Question

помогите пожалуйста найти полный дифференциал второго порядка
u=x+xy+1​

Answer 1

u = x + xy + 1

$d^2u=\dfrac{\partial^2 u}{\partial x^2}dx^2+2\dfrac{\partial^2 u}{\partial x \partial y}dxdy+\dfrac{\partial^2 u}{\partial y^2}dy^2\\ \dfrac{\partial u}{\partial x}=1+y;\ \dfrac{\partial u}{\partial y}=x;\\ \dfrac{\partial^2 u}{\partial x^2}=0;\ \dfrac{\partial^2 u}{\partial y^2}=0;\ \dfrac{\partial}{\partial x}(\dfrac{\partial u}{\partial y})=\dfrac{\partial}{\partial y}(\dfrac{\partial u}{\partial x})=1\\ \\ \Rightarrow\ d^2u = 0dx^2+2dxdy+0dy^2=2dxdy$