[微分] z=f(x,y)=g(r,Θ) 利用隱函數證明
z=f(x,y)=g(r,Θ)利用隱函數證明
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
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