Name: using partial derivatives for tangent planes
Uploaded: 2026-06-08T09:38:16.420Z
Duration: 1 min 19 s
Channel: JoVE

设想你站在一个平缓弯曲的小山上，专注于脚下的一点。在非常小的尺度上，该表面看起来几乎是平坦的。这可以被想象为在该特定点处山丘上放置的一块平坦的板。从数学上讲，这就是切平面。为了确定高度 z 依赖于 x 和 y 的曲面上某一点处的切平面，需在该点处沿曲面作两个截面：一个平行于 xz 平面，另一个平行于 yz 平面。每个切片在曲面上形成一条曲线，每条曲线在该点处都有具有特定斜率的切线。这些斜率来源于偏导数。切平面是唯一包含两条曲线切线的平坦曲面。该平面的方程表示高度的变化是 x 和 y 变化的线性组合。在此方程中，系数为偏导数，代表两条曲线的斜率。在选定的点附近，该线性方程可近似逼近曲面，从而使复杂形状更易于分析。

曲面的切平面

在多变量微积分中，切平面的概念在逼近曲面时起着核心作用。当处理由两个变量函数定义的曲面时（例如 z = f(x, y)），在给定点处的切平面提供了该点附近曲面的最佳线性近似。这种局部线性化使得复杂的非线性几何结构能够通过更简单的平面模型来处理。构造切平面需要沿着保持 x 或 y 为常数的平面，对曲面进行垂直切片。这些切片在曲面上形成曲线，而这些曲线在选定点处的切线则定义了 x 和 y 方向上的方向导数。这些由偏导数 fx 和 fy 给出的方向斜率，决定了切平面的方位。在点 (x0, y0, z0) 处的切平面方程为：\begin{equation*}z-z_0=f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0)\end{equation*}该方程在选定点附近提供了曲面的线性近似，能够精确描述其在微小距离内的行为。

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Partial Derivatives and Gradients

偏导数与梯度

KEY TERMS AND CONCEPTS

关键术语与概念

Tangent Planes to Surfaces

切平面到曲面

using partial derivatives for tangent planes

Partial derivatives help describe a tangent plane for a surface in multivariable calculus. For a surface written as z = f(x, y), the tangent plane at one point gives the best linear approximation near that point. This makes a curved surface easier to study with a simple planar model.

To build the tangent plane, take vertical slices through the surface. Hold x constant in one slice and y constant in another slice. Each slice forms a curve on the surface. The tangent lines to those curves at the chosen point give the directional slopes in the x and y directions.

Those slopes are the partial derivatives fx and fy. They determine the orientation of the tangent plane. If the point on the surface is (x0, y0, z0), the tangent plane is written as z - z0 = fx(x0, y0)(x - x0) + fy(x0, y0)(y - y0). This equation gives a linear approximation of the surface near the chosen point and works well over small distances.

Using Partial Derivatives for Tangent Planes

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functions of two variables

Graphs of Functions in Three Dimensions

Functions of two variables use two inputs to determine one output. The output depends on both independent variables at the same time. They are often written as z = f(x, y), where each ordered pair in the domain gives one unique output value. The domain of a function of two variables is the region in the horizontal plane that contains the input pairs. Each point in this region matches one output value. That output can be shown as a vertical height above the plane. When all of the heights are...

graphs of two-variable functions

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Weather maps show how a function of two variables can describe temperature across a region. In this setting, each place is identified by two coordinates, longitude and latitude. Since one temperature value is assigned to each coordinate pair, the data can be written as z = f(x, y). The variables x and y represent the horizontal position on the map. The variable z represents temperature. This means that every point (x, y) in the region has a matching temperature value. A graph of this...

level curves and contour maps

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functions of three or more variables

Level Surfaces of Multivariable Functions

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limits of multivariable functions

Path Independence in Multivariable Limits

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properties of limits in multivariable calculus

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Limit Laws for Multivariable Functions

Limit laws in multivariable calculus give clear rules for finding limits of functions with more than one variable. These rules help break a complex expression into smaller parts whose limits are already known. They are used when [1m\lim_{(x,y)\to(a,b)} f(x,y)=L[0m and [1m\lim_{(x,y)\to(a,b)} g(x,y)=M[0m exist. The sum law says the limit of a sum equals the sum of the limits. The difference law says the limit of a difference equals the difference of the limits. In notation,...

continuity for functions of multiple variables

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Testing Continuity with Paths in Two Variables

Continuity in multivariable functions describes how a function with two or more input variables changes near a point. It extends the idea of continuity from single-variable calculus into higher dimensions. This idea is important for modeling real-world situations that vary across space. A multivariable function is continuous at a point only if three conditions are met at the same time. The function must be defined at that point. The limit must exist as the input moves toward the point from any...

introduction to partial derivatives

Partial Derivatives in Real-World Models

Partial derivatives help describe how a multivariable function changes when one input changes and the others stay fixed. In many real-world situations, one output depends on more than one input. A high-tech assembly plant is one example. Total production can depend on technician labor and machine capacity at the same time. This relationship can be written as a continuous function P(T, M), where T stands for technician labor input and M stands for machine capacity. When demand rises but the...

interpretations of partial derivatives

Partial Derivatives and Tangent Planes

Partial derivatives describe how a surface changes in one direction at a time. A surface from a function of two variables can be pictured like uneven terrain. Each point on that terrain is located with Cartesian coordinates, and the function gives the height at each pair of horizontal coordinates. To study the surface at one point, you can move in a single direction while holding the other variable fixed. If you move along the x-direction and keep y constant, the change in height gives the...

multivariable functions and higher derivatives

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Partial Derivatives and Surface Curvature

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partial differential equations

Modeling Waves with Space and Time

Partial differential equations help describe waves that change across both space and time. A stone dropped into a still pond creates circular waves that spread outward. The water surface rises and falls at each point as the wave passes, while the shape of the surface also changes across the pond at any fixed moment. This kind of motion needs a mathematical model that tracks more than one variable at once. At one location, the water level changes as time moves forward. That change includes the...

tangent planes to surfaces

Partial derivatives help describe a tangent plane for a surface in multivariable calculus. For a surface written as z = f(x, y), the tangent plane at one point gives the best linear approximation near that point. This makes a curved surface easier to study with a simple planar model. To build the tangent plane, take vertical slices through the surface. Hold x constant in one slice and y constant in another slice. Each slice forms a curve on the surface. The tangent lines to those curves at the...

linear approximations

Tangent Plane Estimates for Two Variables

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multivariable chain rule

Total Derivatives in Connected Variables

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implicit differentiation with partial derivatives

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Finding Tangent Slopes from Implicit Equations

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directional derivatives

Directional Derivatives in Any Direction

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gradient vectors and their applications

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Gradient Direction and Rate of Change

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tangent planes to level surfaces

Gradient Vectors and Tangent Planes

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significance of the gradient vector

Gradient Vector and Direction of Steepest Rise

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local maximum and minimum values

Finding Critical Points in Multivariable Calculus

Multivariable calculus uses critical points to locate local maximum and minimum values on a surface. A local maximum is a point where the function value is higher than at nearby points. A local minimum is a point where the function value is lower than at nearby points. These points are called local extrema, and they matter in optimization problems. Local extrema occur at critical points, where the surface is momentarily flat in all directions. At a critical point, the partial derivatives are...

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lagrange multipliers: two constraints

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key terms and concepts

Learn how partial derivatives and vertical slices determine a tangent plane for z = f(x, y), giving a linear approximation near a point on a surface.

Using Partial Derivatives for Tangent Planes | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Using Partial Derivatives for Tangent Planes

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