数量函数

偏导数与微分

偏导数 将 `n` 元数量函数 `f` 视为其中一个变元 `x^i` 的函数, 其它变元固定, 就得到一元函数 `varphi: x^i mapsto f(x^1, cdots, x^n)`. 定义 `f` 关于变元 `x^i` 的偏导数为 `(del f)/(del x^i) = varphi'(x^i)`, 即 `(del f)/(del x^i)|_(x_0)` `= lim_(h to 0) (f(x_0^1, cdots, x_0^i + h, cdots, x_0^n) - f(x_0^1, cdots, x_0^i, cdots, x_0^n)) / h`.

微分 设 `n` 元数量函数 `f` 在 `x_0 in RR^n` 的邻域上有定义, 若 `f` 在 `x_0` 附近的变化 `f(x_0 + h) - f(x_0)` 近似于一个线性函数 `varphi: h mapsto alpha * h`, 其中 `alpha, h in RR^n`, "`*`" 表示向量内积: `f(x_0 + h) = f(x_0) + alpha * h + o(|h|)`, `quad h to 0`, 即 `lim_(h to 0) (f(x_0+h) - f(x_0) - alpha * h)/|h| = 0`, 就称 `f` 在 `x_0` 可微. 线性函数 `varphi: h mapsto alpha * h` 称为 `f` 在 `x_0` 的微分, 一般用 `"d"f` 表示因变元, `dx` 表示自变元, 记为 `"d"f(x_0) = alpha * dx = alpha_1 dx^1 + cdots + alpha_n dx^n`.

可微必连续 设 `f` 在 `x_0` 处可微, 在 两边令 `h to 0` 就得到 `lim_(h to 0) f(x_0 + h) = f(x_0)`.

可微则偏导数存在 设 `f` 在 `x_0` 处可微, 在 中取 `h = (0, cdots, h_i, cdots, 0)` 就有 `f(x_0 + h) = f(x_0) + alpha_i h_i + o(|h_i|)` 故 `(del f)/(del x^i)(x_0) = lim_(h to 0) (f(x_0 + h) - f(x_0))/h_i = alpha_i`.

偏导连续则可微 设 `f` 在 `x_0` 的邻域上关于各变元存在偏导数, 且这些偏导数在 `x_0` 处连续, 则 `f` 在 `x_0` 可微.

记号简单起见, 以二元函数为例. 在 `(x_0, y_0)` 的邻域上应用一元函数的微分中值定理, `f(x, y) - f(x_0, y_0)` `= f(x, y) - f(x_0, y) + f(x_0, y) - f(x_0, y_0)` `= (del f)/(del x)(xi, y)(x-x_0) + (del f)/(del y)(x_0, eta)(y-y_0)`, 其中 `xi` 在 `x_0` 与 `x` 之间, `eta` 在 `y_0` 与 `y` 之间. 由于偏导数连续, 当 `(x, y) to (x_0, y_0)` 时, `(del f)/(del x)(xi, y) to (del f)/(del x)(x_0, y_0)`, `quad (del f)/(del y)(x_0, eta) to (del f)/(del y)(x_0, y_0)`. 从而 `f(x, y) - f(x_0, y_0)` `= (del f)/(del x)(x_0, y_0)(x-x_0) + o(|x-x_0|)` `+ (del f)/(del y)(x_0, y_0)(y-y_0) + o(|y-y_0|)` `= (del f)/(del x)(x_0, y_0)(x-x_0) + (del f)/(del y)(x_0, y_0)(y-y_0) + o(r)`, 其中 `r = sqrt((x-x_0)^2 + (y-y_0)^2)`, 即 `f` 在 `(x_0, y_0)` 可微.

若 `f` 在定义域 `D` 上关于各变元有连续的偏导数, 就称它连续可微, 记为 `f in C^1(D)`.

