Higher-Order Partial Derivatives of Multivariable Functions | IB数学：多元函数的高阶导数

📚 Higher-Order Partial Derivatives of Multivariable Functions | IB数学：多元函数的高阶导数

In IB Mathematics, after mastering first-order partial derivatives, we move to higher-order derivatives of multivariable functions. These derivatives reveal curvature, concavity, and the local behavior of surfaces, and they are essential for optimisation problems in two or more variables.

在IB数学中，掌握一阶偏导数之后，我们进一步学习多元函数的高阶导数。高阶导数能够揭示曲面的弯曲程度、凹凸性以及局部行为，是解决多变量优化问题的关键工具。

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1. Definition of Second Partial Derivatives | 二阶偏导数的定义

For a function f(x,y), the first-order partial derivatives are f_x and f_y. Taking the derivative of these with respect to x or y gives four second-order partial derivatives:

对于二元函数 f(x,y)，一阶偏导数为 f_x 和 f_y。对它们再分别关于 x 或 y 求导，可以得到四个二阶偏导数：

• f_xx = ∂∂ x(∂ f∂ x) — pure second derivative with respect to x.
• f_xy = ∂∂ y(∂ f∂ x) — first differentiate with respect to x, then with respect to y.
• f_yx = ∂∂ x(∂ f∂ y) — first differentiate with respect to y, then with respect to x.
• f_yy = ∂∂ y(∂ f∂ y) — pure second derivative with respect to y.

Notation varies: f_xy is also written as ^{∂2 f}⁄_{∂ y ∂ x}, where the order of differentiation reads from right to left in the denominator.

记号可能有所不同：f_xy 也可以写作 ^{∂2 f}⁄_{∂ y ∂ x}，此时分母中从右向左依次是求导顺序。

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2. Worked Example: Computing Second Partials | 实例：计算二阶偏导数

Let f(x,y) = x³ y² + sin(xy). First compute:

设 f(x,y) = x³ y² + sin(xy)。先计算一阶偏导数：

f_x = 3x² y² + ycos(xy), f_y = 2x³ y + xcos(xy)

Now compute the four second-order partial derivatives:

再计算四个二阶偏导数：

f_xx = 6x y² - y² sin(xy)

f_xy = 6x² y + cos(xy) - xysin(xy)

f_yx = 6x² y + cos(xy) - xysin(xy)

f_yy = 2x³ - x² sin(xy)

Notice that f_xy = f_yx in this example. This is not a coincidence, provided the function is sufficiently smooth.

注意到本例中 f_xy = f_yx。只要函数足够光滑，这一结果并非偶然。

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3. Clairaut's Theorem: Mixed Partials Commute | 克莱罗定理：混合偏导可交换

Clairaut's Theorem states: If f is defined on an open region containing a point (a,b), and both f_xy and f_yx are continuous on that region, then at the point (a,b),

克莱罗定理指出：若 f 在包含点 (a,b) 的某个开区域内定义，且 f_xy 和 f_yx 在该区域内连续，则在点 (a,b) 处，

f_xy(a,b) = f_yx(a,b)

In essence, under continuity assumptions, the order of differentiation does not affect the result. This greatly simplifies calculations in IB exam problems.

本质上，在连续性假设下，求导顺序不影响最终结果。这大大简化了IB考试中的计算。

If the mixed partials are not continuous, equality may fail. However, IB functions are almost always polynomials, exponentials, trigonometric, or rational functions with no singularities in the domain, so Clairaut's theorem applies.

如果混合偏导数不连续，等式可能不成立。不过IB中的函数通常是多项式、指数函数、三角函数或在定义域内无奇异点的有理函数，因此克莱罗定理几乎总是适用。

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4. Higher-Order Derivatives Beyond Second Order | 高于二阶的偏导数

Third-order and higher partial derivatives are defined iteratively. For example, f_xxy means: differentiate f with respect to x, then x again, then y.

三阶及更高阶的偏导数可通过迭代定义。例如，f_xxy 表示：先对 f 关于 x 求导，再对 x 求导，最后对 y 求导。

• Notation: f_xxx = ^{∂3 f}⁄_{∂ x³}, f_xyy = ^{∂3 f}⁄_{∂ y² ∂ x}.
• 记号： f_xxx = ^{∂3 f}⁄_{∂ x³}，f_xyy = ^{∂3 f}⁄_{∂ y² ∂ x}。

When all partial derivatives up to order n are continuous, mixed partials of the same total order are equal regardless of differentiation order. For instance, f_xyy = f_yxy = f_yyx.

