘ dy dr r dΘ (d)Use the result in part (c) to show that dz dz dz 1 dz (__)^2 + (__)^2 = (__)^2 + ___ (____)^2 dx dy dr r^2 dΘ (e)Use the result in part(c) to show that if z=f(x,y) satisfies Laplace's equation d^2z d^2z ______ + ______ = 0 dx^2 dy^2 then z=(r,Θ) satisfies the equation d^2z 1 d^2z 1 dz ____ + ___ _____ + ___ ____ = 0 dr^2 r^2 dΘ^2 r dr and conversely.The latter equation is called the polar form of Laplace's equation 如果打的太麻烦可以寄电子档给我,我的雅虎信箱:[email protected] 愿意的话也可以加我及时通:x213235x 5/24日约在台北市交我....车费我出 --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 61.231.8.153