F B I W A I N I N G ! \color{red}FBI \quad WAINING ! FBIWAINING!
设 f , g 二阶连续可微, u = y f ( x y ) + x g ( y x ) , 求 x ∂ 2 u ∂ x 2 + y ∂ 2 u ∂ x ∂ y \color{green}设f,g二阶连续可微,u=y f(\frac{x}{y}) + xg(\frac{y}{x}),求x\frac{\partial^2u}{\partial x^2} + y\frac{\partial^2u}{\partial x \partial y} 设f,g二阶连续可微,u=yf(yx)+xg(xy),求x∂x2∂2u+y∂x∂y∂2u
解:
∂ u / ∂ x = f x ′ ( x y ) + g ( y x ) − y x g x ′ ( y x ) \partial u/\partial x = f'_x(\frac{x}{y}) + g(\frac{y}{x}) - \frac{y}{x}g'_x(\frac{y}{x}) ∂u/∂x=fx′(yx)+g(xy)−xygx′(xy)
∂ 2 u ∂ x 2 = 1 y f x x ′ ′ ( x y ) − y x 2 g x ′ ( y x ) + y x 2 g x ′ ( y x ) + y 2 x 3 g x x ′ ′ ( y x ) = 1 y f x x ′ ′ ( x y ) + y 2 x 3 g x x ′ ′ ( y x ) \frac {\partial ^2u}{\partial x^2} = \frac{1}{y} f''_{xx}(\frac{x}{y}) - \frac{y}{x^2}g'_x(\frac{y}{x}) +\frac{y}{x^2}g'_x(\frac{y}{x}) + \frac{y^2}{x^3}g''_{xx}(\frac{y}{x})\\ = \frac{1}{y} f''_{xx}(\frac{x}{y}) + \frac{y^2}{x^3}g''_{xx}(\frac{y}{x}) ∂x2∂2u=y1fxx′′(yx)−x2ygx′(xy)+x2ygx′(xy)+x3y2gxx′′(xy)=y1fxx′′(yx)+x3y2gxx′′(xy)
∂ 2 u ∂ x ∂ y = − x y 2 f x y ′ ′ ( x y ) − 1 x g y ′ ( y x ) + 1 x g x ′ ( y x ) − y x 2 g x x ′ ′ ( y x ) = − x y 2 f x y ′ ′ ( x y ) − y x 2 g x x ′ ′ ( y x ) \frac {\partial ^2u}{\partial x \partial y} = -\frac{x}{y^2} f''_{xy}(\frac{x}{y}) - \frac{1}{x}g'_y(\frac{y}{x}) +\frac{1}{x}g'_x(\frac{y}{x}) - \frac{y}{x^2}g''_{xx}(\frac{y}{x})\\ = -\frac{x}{y^2} f''_{xy}(\frac{x}{y}) - \frac{y}{x^2}g''_{xx}(\frac{y}{x}) ∂x∂y∂2u=−y2xfxy′′(yx)−x1gy′(xy)+x1gx′(xy)−x2ygxx′′(xy)=−y2xfxy′′(yx)−x2ygxx′′(xy)
x ∂ 2 u ∂ x 2 + y ∂ 2 u ∂ x ∂ y = x y f x x ′ ′ ( x y ) + y 2 x 2 g x x ′ ′ ( y x ) − x y f x y ′ ′ ( x y ) − y 2 x 2 g x x ′ ′ ( y x ) = 0 x\frac{\partial^2u}{\partial x^2} + y \frac{\partial^2u}{\partial x \partial y} = \\ \frac{x}{y} f''_{xx}(\frac{x}{y}) + \frac{y^2}{x^2}g''_{xx}(\frac{y}{x}) -\frac{x}{y} f''_{xy}(\frac{x}{y}) - \frac{y^2}{x^2}g''_{xx}(\frac{y}{x}) = 0 x∂x2∂2u+y∂x∂y∂2u=yxfxx′′(yx)+x2y2gxx′′(xy)−yxfxy′′(yx)−x2y2gxx′′(xy)=0文章来源:https://www.toymoban.com/news/detail-670961.html
注意:下标应该是改成对 y x 求导,而不是对 x 和对 y , 此处的 g x x ′ ′ 与 g x y ′ ′ 是一样的,最后结果为 0 \color{red}注意:下标应该是改成对\frac{y}{x}求导,而不是对x和对y,\\ 此处的g''_{xx}与g''_{xy}是一样的,最后结果为0 注意:下标应该是改成对xy求导,而不是对x和对y,此处的gxx′′与gxy′′是一样的,最后结果为0文章来源地址https://www.toymoban.com/news/detail-670961.html
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