用二阶偏导数来判定函数f(x,y)在其驻点(x,y_0)处的极值,有时可能有判别式f_(xy)~2(x_0,y_0)-f_(xx)(X_0,y)·f_y(x,y_0)等于零的情况.这时,原来的判别法失效,从而需要作出进一步的考察.为此,本文特给出一种利用一般的高阶偏导数的...用二阶偏导数来判定函数f(x,y)在其驻点(x,y_0)处的极值,有时可能有判别式f_(xy)~2(x_0,y_0)-f_(xx)(X_0,y)·f_y(x,y_0)等于零的情况.这时,原来的判别法失效,从而需要作出进一步的考察.为此,本文特给出一种利用一般的高阶偏导数的判别方法.设函数f(x,y)在点(x,y_0)处可展开成n阶泰勒公式,并将其写成△f=P(h,k)+ε.式中P_n(h,k)=sum from m=1 to n(1/(m+1)!)(h((?)/(?)x)+(k(?)/(?)y))^(m 1)f(x,y_0);当ρ趋于零时ε趋于零.同时还设函数f(x,y)在点(x,y_0)处所有阶数不大于某个正整数N的偏导数都等于零,或在点(x,y_0)的某个邻域内所有阶数大于N+1的偏导数都恒等于零.那末,二元函数极值的高阶偏导数判别法可简单地归结为:若P_N(h,k)恒正或恒负,则f(x,y)在点(x_0,y_0)取得极值;若P_N(h,k)有正有负,则f(x,y)在点(x_0,y_0)处不取极值.展开更多
There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle th...There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation （or Helmholtz equation） arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.展开更多
文摘用二阶偏导数来判定函数f(x,y)在其驻点(x,y_0)处的极值,有时可能有判别式f_(xy)~2(x_0,y_0)-f_(xx)(X_0,y)·f_y(x,y_0)等于零的情况.这时,原来的判别法失效,从而需要作出进一步的考察.为此,本文特给出一种利用一般的高阶偏导数的判别方法.设函数f(x,y)在点(x,y_0)处可展开成n阶泰勒公式,并将其写成△f=P(h,k)+ε.式中P_n(h,k)=sum from m=1 to n(1/(m+1)!)(h((?)/(?)x)+(k(?)/(?)y))^(m 1)f(x,y_0);当ρ趋于零时ε趋于零.同时还设函数f(x,y)在点(x,y_0)处所有阶数不大于某个正整数N的偏导数都等于零,或在点(x,y_0)的某个邻域内所有阶数大于N+1的偏导数都恒等于零.那末,二元函数极值的高阶偏导数判别法可简单地归结为:若P_N(h,k)恒正或恒负,则f(x,y)在点(x_0,y_0)取得极值;若P_N(h,k)有正有负,则f(x,y)在点(x_0,y_0)处不取极值.
文摘There is a large class of problems in the field of fluid structure interaction where higher-order boundary conditions arise for a second-order partial differential equation. Various methods are being used to tackle these kind of mixed boundary-value problems associated with the Laplace’s equation （or Helmholtz equation） arising in the study of waves propagating through solids or fluids. One of the widely used methods in wave structure interaction is the multipole expansion method. This expansion involves a general combination of a regular wave, a wave source, a wave dipole and a regular wave-free part. The wave-free part can be further expanded in terms of wave-free multipoles which are termed as wave-free potentials. These are singular solutions of Laplace’s equation or two-dimensional Helmholz equation. Construction of these wave-free potentials and multipoles are presented here in a systematic manner for a number of situations such as two-dimensional non-oblique and oblique waves, three dimensional waves in two-layer fluid with free surface condition with higher order partial derivative are considered. In particular, these are obtained taking into account of the effect of the presence of surface tension at the free surface and also in the presence of an ice-cover modelled as a thin elastic plate. Also for limiting case, it can be shown that the multipoles and wave-free potential functions go over to the single layer multipoles and wave-free potential.
