高等数学-多元微分学习题
考察多元函数定义、极限存在、连续性、可微、可偏导、泰勒定理
多元函数的定义
例1
image-20200702122423759例2
image-20200702122611230偏导数的定义
偏导数是把多元函数其中一元看作变量，其他元看作常数后，求函数的变化率（求导）
例1
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image-20200713145725465例7
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image-20200713162250977极值的定义
例1
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image-20200714171451504连续不一定可偏导，可偏导不一定连续
例1 连续不一定可偏导，可偏导不一定连续
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image-20200711093943298可微必连续
例1 证明可微必连续
可微必可导
例1 证明可微必可偏导
求证：如果函数 \(z=f(x, y)\) 在点 \((x, y)\) 可微分,那么该函数在点\((x, y)\) 的偏导数 \(\frac{\partial z}{\partial x}\) 与 \(\frac{\partial z}{\partial y}\) 必定存在 \(,\) 且函数 \(z=f(x, y)\) 在点 \((x, y)\) 的全微分为\(\mathrm{d} z=\frac{\partial z}{\partial x} \Delta x+\frac{\partial z}{\partial y} \Delta y\)
image-20200701153847161例2 可微必可偏导
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image-20200711002115138连续可偏导必可微
例1 证明连续可偏导必可微
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image-20200701160157106例2
image-20200713122053068连续性、可偏导、可微的判断
连续性判断： 用连续定义判断：求该点处的极限，若极限值=对应点值，则连续。否则存在某路径极限值\(\neq\)对应点值，在该点不连续。
可偏导性的判断： 用偏导数的定义判断/求偏导数
可微的判断： 1）可微的定义判断 2）函数连续可偏导必可微 3）可微必可偏导、可微必连续的逆否命题：不可偏导一定不可微、不连续一定不可微
例1 函数连续性、可偏导性、可微的判断
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image-20200711094523571例2 函数连续性、可偏导性、可微性的判断
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image-20200711094948922例3 函数连续性的判断
image-20200712233234132例4 函数连续性、可偏导性的判断
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image-20200712234115590例5 函数连续性、可偏导性、可微性的判断
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image-20200713112654421例6 可微的判断
2020-7-13-001例7 可微的判断
2020-7-13-002例8 可微的判断
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image-20200713143240883二元函数泰勒定理
例1
image-20200714173833094极限的证明与计算
例1
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image-20200627234814941例3
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image-20200709115355220例4
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2020-7-12-001偏导数的计算
普通多元函数偏导数的计算
例1
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image-20200712233907028例9
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image-20200713144708235多元复合函数偏导数的计算
例1
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image-20200630211557621例9
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image-20200709115812472例10
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image-20200709120026906例12
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image-20200709120119794例13
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注意一下这里偏导数代入的是不同的值
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与变限积分函数复合，求偏导
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image-20200711101524524例27
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image-20200711142540183例28
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image-20200711145114756例29 换元/换坐标系
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image-20200711174416018例30 换元/换坐标系
拉普拉斯算子在直角坐标系与柱坐标系中的转换
参考：拉普拉斯算子的百度百科
高数同济第七版P82的证明：
还有csdn上博主的证明（与高数同济7的证明类似）：https://blog.csdn.net/u013102281/article/details/70800631
例31
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image-20200713145111578例32 换元求偏导
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image-20200713155710640例41
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image-20200713164329698多元隐函数及方程组的偏导数的计算
例1
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image-20200713172803624全微分的计算
已知函数或隐函数求全微分
求多元函数的全微分，可以对函数两边取微分，直接得全微分。 如果是多元隐函数，或者方程组，可以对方程两边求各自由元的偏导数。利用偏导数得全微分。
例1
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image-20200713165427769已知部分偏导信息求全微分
例1
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image-20200710231054533已知极限求全微分
例1
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image-20200709141659190已知全微分求参数
例1
image-20200713151304137多元微分学代数应用：求多元函数极值
极值/最值问题，其实就是最优化问题。
注意：极值问题，可能取值的的位置，包括驻点、不可导点、边界点。
无条件极值的定义域为开区域，考虑的是定义域内的驻点是否取极值。 条件极值多了约束，考虑的是有约束的情况下是否取极值。（如果约束正好是对应无条件极值的边界，则条件极值考虑的是边界点是否取极值）
例如求 \(f(x, y)\) 在区域 \(D=\left\{(x, y) \mid x^{2}+4 y^{2} \leqslant 4\right\}\) 上的极值/最值。 可以拆成两部分来求： 在区域 \(D_1=\left\{(x, y) \mid x^{2}+4 y^{2} < 4\right\}\) 上找到所有的驻点判断是否取极值（求非条件极值） 在区域 \(D_2=\left\{(x, y) \mid x^{2}+4 y^{2} = 4\right\}\) 上，即给定约束\(x^{2}+4 y^{2} = 4\)的条件下，求函数的极值（求条件极值）
实际问题中，很多问题只在开区域内取最值，这个时候，可以不考虑在边界上取极值的情况（不用考虑条件极值）
无条件极值
无条件极值要求定义域为开区域
例1
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image-20200713174923357条件极值
例1
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image-20200714110146199多元微分学几何/向量分析应用：
空间曲线切向量和曲面法向量
空间曲线切向量的计算
切向量为\((x_t^\prime, y_t^\prime, z_t^\prime)\)或者\((1,\frac{dy}{dx}, \frac{dz}{dx})\)
参考答案是代隐函数方程组求偏导的Jacobi公式。实际上，求解过程一般就是解\(\frac{dy}{dx}, \frac{dz}{dx}\)的线性方程组的过程。
例1
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image-20200710000906166空间曲面法向量的计算
空间曲面\(F(x,y,z)\)上任一点法向量为\((F_x,F_y,F_z)\)
例1
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image-20200714172733799例7
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image-20200714173216173向导数与梯度的计算
方向导数的计算
（方向导数是个值哦，不是向量；只是因为和梯度联系紧密，放到了这里）
例1
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image-20200714145523955梯度的计算
例1
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