IB Math: The Two-Dimensional Gradient Vector Explained | IB数学：二维梯度向量详解

📚 IB Math: The Two-Dimensional Gradient Vector Explained | IB数学：二维梯度向量详解

The two-dimensional gradient vector is one of the most important ideas that connects calculus with geometry in IB Mathematics. It takes a scalar function of two variables and produces a vector whose components describe how the function changes in the x-direction and the y-direction. Once you understand this single idea, many problems about rates of change, level curves, and optimization become much clearer.

二维梯度向量是 IB 数学中连接微积分与几何的最重要概念之一。它把一个含两个变量的标量函数变成一个向量，其分量分别描述函数在 x 方向和 y 方向的变化。理解了这一个概念，关于变化率、等高线和最优化问题的许多题目都会变得清晰很多。

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1. What Is the Gradient Vector? | 什么是梯度向量？

For a function f(x, y), the gradient vector, written as ∇f, is formed by collecting the two first partial derivatives into a single vector. The symbol ∇ is called "nabla", and it turns a scalar function into a vector field.

对于一个函数 f(x, y)，梯度向量写作 ∇f，它由两个一阶偏导数组成一个向量。符号 ∇ 被称为“纳布拉”，它把一个标量函数变成一个向量场。

∇f(x, y) = (∂f/∂x, ∂f/∂y)

We often also write ∇f = (f_x, f_y), where f_x means the partial derivative with respect to x and f_y means the partial derivative with respect to y. Each component is the rate of change of f along one coordinate direction.

我们也常把梯度写作 ∇f = (f_x, f_y)，其中 f_x 表示对 x 的偏导数，f_y 表示对 y 的偏导数。每一个分量都表示 f 沿某一个坐标方向的变化率。

+ 3y, then the partial derivatives are ∂f/∂x = 2xy and ∂f/∂y = x² + 3, so the gradient is ∇f = (2xy, x² + 3).

例如，若 f(x, y) = x²y + 3y，则偏导数为 ∂f/∂x = 2xy，∂f/∂y = x² + 3，因此梯度为 ∇f = (2xy, x² + 3)。

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2. Partial Derivatives: The Building Blocks | 偏导数：梯度的基本成分

only. To find ∂f/∂y, we treat x as a constant and differentiate f with respect to y only.

在使用梯度向量之前，我们需要清楚偏导数的定义。求 ∂f/∂x 时，我们把 y 看作常数，仅对 x 求导；求 ∂f/∂y 时，我们把 x 看作常数，仅对 y 求导。

∂f/∂x = lim_Δx→0 [f(x + Δx, y) − f(x, y)] / Δx

∂f/∂y = lim_Δy→0 [f(x, y + Δy) − f(x, y)] / Δy

∂f/∂y is the slope of the curve obtained by cutting the surface with a plane x = constant.

从几何角度看，∂f/∂x 是用平面 y = 常数去截曲面 z = f(x, y) 所得曲线的斜率；类似地，∂f/∂y 是用平面 x = 常数所得曲线的斜率。

This is why the gradient vector is two-dimensional: it combines the two independent slope directions in the xy-plane into one vector.

这就是为什么梯度向量是二维的：它把 xy 平面中两个相互独立的方向上的斜率合成为一个向量。

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3. How to Calculate the Gradient: Worked Examples | 如何计算梯度：典型例题

hem carefully. Suppose f(x, y) = 3x²y − 4y³. Let us find the gradient at the point (1, 2).

学习梯度向量最可靠的方法就是仔细计算。设 f(x, y) = 3x²y − 4y³，我们来求它在点 (1, 2) 处的梯度。

∂f/∂x = 6xy, ∂f/∂y = 3x² − 12y²

both partial derivatives: ∂f/∂x = 6(1)(2) = 12 and ∂f/∂y = 3(1)² − 12(2)² = 3 − 48 = −45.

在 (1, 2) 处，代入 x = 1，y = 2：∂f/∂x = 6(1)(2) = 12，∂f/∂y = 3(1)² − 12(2)² = 3 − 48 = −45。

Therefore the gradient vector at that point is ∇f(1, 2) = (12, −45).

因此该点处的梯度向量为 ∇f(1, 2) = (12, −45)。

x, y) = ln(x² + y²). Then ∂g/∂x = 2x/(x² + y²) and ∂g/∂y = 2y/(x² + y²). At (1, 0), we get ∇g = (2, 0).

再看一个例子，设 g(x, y) = ln(x² + y²)，则 ∂g/∂x = 2x/(x² + y²)，∂g/∂y = 2y/(x² + y²)。在点 (1, 0) 处，得到 ∇g = (2, 0)。

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4. Geometric Meaning: Direction of Steepest Ascent | 几何意义：最陡上升方向

radient vector has a beautiful geometric interpretation. At a point (x₀, y₀), the gradient ∇ magine standing on a hillside modeled by z = f(x, y), the gradient points directly uphill.

梯度向量具有非常优美的几何意义。在点 (x₀, y₀) 处，梯度 ∇f(x₀, y₀) 指向 f 增长最快的方向。如果你站在由 z = f(x, y) 所模拟的山坡上，梯度指向正上坡方向。

Its magnitude gives the steepness of that fastest increase. The larger |∇f| is, the more rapidly f changes.