当所有直到 n 阶的偏导数连续时，相同总阶数的混合偏导数与求导顺序无关。例如，f_xyy = f_yxy = f_yyx。

In IB, third-order derivatives appear occasionally in Taylor polynomials and in verifying solutions to partial differential equations.

在IB中，三阶导数偶尔出现在泰勒多项式以及验证偏微分方程解的过程中。

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5. The Hessian Matrix | 黑塞矩阵

The Hessian matrix of a function f(x,y) is the 2 × 2 matrix of second-order partials:

函数 f(x,y) 的黑塞矩阵是由二阶偏导数构成的 2 × 2 矩阵：

H = (f_xx, f_xy; f_yx, f_yy)

By Clairaut's theorem, when mixed partials are equal, the Hessian is symmetric. The determinant and trace of this matrix are used to classify critical points.

由克莱罗定理，当混合偏导相等时，黑塞矩阵是对称的。该矩阵的行列式和迹用于判定临界点的类型。

For a function of three variables f(x,y,z), the Hessian is a 3 × 3 matrix:

对于三元函数 f(x,y,z)，黑塞矩阵是 3 × 3 矩阵：

H = (f_xx, f_xy, f_xz; f_yx, f_yy, f_yz; f_zx, f_zy, f_zz)

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6. Second Derivative Test for Local Extrema | 极值的二阶导数判别法

To classify a critical point (a,b) of a differentiable function f(x,y), compute the Hessian determinant:

为了对可微函数 f(x,y) 的临界点 (a,b) 进行分类，计算黑塞行列式：

D = f_xx(a,b) f_yy(a,b) - [ f_xy(a,b) ]²

• If D > 0 and f_xx(a,b) > 0, then (a,b) is a local minimum.
• 如果 D > 0 且 f_xx(a,b) > 0，则 (a,b) 是局部极小值点。
• If D > 0 and f_xx(a,b) < 0, then (a,b) is a local maximum.
• 如果 D > 0 且 f_xx(a,b) < 0，则 (a,b) 是局部极大值点。
• If D < 0, then (a,b) is a saddle point.
• 如果 D < 0，则 (a,b) 是鞍点。
• If D = 0, the test is inconclusive.
• 如果 D = 0，判别法失效。

This test is essentially the IB version of the second derivative test for single-variable functions, extended to two dimensions.

该判别法本质上是单变量函数二阶导数判别法在二维情形的推广。

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7. Example of the Second Derivative Test | 二阶导数判别法示例

Find and classify the critical points of f(x,y) = x² + y² - 2x - 4y + 5.

求函数 f(x,y) = x² + y² - 2x - 4y + 5 的临界点并分类。

First-order partials:

一阶偏导数为：

f_x = 2x - 2, f_y = 2y - 4

Setting f_x = 0 and f_y = 0 gives the critical point (1,2).

令 f_x = 0、f_y = 0，得到临界点 (1,2)。

Second-order partials:

二阶偏导数为：

f_xx = 2, f_yy = 2, f_xy = 0

Thus D = 2 × 2 - 0² = 4 > 0, and f_xx = 2 > 0, so (1,2) is a local minimum.

因此 D = 2 × 2 - 0² = 4 > 0，且 f_xx = 2 > 0，所以 (1,2) 是局部极小值点。

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8. Higher-Order Chain Rule and Notation | 高阶链式法则与记号

When variables are transformed, e.g., using polar coordinates x = rcosθ, y = rsinθ, higher-order derivatives require careful application of the chain rule.

当变量发生变换时，例如使用极坐标 x = rcosθ、y = rsinθ，高阶导数的计算需要谨慎应用链式法则。

For a function u = f(x,y), the first-order derivatives transform as:

对于函数 u = f(x,y)，一阶导数的变换公式为：

∂ u∂ r = f_x cosθ + f_y sinθ, ∂ u∂ θ = -f_x rsinθ + f_y rcosθ

To find ^{∂2 u}⁄_{∂ r²}, apply the first-order operator again:

为了求 ^{∂2 u}⁄_{∂ r²}，再次应用一阶算子：

^{∂2 u}⁄_{∂ r²} = ∂∂ r(f_x cosθ + f_y sinθ)

Because f_x and f_y are themselves functions of x and y, their r-derivatives involve f_xx, f_xy, and f_yy. Such expansions are rarely required inIB at standard level, but appear in HL calculus options.

