📚 Part 5: Multivariable Calculus | 第五部分：多变量微积分
Multivariable calculus extends the principles of differentiation and integration to functions of several variables. This branch of mathematics is essential for modelling systems in physics, engineering, economics, and data science, where quantities depend on multiple inputs. In this article, we will explore key topics such as partial derivatives, gradients, constrained optimization, and multiple integrals, providing a rigorous yet accessible overview aligned with IB Higher Level requirements.
多变量微积分将微分与积分的原理推广到多元函数。这一数学分支对于物理、工程、经济学和数据科学等领域的建模至关重要，在这些领域中量值往往依赖于多个输入。本文将围绕偏导数、梯度、约束优化以及多重积分等核心主题展开，提供严谨且易于理解的综述，密切贴合IB高水平课程的要求。
1. Functions of Several Variables | 多元函数
more variables, e.g., f(x, y) or f(x, y, z). The domain of such a function is a subset of ℝ² or ℝ d; holding y constant gives a parabola in x. Level curves, or contour lines, join points where the function takes the same value, providing an alternative way to visualise two-variable functions.
在单变量微积分中，我们处理只有一个自变量的函数，如 f(x)。多变量微积分则研究二元或更多元函数，例如 f(x, y) 或 f(x, y, z)。这类函数的定义域是 ℝ² 或 ℝ³ 的子集，其图像通常是三维空间中的曲面。为了理解 f(x, y) 的行为，可以固定一个变量并观察截面曲线。例如，f(x, y) = x² + y² 表示一个抛物面；保持 y 不变就会得到关于 x 的抛物线。等高线（等值线）将函数值相同的点连接起来，是可视化二元函数的另一种方式。
2. Partial Derivatives | 偏导数
ivative with respect to x is denoted ∂f/∂x or fₓ, and is defined by the limit: ∂f/∂x = limh→0 artial derivatives give the rate of change of the function along the coordinate axes and serve as the foundation for the gradient vector and linear approximations.
多元函数的偏导数是对其中一个变量求导，而将其他变量视为常数。对于 z = f(x, y)，关于 x 的偏导数记为 ∂f/∂x 或 fₓ，其定义为极限：∂f/∂x = limh→0 [f(x+h, y) − f(x, y)] / h。计算偏导数时，只需将其他变量视作常数进行微分。例如，若 f(x, y) = x³y + sin(x y)，则 ∂f/∂x = 3x² y + y cos(x y)，而 ∂f/∂y = x³ + x cos(x y)。偏导数给出了函数沿坐标轴方向的变化率，是梯度向量和线性近似的基础。
3. Higher-Order Partial Derivatives | 高阶偏导数
s are equal: ∂²f/∂x∂y = ∂²f/∂y∂x. This symmetry greatly simplifies calculations when it holds. Higher-order partials can be computed repeatedly, enabling Taylor expansions of functions of several variables
Published by TutorHao | IB Mathematics Revision Series | aleveler.com
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