混合偏导数

一阶偏导可微则混合偏导相等 (Young) 设二元函数 `f` 在 `P_0(x_0, y_0)` 的邻域上存在偏导数, 且 `f_x`, `f_y` 都在 `P_0` 可微, 则 `f_(x y)(x_0, y_0) = f_(y x)(x_0, y_0)`.

构造函数 `g(h) = f(x_0+h,y_0+h) - f(x_0+h,y_0) - f(x_0,y_0+h) + f(x_0,y_0)`, 下证 `g(h) = f_(y x)(x_0,y_0) h^2 + o(h^2)` `= f_(x y)(x_0,y_0) h^2 + o(h^2)`, `quad h to 0`, 从而得到定理的结论. 事实上对固定的充分小的 `h`, 令 `varphi(x) = f(x, y_0 + h) - f(x, y_0)`, 则由微分中值定理, `g(h) = varphi(x_0+h) - varphi(x_0)` `= varphi'(x_0 + theta h) h` `= [f_x(x_0 + theta h, y_0 + h) - f_x(x_0 + theta h, y_0)]h`. 注意在 `P_0` 处 `f_x` 可微, 故成立 `(g(h))/h` `= (f_x + f_(x x)theta h + f_(y x) h + o(h))` `- (f_x + f_(x x)theta h + o(h))` `= f_(y x) h + o(h)` `quad h to 0`. 同理考虑函数 `psi(y) = f(x_0 + h, y) - f(x_0, y)` 可得另一个等式.

混合偏导连续则相等 (Schwartz) 设二元函数 `f` 在 `P_0(x_0, y_0)` 的邻域上存在二阶混合偏导数 `f_(x y)`, `f_(y x)`, 且两个混合偏导数在 `P_0` 连续, 则 `f_(x y)(x_0, y_0) = f_(y x)(x_0, y_0)`.

记 `g(h, k) = f(x_0+h,y_0+k) - f(x_0+h,y_0) - f(x_0,y_0+k) + f(x_0,y_0)`, 则 `f_(x y)(x_0,y_0) = lim_(h to 0) lim_(k to 0) (g(h, k))/(h k)`,
`f_(y x)(x_0,y_0) = lim_(k to 0) lim_(h to 0) (g(h, k))/(h k)`. 下面证明这两个累次极限可交换. 对固定的充分小的 `k`, 令 `varphi(x) = f(x, y_0 + k) - f(x, y_0)`, 运用两次微分中值定理有 `g(h, k)` `= varphi(x_0+h) - varphi(x_0)` `= varphi'(x_0 + theta_1 h) k` `= [f_x(x_0 + theta_1 h, y_0 + k) - f_x(x_0 + theta_1 h, y_0)] k` `= f_(y x)(x_0 + theta_1 h, y_0 + theta_2 k) h k`. 令 `(h, k) to (0, 0)`, 由 `f_(y x)` 在 `P_0` 连续得 `lim_((h,k) to (0,0)) g(h, k)/(h k) to f_(y x)(x_0, y_0)`. 同理考虑函数 `psi(y) = f(x_0+h, y) - f(x_0, y)` 可得 `lim_((h,k) to (0,0)) g(h, k)/(h k) to f_(x y)(x_0, y_0)`. 因此两个混合偏导数在 `(x_0, y_0)` 相等.

1. `k` 阶可微 若 `n` 元数量函数 `f` 在 `x_0` 的邻域上存在各个 `k-1` 阶偏导数 (`k ge 2`), 且所有 `k-1` 阶偏导数在 `x_0` 可微, 则称 `f` 在 `x_0` 是 `k` 阶可微的.
2. `k` 阶可微蕴涵 `k-1` 阶可微 若 `f` 在 `x_0` 处二阶可微, 由定义 `f` 的所有一阶偏导数连续, 从而 `f` 在 `x_0` 处可微. 一般地, `k` 阶可微蕴涵 `k-1` 阶可微.
3. `k` 阶连续可微 若 `f` 在定义域 `D` 上有 `k` 阶偏导数, 且所有 `k` 阶偏导数在 `D` 上连续, 于是 `f` 在 `D` 上的 `k-1` 阶偏导数均可微, 我们称 `f` 在 `D` 上 `k` 阶连续可微, 记为 `f in C^k(D)`. `k` 阶连续可微蕴涵 `k` 阶可微.

