多元微分学

全微分 dz

$$\mathrm{d}z = \frac{\partial z}{\partial x} \Delta x + \frac{\partial z}{\partial x} \Delta y = \frac{\partial z}{\partial x} \mathrm{d} x + \frac{\partial z}{\partial x} \mathrm{d} y$$

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$$z = f[u(t), v(t)]$$

求$\frac{\mathrm{d} z}{\mathrm{d} t}$

$$\frac{\mathrm{d} z}{\mathrm{d} t} = \frac{\partial z}{\partial u}\frac{\mathrm{d} u}{\mathrm{d} t} + \frac{\partial z}{\partial v}\frac{\mathrm{d} v}{\mathrm{d} t}$$

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$$z = f[u(x, y), v(x, y)]$$

求$\frac{\partial z}{\partial x}$

$$\frac{\partial z}{\partial x} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial x} + \frac{\partial z}{\partial v} \frac{\partial v}{\partial x}$$

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$$z = f[u(x, y), v(y)]$$

求$\frac{\partial z}{\partial y}$

$$\frac{\partial z}{\partial y} = \frac{\partial z}{\partial u} \frac{\partial u}{\partial y} + \frac{\partial z}{\partial v} \frac{\partial v}{\mathrm{d} y}$$

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$$F(x,y), \frac{\mathrm{d} y}{\mathrm{d} x}$$ $$\frac{\mathrm{d} y}{\mathrm{d} x} = - \frac{F_x'}{F_y'}$$

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$$F(x,y,z), \frac{\partial z}{\partial x}$$ $$\frac{\partial z}{\partial x} = - \frac{F_x'}{F_z'}$$

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f(x,y)在 P 点取极值的必要条件是

$$f_x'(x_0, y_0) = 0, f_y'(x_0, y_0) = 0$$

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f(x,y)在 P 点取极值的充分条件是

$$\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC < 0$$

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$$\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC < 0, A < 0$$

f(x,y)在该点取极大值

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$$\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC < 0, A > 0$$

f(x,y)在该点取极小值

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$$\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC > 0$$

f(x,y)在该点非极值

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$$\begin{cases} f_{xx}''(x_0, y_0) = A \\ f_{xy}''(x_0, y_0) = B \\ f_{yy}''(x_0, y_0) = C \\ \end{cases}, B^2-AC = 0$$

无法判断

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

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

$$F(x,y,z,\lambda, \mu), \begin{cases} F_x' = f_x' + \lambda\phi_x' + \mu\psi_x' = 0 \\ F_y' = f_y' + \lambda\phi_y' + \mu\psi_y' = 0 \\ F_z' = f_z' + \lambda\phi_z' + \mu\psi_z' = 0 \\ F_\lambda' = \phi(x, y, z) = 0 \\ F_\mu' = \psi(x, y, z) = 0 \\ \end{cases}$$

带入备选点如(1,1,1),(2,2,2)取最大值和最小值为所求