梯度向量的模表示最陡上升的剧烈程度。|∇f| 越大，f 变化得越快。

|∇f| = √((∂f/∂x)² + (∂f/∂y)²)

The opposite direction, −∇f, points directly downhill. Moving perpendicular to the gradient means staying on a level curve, where f does not change at all.

相反方向 −∇f 指向正下坡方向。沿着与梯度垂直的方向移动，意味着沿等高线移动，此时 f 完全不改变。

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5. Gradient and Level Curves: Normal and Tangent Lines | 梯度与等高线：法线与切线

(x, y) = c, with c constant. In the xy-plane, different values of c give different curves.

f(x, y) 的等高线是满足 f(x, y) = c 的点的集合，其中 c 为常数。在 xy 平面中，不同的 c 对应不同的曲线。

at the gradient vector is perpendicular to the level curve at every point. In other words, ∇f(x₀, y₀) is a normal vector to the curve f(x, y) = c at (x₀, y₀).

一个重要定理指出：梯度向量在每一点处都垂直于等高线。换句话说，∇f(x₀, y₀) 是曲线 f(x, y) = c 在 (x₀, y₀) 处的一个法向量。

rotating the gradient by 90 degrees. For example, a tangent vector is T = (−∂f/∂y, ∂f/∂x).

因此，把梯度旋转 90° 就可以得到等高线的一个切向方向。例如，切向量可取 T = (−∂f/∂y, ∂f/∂x)。

Consider f(x, y) = x² + y² at (3, 4) on the level curve x² + y² = 25. The gradient is ∇ nd the tangent line equation is 6(x − 3) + 8(y − 4) = 0, which simplifies to 3x + 4y = 25.

考虑等高线 x² + y² = 25 上的点 (3, 4)，其中 f(x, y) = x² + y²。梯度为 ∇f = (2x, 2y) = (6, 8)。一个切向量为 T = (−8, 6)，切线方程为 6(x − 3) + 8(y − 4) = 0，化简得 3x + 4y = 25。

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6. Directional Derivatives: Rates of Change in Any Direction | 方向导数：任意方向上的变化率

Sometimes we need the rate of change of f in a direction that is not horizontal or vertical. This is called the directional derivative. If u is a unit vector, then the directional derivative of f in the direction of u is given by the dot product of the gradient and u.

有时我们需要求 f 在既不是水平也不是垂直的方向上的变化率，这就是方向导数。若 u 是单位向量，则 f 在 u 方向上的方向导数等于梯度与 u 的点积。

D_uf = ∇f · u

If the direction is given by a non-unit vector v, we must first normalize it: u = v / |v|. After normalization, the directional derivative equals ∇f · (v/|v|).

如果方向用非单位向量 v 给出，我们必须先让其单位化：u = v / |v|。单位化后，方向导数等于 ∇f · (v/|v|)。

For example, let f(x, y) = x² + y² and take the point (1, 2). Then ∇f = (2, 4). In the direction u = (1/√2, 1/√2), we get D_uf = (2, 4) · (1/√2, 1/√2) = 6/√2 = 3√2.

例如，设 f(x, y) = x² + y²，取点 (1, 2)，则 ∇f = (2, 4)。在方向 u = (1/√2, 1/√2) 上，得到 D_uf = (2, 4) · (1/√2, 1/√2) = 6/√2 = 3√2。

This result makes sense because the dot product selects the component of the gradient that lies in the chosen direction.

这个结果是合理的，因为点积提取了梯度在所选方向上的分量。

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7. The Chain Rule and Parametric Curves | 链式法则与参数曲线

hen f(x, y) becomes a function of t, and the rate of change df/dt can be expressed using the gradient vector.

在 IB 数学中，我们经常研究 x 和 y 都是参数 t 的函数的曲线，例如 x = g(t)，y = h(t)。此时 f(x, y) 成为 t 的函数，df/dt 可以用梯度向量表示。

df/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt) = ∇f · (dx/dt, dy/dt)

This is the multivariable chain rule in two dimensions. The vector (dx/dt, dy/dt) is the velocity vector of the parametric curve, so df/dt is the dot product of velocity with the gradient.

这就是二维情形的多元链式法则。向量 (dx/dt, dy/dt) 是参数曲线的速度向量，因此 df/dt 是速度向量与梯度的点积。

level curve, then df/dt = 0 because the gradient is perpendicular to the velocity vector.

如果曲线在每个点都沿着梯度方向移动，那么 f 沿这条曲线增长最快。如果曲线沿等高线移动，则 df/dt = 0，因为梯度与速度向量垂直。

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8. Critical Points and Optimisation | 临界点与最优化

The gradient also plays a central role in finding local maxima, local minima, and saddle points of a function f(x, y). At any stationary point, the function is locally flat, so both partial derivatives must be zero.

梯度在寻找函数 f(x, y) 的局部最大值、局部最小值和鞍点中也起着核心作用。在任意驻点处，函数局部平坦，因此两个偏导数都必须为零。

∇f(x₀, y₀) = (0, 0)

So to locate candidates for maxima or minima, we solve the system ∂f/∂x = 0 and ∂f/∂y = 0.

因此，要寻找极大值或极小值的候选点，我们求解方程组 ∂f/∂x = 0 和 ∂f/∂y = 0。

x − 4y + 5. Then ∂f/∂x = 2x − 2 and ∂f/∂y = 2y − 4. Setting both equal to zero gives x = 1

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