由于 f_x 和 f_y 本身是 x 和 y 的函数，它们关于 r 的导数将涉及 f_xx、f_xy 和 f_yy。此类展开在IB标准水平中不常出现，但在高级水平微积分选修部分可能出现。

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9. Taylor Expansion with Higher-Order Partial Derivatives | 含高阶偏导数的泰勒展开

The second-order Taylor expansion of f(x,y) near a point (a,b) uses all second-order partials:

函数 f(x,y) 在点 (a,b) 附近的二阶泰勒展开使用了所有二阶偏导数：

f(a+h,b+k) ≈ f(a,b) + f_x(a,b)h + f_y(a,b)k + 12[ f_xx h² + 2 f_xy hk + f_yy k² ]

This formula generalizes the single-variable Taylor series. The quadratic term is often written in matrix form using the Hessian:

该公式是单变量泰勒级数的推广。二次项常借助黑塞矩阵写成矩阵形式：

12 (h, k) H(a,b) (h, k)ᵀ

This expansion is used in numerical approximation and in proving the second derivative test.

此展开用于数值逼近以及证明二阶导数判别法。

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10. Common Pitfalls in IB Exams | IB考试常见易错点

• Order of differentiation: When using ^{∂2 f}⁄_{∂ y ∂ x}, always differentiate with respect to x first, then y. In subscript notation f_xy, differentiate with respect to x first as well.
• 求导顺序： 使用 ^{∂2 f}⁄_{∂ y ∂ x} 时，总是先对 x 求导，再对 y 求导。在下标记号 f_xy 中，也是先对 x 求导。
• Forgetting the mixed term: In Taylor expansion, the term 2f_xyhk is often omitted. Remember it carries a factor of 2.
• 遗漏混合项： 在泰勒展开中，2f_xyhk 这一项经常被忽略。注意它带有系数2。
• Sign errors in Hessian determinant: The off-diagonal term is squared and subtracted: D = f_xxf_yy - f_xy². Do not use f_xyf_yx unless you have verified they are equal.
• 黑塞行列式中的符号错误： 非对角线项要平方并相减：D = f_xxf_yy - f_xy²。除非已确认两者相等，否则不要写成 f_xyf_yx。

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11. Worked IB-Style Problem | 例题（IB风格）

Consider f(x,y) = e^xy + ln(x² + y² + 1). Find f_xy at (0,0).

考虑函数 f(x,y) = e^xy + ln(x² + y² + 1)。求 f_xy 在 (0,0) 处的值。

First compute f_x:

先求 f_x：

f_x = y e^xy + ^2x⁄_{x² + y² + 1}

Then differentiate with respect to y:

再对 y 求导：

f_xy = e^xy + xy e^xy - ^{2y · 2x}⁄_{(x² + y² + 1)²}

Evaluate at (0,0):

在 (0,0) 处取值：

f_xy(0,0) = 1 + 0 - 0 = 1

Alternatively, compute f_yx first and obtain the same result by Clairaut's theorem.

或者先计算 f_yx，根据克莱罗定理会得到相同结果。

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12. Summary and Exam Tips | 总结与考试技巧

• Always state that the function is smooth (partials are continuous) when using Clairaut's theorem.
• 使用克莱罗定理时，务必说明函数光滑（偏导数连续）。
• Practice computing f_xy both ways to catch mistakes.
• 练习分别按两种顺序计算 f_xy，以检查错误。
• In the second derivative test, remember the condition f_xx > 0 for minimum and f_xx < 0 for maximum; if f_xx = 0, the test cannot distinguish, so further analysis is needed.
• 在二阶导数判别法中，记住极小值要求 f_xx > 0、极大值要求 f_xx < 0；若 f_xx = 0，判别法无法区分，需要进一步分析。
• For HL students, be comfortable with the Hessian matrix form and Taylor expansion in vector notation.
• 对于高级水平（HL）学生，要熟悉黑塞矩阵形式以及向量记法下的泰勒展开。

Mastering higher-order partial derivatives builds a strong foundation for multivariable optimisation, differential geometry, and partial differential equations — topics that are central to university mathematics.

掌握多元函数的高阶导数为多变量优化、微分几何和偏微分方程打下坚实基础——这些内容在大学数学中至关重要。

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