多元微分学

全微分 dz

dz=∂z∂xΔx+∂z∂xΔy=∂z∂xdx+∂z∂xdy\mathrm{d}z = \frac{\partial z}{\partial x} \Delta x + \frac{\partial z}{\partial x} \Delta y = \frac{\partial z}{\partial x} \mathrm{d} x + \frac{\partial z}{\partial x} \mathrm{d} y
z=f[u(t),v(t)]z = f[u(t), v(t)]

求dzdt\frac{\mathrm{d} z}{\mathrm{d} t}

dzdt=∂z∂ududt+∂z∂vdvdt\frac{\mathrm{d} z}{\mathrm{d} t} = \frac{\partial z}{\partial u}\frac{\mathrm{d} u}{\mathrm{d} t} + \frac{\partial z}{\partial v}\frac{\mathrm{d} v}{\mathrm{d} t}
z=f[u(x,y),v(x,y)]z = f[u(x, y), v(x, y)]

求∂z∂x\frac{\partial z}{\partial x}

∂z∂x=∂z∂u∂u∂x+∂z∂v∂v∂x\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}
z=f[u(x,y),v(y)]z = f[u(x, y), v(y)]

求∂z∂y\frac{\partial z}{\partial y}

∂z∂y=∂z∂u∂u∂y+∂z∂v∂vdy\frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial z}{\partial v} \frac{\partial v}{\mathrm{d} y}
F(x,y),dydxF(x,y), \frac{\mathrm{d} y}{\mathrm{d} x} dydx=−Fx′Fy′\frac{\mathrm{d} y}{\mathrm{d} x} = - \frac{F_x'}{F_y'}
F(x,y,z),∂z∂xF(x,y,z), \frac{\partial z}{\partial x} ∂z∂x=−Fx′Fz′\frac{\partial z}{\partial x} = - \frac{F_x'}{F_z'}

f(x,y)在 P 点取极值的必要条件是

fx′(x0,y0)=0,fy′(x0,y0)=0f_x'(x_0, y_0) = 0, f_y'(x_0, y_0) = 0

f(x,y)在 P 点取极值的充分条件是

{fxx′′(x0,y0)=Afxy′′(x0,y0)=Bfyy′′(x0,y0)=C,B2−AC<0\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC < 0
{fxx′′(x0,y0)=Afxy′′(x0,y0)=Bfyy′′(x0,y0)=C,B2−AC<0,A<0\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC < 0, A < 0

f(x,y)在该点取极大值

{fxx′′(x0,y0)=Afxy′′(x0,y0)=Bfyy′′(x0,y0)=C,B2−AC<0,A>0\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC < 0, A > 0

f(x,y)在该点取极小值

{fxx′′(x0,y0)=Afxy′′(x0,y0)=Bfyy′′(x0,y0)=C,B2−AC>0\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC > 0

f(x,y)在该点非极值

{fxx′′(x0,y0)=Afxy′′(x0,y0)=Bfyy′′(x0,y0)=C,B2−AC=0\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC = 0

无法判断

{ϕ(x,y,z)=0ψ(x,y,z)=0\begin{cases} \phi(x,y,z) = 0 \\ \psi(x,y,z) = 0 \\ \end{cases}

求 u=f(x,y,z)在上面条件下的最值，列出拉格朗日乘数公式

F(x,y,z,λ,μ),{Fx′=fx′+λϕx′+μψx′=0Fy′=fy′+λϕy′+μψy′=0Fz′=fz′+λϕz′+μψz′=0Fλ′=ϕ(x,y,z)=0Fμ′=ψ(x,y,z)=0F(x,y,z,\lambda, \mu), \begin{cases} F_x' = f_x' + \lambda\phi_x' + \mu\psi_x' = 0 \\ F_y' = f_y' + \lambda\phi_y' + \mu\psi_y' = 0 \\ F_z' = f_z' + \lambda\phi_z' + \mu\psi_z' = 0 \\ F_\lambda' = \phi(x, y, z) = 0 \\ F_\mu' = \psi(x, y, z) = 0 \\ \end{cases}

带入备选点如(1,1,1),(2,2,2)取最大值和最小值为所求