由 Young 定理得: 若 `f` 在 `x_0` 处 `k` 阶可微, 则它在该点直到 `k` 阶的所有混合偏导数都与次序无关.

隐函数

[来自 同济高数第七版] 已知两个连续可微函数 `f, F`, 满足 `y = f(x, t)`, 且 `F(x, y, t) = 0` 确定一个隐函数 `t(x, y)`. 求 `dy/dx`.

由隐函数定理知 `(del t)/(del x) = -F_1' // F_3'`, `(del t)/(del y) = -F_2' // F_3'`. 使用全微分的链式法则, `dy/dx = f_1' + f_2' dt/dx` `= f_1' + f_2'((del t)/(del x) + (del t)/(del y) dy/dx)`. 解得 `dy/dx = (f_1' + f_2' (del t)/(del x))/(1 - f_1'(del t)/(del y))` `= (f_1' F_3' - f_2' F_1')/(F_3' + f_2' F_2')`.

齐次函数

齐次函数的 Euler 定理 设 `f(bm x)` 是 `D sube RR^n` 到 `RR` 的可微函数, `alpha in RR`, 则
1. 若 `f` 是 `alpha` 次齐次函数, 即 `f` 满足以下函数方程: `f(k bm x) = k^alpha f(bm x)`, `quad AA k in RR`, 则 `(del f)/(del bm x) := ((del f)/(del x_1), cdots, (del f)/(del x_n))` 是 `alpha-1` 次齐次函数.
2. `f` 是齐次函数当且仅当它满足下面的偏微分方程 (Euler 方程): `(del f)/(del bm x) * bm x = alpha f(bm x)`.

1. 在函数方程 两边对 `bm x` 求导, `k (del f)/(del bm x) (k bm x) = k^alpha (del f)/(del bm x)`, 这指出 `(del f)/(del bm x)` 是 `alpha-1` 次齐次的.
2. 先设 `f` 为 `alpha` 次齐次函数. 在函数方程 两边对 `k` 求导, `(del f)/(del bm x)(k bm x) * bm x = alpha k^(alpha-1) f(bm x)`, 取 `k = 1` 即得 .
   再设 Euler 方程成立, 取 `bm x = k bm x` 有 `(del f)/(del bm x)(k bm x) * (k bm x) = alpha f(k bm x)`, 于是 `"d"/("d"k) f(k bm x)` `= (del f)/(del bm x)(k bm x) * bm x` `= alpha/k f(k bm x)`. 这是关于函数 `g(k) = f(k bm x)` 的常微分方程, 通解为 `f(k bm x) = c k^alpha`. 取 `k = 1` 即得 `c = f(bm x)`, 因此 `f(k bm x) = k^alpha f(bm x)`.

`f(x, y) = x^2 + 3 x y + y^2` 是二次齐次函数, 我们有 `(del f)/(del x) x + (del f)/(del y) y = 2 f`.

[来自 我是得不到的i] 若 `f(x, y)` 是可微的零阶齐次函数, `y != 0`, 则存在一元可微函数 `varphi` 使得 `f = varphi(x/y)`.

设 `u = x/y`, `v = y`, 换元得 `f(x, y) = varphi(u, v)`. 下证 `(del varphi)/(del v) = 0`, 从而 `varphi` 仅依赖于 `u`. 事实上, 由 Euler 方程 `0 = x (del f)/(del x) + y (del f)/(del y)` `= x ((del varphi)/(del u) 1/y + (del varphi)/(del v) * 0) + y ((del varphi)/(del u) (-x/y^2) + (del varphi)/(del v))` `= y (del varphi)/(del v)`.

向量函数

Jacobi 矩阵 的意义是向量对向量的偏导数, 它表示向量微元间的线性变换. 我们从一个简单例子入手: 设 `f(x,y)` 二阶连续可微, 令 `x = a u + b v`, `y = c u + d v`, `a, b, c, d` 为常数, 则 `(del f)/(del u) = a(del f)/(del x) + c(del f)/(del y)`, `quad (del f)/(del v) = b(del f)/(del x) + d(del f)/(del y)`. 如果记 `del/(del u) = a del/(del x) + c del/(del y)`, `del/(del v) = b del/(del x) + d del/(del y)`, 容易验证 `del^2/(del u^2) = (a del/(del x) + c del/(del y))^2` `= a^2 del^2/(del x^2) + 2a c del^2/(del x del y) + c^2 del^2/(del y^2)`,
`del^2/(del u del v) = (a del/(del x) + c del/(del y)) (b del/(del x) + d del/(del y))` `= a b del^2/(del x^2) + (a d+b c) del^2/(del x del y) + c d del^2/(del y^2)`,
`del^2/(del v^2) = (b del/(del x) + d del/(del y))^2` `= b^2 del^2/(del x^2) + 2b d del^2/(del x del y) + d^2 del^2/(del y^2)`. 以上结论可用矩阵简洁地表示: 注意虽然 `x = x(u, v)`, `y = y(u, v)` 可能为非线性变换, 但在 `f` 可微的条件下, 向量微元 `(dx, dy)` 和 `("d"u, "d"v)` 之间的变换是线性的. 这个线性变换记为: `[dx;dy] = J ["d"u;"d"v]`, `quad J = (del(x, y))/(del(u,v))`. 定义数量函数对向量的偏导数, `del/(del(u, v)) = (del/(del u), del/(del v))`, `del/(del(x, y)) = (del/(del x), del/(del y))`, 则有 `del/(del(u, v)) = del/(del(x,y)) (del(x,y))/(del(u,v))` `= del/(del(x,y)) J`,
`H_((u,v)) = [del^2/(del u^2), del^2/(del u del v); del^2/(del v del u), del^2/(del v^2)]` `= (del/(del u), del/(del v))^T (del/(del u), del/(del v))` `= J^T (del/(del x), del/(del y))^T (del/(del x), del/(del y)) J` `= J^T H_((x,y)) J`,
`dx^2 + dy^2 = ("d"u, "d"v) J^T J ["d"u; "d"v]`, 等等. 矩阵 `J` 称为 Jacobi 矩阵, `H` 称为 Hessian 矩阵.

设 `f(x, y)` 二阶连续可微, 且满足方程 `a(del^2 f)/(del x^2) + b(del^2 f)/(del x del y) + c(del^2 f)/(del y^2) = 0`, 其中 `a, b, c` 为常数, `a != 0`. 若特征方程 `a z^2 + b z + c = 0` 有两个不同的根 `alpha, beta`, 作变元代换 `u = alpha x + y`, `v = beta x + y`, 验证 `(del^2 f)/(del u del v) = 0`.

`(del f)/(del x) = alpha (del f)/(del u) + beta (del f)/(del v)`, `quad (del f)/(del y) = (del f)/(del u) + (del f)/(del v)`,
`(del^2 f)/(del x^2) = alpha^2 (del^2 f)/(del u^2) + 2 alpha beta (del^2 f)/(del u del v) + beta^2 (del^2 f)/(del v^2)`,
`(del^2 f)/(del x del y) = alpha (del^2 f)/(del u^2) + (alpha+beta) (del^2 f)/(del u del v) + beta (del^2 f)/(del v^2)`,
`(del^2 f)/(del y^2) = (del f^2)/(del u^2) + 2(del^2 f)/(del u del v) + (del^2 f)/(del v^2)`. 代入方程, 并利用 Vieta 定理得 `0 = [a(2alpha beta) + b(alpha+beta) + 2c] (del^2 f)/(del u del v)` `= (4a c - b^2)/a (del^2 f)/(del u del v)`. 由于特征方程有两根, 判别式 `b^2 - 4 a c != 0`, 最终得到 `(del^2 f)/(del u del v) = 0`.

设 `f` 二阶连续可微, `z = f(x+y, x y)`. 求 `pp^2 z (x y)`.

记 `u = x+y`, `v = x y`, 则 `pp z x` `= pp z u pp u x + pp z v pp v x` `= pp z u + pp z v * y`, `pp^2 z (x y)`
`= del/(del y)(pp z u + pp z v * y)`
`= del/(del u)(pp z u) pp u y + del/(del v)(pp z u) pp v y` `+ del/(del y)(pp z v) y + pp z v`
`= pp^2 z u^2 + pp^2 z (u v) * (x+y) + pp^2 z v^2 * x y + pp z v`.

设 `f` 二阶连续可微, `z = f(x+y, f(x, y))`. 求 `pp^2 z (x y)`.

记 `u = x+y`, `v = f(x, y)`, 则 `pp z y` `= pp z u pp u y + pp z v pp v y` `= pp z u + pp z v pp v y`,
`pp^2 z (x y)` `= pp^2 z u^2 pp u x + pp^2 z (v u) pp v x` `+ (pp^2 z (u v) pp u x + pp^2 z v^2 pp v x ) pp v y` `+ pp z v pp^2 v (x y)` `= f_(11)''(u, v) + f_(12)''(u, v) f_1'(x, y)` `+ (f_(21)''(u, v) + f_(22)''(u, v) f_1'(x, y))f_2'(x, y)` `+ f_2'(u, v) f_(21)''(x, y)`. 再将 `u, v` 用 `x, y` 代回, 即得到最终结果.

讨论二元数量函数的极坐标表示和三元数量函数的球坐标表示.

1. 设 `u(x, y)` 可微, 令 `x = r cos theta`, `y = r sin theta`, 则 `(del u)/(del r) = (del u)/(del x) cos theta + (del u)/(del y) sin theta`,
   `(del u)/(del theta) = r (-(del u)/(del x) sin theta + (del u)/(del y) cos theta)`. 代换的 Jacobi 行列式为 `|(del x)/(del r), (del x)/(del theta); (del y)/(del r), (del y)/(del theta)| = r`. 解线性方程组得 `grad u = ((del u)/(del x), (del u)/(del y))` `= ((del u)/(del r) cos theta - 1/r (del u)/(del theta) sin theta, (del u)/(del r) sin theta + 1/r (del u)/(del theta) cos theta)` `= (x/r, y/r) (del u)/(del r) + 1/r (-y/r, x/r) (del u)/(del theta)` `:= (del u)/(del r) bm e_r + 1/r (del u)/(del theta) bm e_theta`. 其中 `bm e_r = ((x","y))/r` 表示极径 `r` 方向的单位向量, `bm e_theta = ((-y","x))/r` 表示极角 `theta` 方向的单位向量.
2. 设 `u(x, y, z)` 可微, 令 `x = r cos theta cos varphi`, `y = r sin theta cos varphi`, `z = r sin varphi`, 则 `(del u)/(del r) = (del u)/(del x) cos theta cos varphi` `+ (del u)/(del y) sin theta cos varphi` `+ (del u)/(del z) sin varphi`,
   `(del u)/(del theta) = -(del u)/(del x) r sin theta cos varphi` `+ (del u)/(del y) r cos theta cos varphi`,
   `(del u)/(del varphi) = -(del u)/(del x) r cos theta sin varphi` `- (del u)/(del y) r sin theta sin varphi` `+ r cos varphi`. 代换的 Jacobi 行列式为 `|(del x)/(del r), (del x)/(del theta), (del x)/(del varphi); (del y)/(del r), (del y)/(del theta), (del y)/(del varphi); (del z)/(del r), (del z)/(del theta), (del z)/(del varphi)|` `= r^2 cos varphi`. 解线性方程组得 `grad u = ((del u)/(del x), (del u)/(del y), (del u)/(del z))`
   `= 1/(r^2 cos varphi) ((del u)/(del r) r^2 cos theta cos^2 varphi - (del u)/(del theta) r sin theta - (del u)/(del varphi) r cos theta cos varphi sin varphi; (del u)/(del r) r^2 sin theta cos^2 varphi + (del u)/(del theta) r cos theta - (del u)/(del varphi) r sin theta cos varphi sin varphi; (del u)/(del r) r^2 cos varphi sin varphi +(del u)/(del varphi) r cos^2 varphi)`
   `= (del u)/(del r) (cos theta cos varphi, sin theta cos varphi, sin varphi)`
   `+ 1/(r cos varphi) (del u)/(del theta) (-sin theta, cos theta, 0)`
   `+ 1/r (del u)/(del varphi) (-cos theta sin varphi, -sin theta sin varphi, cos varphi)`
   `:= (del u)/(del r) bm e_r + 1/(r cos varphi) (del u)/(del theta) bm e_theta + 1/r (del u)/(del varphi) bm e_varphi`. 以 `bm alpha_1` 记与 `bm alpha` 同向的单位向量, 其中 `bm e_r = (x_r, y_r, z_r)_1 = (cos theta cos varphi, sin theta cos varphi, sin varphi) = ((x","y","z))/r`,
   `bm e_theta = (x_theta, y_theta, z_theta)_1 = (-sin theta, cos theta, 0) = ((-y","x","0))/sqrt(x^2+y^2)`,
   `bm e_varphi = (x_varphi, y_varphi, z_varphi)_1 = (-cos theta sin varphi, -sin theta sin varphi, cos varphi)` `= ((-x z","-y z","x^2+y^2))/(r sqrt(x^2+y^2))`. 三者分别表示 `r, theta, varphi` 方向的单位向量. (注: 如果作变换 `x = r cos theta sin varphi`, `y = r sin theta sin varphi`, `z = r cos varphi`, 上面的行列式应该等于 `-r^2 sin varphi`)

分别在柱坐标与球坐标下计算 `"d"l^2 = dx^2 + dy^2 + dz^2`.

1. 柱坐标 `x = r cos theta`, `y = r sin theta` 下, `dx = cos theta "d"r - r sin theta "d"theta`, `quad dy = sin theta "d"r + r cos theta "d"theta`. 于是 `dx^2 + dy^2 + dz^2 = "d"r^2 + r^2 "d"theta^2 + dz^2`. 也可以由 Jacobi 矩阵 `bm J = (del(x, y))/(del(r, theta))` 直接得到 `dx^2 + dy^2 = ["d"r, "d"theta] bm(J' J) ["d"r; "d"theta]`.
2. 对球坐标 `x = r sin varphi cos theta`, `y = r sin varphi sin theta`, `z = r cos varphi` 也作类似 1. 的计算, 得 `"d"l^2 = "d"r^2 + r^2(sin varphi"d"theta^2 + "d"varphi^2)`.

多重指标

设 `alpha = (alpha_1, alpha_2, cdots, alpha_n)`, `beta = (beta_1, beta_2, cdots, beta_n)` 是非负整数组成的 `n` 维向量, `x = (x_1, x_2, cdots, x_n) in RR^n`. 定义 `|alpha| = alpha_1 + alpha_2 + cdots + alpha_n`,
`alpha! = alpha_1! alpha_2! cdots alpha_n!`,
`alpha +- beta = (alpha_1 +- beta_1, cdots, alpha_n +- beta_n)`,
`beta le alpha iff beta_1 le alpha_1, beta_2 le alpha_2, cdots, beta_n le alpha_n`,
`(alpha;beta) = (alpha_1;beta_1)(alpha_2;beta_2)cdots(alpha_n;beta_n)`,
`x^alpha = x_1^(alpha_1) x_2^(alpha_2) cdots x_n^(alpha_n)`, 有了这套多重指标的记号, 就可大大简化我们的书写. 设 `|alpha| = k`, `n` 元数量函数 `f` 在定义域上 `k` 次可微, 则 `f` 直到 `k` 阶的偏导数都与求导次序无关, 可以简记 `del^alpha = (del^|alpha|)/(del x_1^(alpha_1) cdots del x_n^(alpha_n))`

多重二项公式 `(x + y)^alpha = sum_(beta le alpha) (alpha; beta) x^(alpha-beta) y^beta`, 其中 `x, y in RR^n`, `alpha, beta` 是 `n` 重指标.

左边等于 `(x_1+y_1)^(alpha_1) cdots (x_n+y_n)^(alpha_n)` `= sum_(beta_1 le alpha_1) (alpha_1;beta_1)x_1^(alpha_1-beta_1) y_1^(beta_1) cdots sum_(beta_n le alpha_n) (alpha_n;beta_n)x_n^(alpha_n-beta_n) y_n^(beta_n)` `= sum_(beta_1 le alpha_1) cdots sum_(beta_n le alpha_n) (alpha_1;beta_1) cdots (alpha_n;beta_n) x_1^(alpha_1-beta_1) cdots x_n^(alpha_n-beta_n) y_1^(beta_1) cdots y_n^(beta_n)` 等于右边.

用 `epsi_i` 表示第 `i` 个分量为 1, 其余分量都为 0 的 `n` 重指标, 用 `0` 表示各分量全为 0 的 `n` 重指标, 则
1. `|epsi_i| = 1`; 若 `alpha le epsi_i`, 则 `alpha = epsi_i` 或 `alpha = 0`;
2. Pascal 恒等式 `(alpha-epsi_i;beta-epsi_i) + (alpha-epsi_i;beta) = (alpha; beta)`.

Leibniz 公式 `del^alpha(f g) = sum_(beta le alpha) (alpha;beta) del^(alpha-beta) f del^beta g`.

对 `|alpha|` 进行归纳. 若 `|alpha| = 1`, 由乘积的求导法则知结论成立; 现在设 `|alpha| gt 1`, `epsi_i le alpha`, 记 `gamma = alpha-epsi_i`, 则左边等于 `del^(epsi_i) del^gamma(f g)` `= del^(epsi_i) sum_(beta le gamma) (gamma;beta) del^(gamma-beta) f del^beta g` `= sum_(beta le gamma) (alpha-epsi_i;beta) del^(alpha-beta) f del^beta g` `+ sum_(epsi_i le beta le alpha) (alpha-epsi_i;beta-epsi_i) del^(alpha-beta) f del^beta g` 等于右边.

多元 Taylor 公式

`f(x) = sum_(|alpha| le n) 1/alpha! del^alpha f(x) h^alpha + R_(n+1)`
`= sum_(k le n) 1/k! (h * grad)^k f(a) + R_(n+1)` 其中余项有三种形式: `R_(n+1) = sum_(|alpha| = n+1) 1/alpha! del^alpha f(a + theta h) h^alpha` `= 1/(n+1)! (h * grad)^(n+1) f(a + theta h)`, `theta in (0, 1)` (Lagrange 余项) `R_(n+1) = sum_(|alpha| = n+1) (n+1) h^alpha/alpha! int_0^1 (1-t)^n del^alpha f(a + t h) dt` `= 1/n! int_0^1 (1-t)^n (h * grad)^(n+1) f(a + t h) dt` (积分余项) `R_(n+1) = O(|h|^(n+1))` (Peano 余项)