4.7 偏导数

多元函数偏导与全微分

例：设函数$z= z(x, y)$由方程$z=e^{2x-3y} + 2y$确定，求$3\left ( \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \right )|_{x=3, y=2}$的值

[]:f=Lambda((x, y), exp(2*x-3*y)+2*y)

(3*f(x, y).diff(x)+f(x, y).diff(y)).subs(x, 3).subs(y, 2)

[]:

例：求函数$z=\cos \sqrt{x^2 + y^2}$的偏导数$\frac{\partial z}{\partial x}$, $\frac{\partial z}{\partial y}$, $\frac{\partial^{2} z}{\partial x \partial y}$，全微分$dz$以及$dz|_{x=1, y=2}$

[]:f=Lambda((x, y), cos(sqrt(x\*\*2+y\*\*2)))*

f(x, y).diff(x), f(x, y).diff(y), f(x, y).diff(x, y)*

*[]:*![](../media/cc4a473aa8a3a2fd18312a0143a6fa1e.png)

*[]:float(f(x, y).diff(x).subs(x,1).subs(y, 2)), float(f(x, y).diff(y).subs(x,
1).subs(y, 2))*

[]:

例：设函数$f(u)$可微，且$f'(0)=\frac{1}{2}$，求$z=f(4x^2-y^2)$在点$(1,2)$处的全微分$dz|_{(1, 2)}$。

[]:f=Function('f')

x, y =symbols('x y')

f(4\*x\*\*2-y\*\*2).diff(x).subs(x, 1).subs(y, 2),
f(4\*x\*\*2-y\*\*2).diff(y).subs(x, 1).subs(y, 2)

[]: ![](../media/45ecafafc2c5ebd409fe94d3be3754d6.png)

例：设$z=(x^2+y^2)e^{-\arctan \frac{y}{x}}$，求$dz$与$\frac{\partial^2 z}{\partial x \partial y}$。

[]:f=Lambda((x, y), (x\*\*2+y\*\*2)\*exp(-atan(y/x)))

f(x, y).diff(x).simplify(), f(x, y).diff(y).simplify(), f(x, y).diff(x,
y).simplify()

[]:

多元函数极值与最值

例：求函数$f(x,y)=x^2+y^2$的极值

[]:f=Lambda((x, y), x\*\*2+y\*\*2)

linsolve([f(x, y).diff(x), f(x, y).diff(y)], (x, y))

[]:

[]:(f(x, y).diff(x, y))\*\*2-f(x, y).diff(x, x)-f(x, y).diff(y, y)

[]:

[]:f(x, y).diff(x, x).subs(x, 0)

[]: