Name: weather maps as two-variable surfaces
Uploaded: 2026-06-08T09:38:41.450Z
Duration: 1 min 27 s
Channel: JoVE

标准天气图展示了温度如何在一个广阔区域（如美国）内变化。该地图上的每个特定位置均由一对独特的地理坐标定义：经度和纬度。当地图上的位置从一个城市转移到另一个城市时，相应的温度值也随之变化。为了从数学上分析这种关系，引入了一个清晰的结构：经度和纬度坐标作为两个输入，分别标记为 x 和 y，而对应的温度值作为单一输出，标记为 z。然后通过将温度 z 分配给垂直轴，将这种关系绘制到三维图上，其中较高温度显示得更高，较低温度显示得更平坦。由于温度在各点之间是逐渐变化的，将所有这些垂直数值连接起来会生成一个平滑、连续的三维曲面，该曲面由函数 z 等于 f(x, y) 定义。这个三维图形将数学方程与真实空间联系起来，把数字转化为易于观察和理解的形状，使我们能够一眼就直观地了解整个区域的气候状况。

双变量函数的图像

天气图是双变量函数的一个实际例子。在一个广阔的区域（例如美国）内，气温随地点不同而变化。每个地点都可以用两个地理坐标来确定：经度和纬度。由于每个坐标对都对应一个唯一的温度值，这种情况在数学上可以表示为具有两个输入和一个输出的函数。在数学表示中，经度和纬度可以标记为 x 和 y，而温度可以标记为 z。该关系式写作 z = f(x, y)。这意味着对于区域内的每一点（x,y），该函数都会分配一个对应的温度值 z。例如，两个具有不同经度和纬度坐标的城市可能因在地图上的位置不同而具有不同的温度。为了直观展示这种关系，将坐标对（x, y）置于水平平面内，温度值 z 沿垂直方向绘制。较高的温度在图中表现为较高的点，而较低的温度则更接近水平平面。当所有这些温度值连接在一起时，便形成一个三维曲面。由于温度在空间上通常逐渐变化，因此生成的曲面通常平滑且连续。该曲面直观地展示了温度在区域内的变化情况。曲面较陡的部分表示在短距离内温度变化较快的区域，而较平坦的区域则表示温度相对稳定的区域。这种图形将现实世界的地理数据与数学结构相连接。它展示了双变量函数如何将坐标信息转化为曲面，从而更易于解读和比较区域温度模式。

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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

Graphs of Two-Variable Functions

两元函数图

weather maps as two-variable surfaces

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 relationship places the coordinate pair (x, y) in the horizontal plane and the temperature value z above it. Higher temperatures appear as higher points on the graph. Lower temperatures appear closer to the horizontal plane.

When the temperature values are connected, they form a three-dimensional surface. Because temperature often changes gradually over space, this surface is usually smooth and continuous. Steeper parts of the surface show places where temperature changes quickly over short distances, while flatter parts show areas where temperature stays more constant. This kind of graph links geographic data to a clear mathematical shape and makes regional temperature patterns easier to compare.

Weather Maps as Two-Variable Surfaces

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

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

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properties of limits in multivariable calculus

properties-of-limits-in-multivariable-calculus

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

continuity-for-functions-of-multiple-variables

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

multivariable-functions-and-higher-derivatives

Partial Derivatives and Surface Curvature

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Modeling Waves with Space and Time

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tangent planes to surfaces

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linear approximations

Tangent Plane Estimates for Two Variables

Tangent plane estimates help approximate a differentiable function of two variables near a known point. Instead of using the curved surface directly, the function is replaced by its tangent plane. This gives a fast estimate for values close to the point being studied. For the function f(x,y)=x^2+3y^2, the point (2, 1) is used as the reference. The exact value there is f(2,1)=2^2+3(1)^2=4+3=7. The linear approximation near (a,b) is L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b). To build the...

multivariable chain rule

Total Derivatives in Connected Variables

Total derivatives in connected variables show how one change can affect several related quantities. Here, z depends on two intermediate variables, x and y, and both x and y change with a third variable, t. The connection is indirect, because t does not appear in the formula for z itself. The goal is to find the total derivative of z with respect to t, written as dz/dt. When the functions are differentiable, the change in z can be split into two parts. One part comes from how z changes with x...

implicit differentiation with partial derivatives

implicit-differentiation-with-partial-derivatives

Finding Tangent Slopes from Implicit Equations

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directional derivatives

Directional Derivatives in Any Direction

Directional derivatives measure how a surface changes in any chosen direction. In multivariable calculus, partial derivatives show change along one coordinate at a time. For a function of two variables, one partial derivative gives the slope in the x-direction, and the other gives the slope in the y-direction. These single-direction slopes are useful for local behavior. However, many real movements do not follow the coordinate axes. Fluid flow, heat transfer, and motion across uneven terrain...

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Steepest Ascent and the Gradient Vector

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maximizing the directional derivative

Gradient Direction and Rate of Change

The directional derivative shows how a multivariable function changes at a point when you move in a chosen direction. That direction is written as a unit vector, so only the direction matters and not the length of the vector. Changing the direction changes the rate of change, which is why the directional derivative depends strongly on the path you choose. The directional derivative is found using the gradient vector and a unit direction vector. The gradient tells you both the direction and the...

tangent planes to level surfaces

Gradient Vectors and Tangent Planes

Gradient vectors and tangent planes help describe the shape of a level surface at a point. A level surface is the set of all points in space where a function of three variables has the same fixed value. To understand the local geometry near one point, it is not enough to know the coordinates alone. You also need to know how the surface tilts there. One way to study that tilt is to follow a path that stays on the level surface and passes through the point. This path can be written as a...

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...

lagrange multipliers: one constraint

Optimizing a Fence Pen with Lagrange Multipliers

Lagrange multipliers help solve constrained optimization problems. In these problems, you maximize or minimize a quantity while keeping a fixed condition. A common example is a rectangular pen built against a barn wall using 100 meters of fencing. The barn wall makes one side of the pen, so only the other three sides need fencing. The goal is to find the dimensions that give the greatest area. Let L be the length parallel to the wall and W be the width perpendicular to the wall. Then the area...

lagrange multipliers: two constraints

Lagrange Multipliers on an Intersection Curve

Lagrange multipliers with two constraints help find a maximum or minimum when a function must satisfy two conditions at once. The goal may be to optimize cost, area, distance, or energy, but only among solutions that meet both requirements. In many problems, the constraints represent fixed volume, limited resources, or prescribed dimensions. For a function of three variables, each constraint makes a surface in three-dimensional space. A feasible point is a point that satisfies both constraints.

lagrange multipliers: problem solving

Optimizing Silo Design with Lagrange Multipliers

A silo with a cylindrical base and a hemispherical roof can be sized with Lagrange multipliers. This calculus method helps find the best design when the surface area must stay fixed. In this problem, the goal is to maximize storage volume while using a set amount of material. The model uses two main variables: the radius r of the base and the height h of the cylindrical section. The total volume comes from adding the volume of the cylinder and the hemisphere. The fixed surface area includes...

key terms and concepts

See how longitude and latitude can model temperature as a function of two variables, then graph the data as a smooth 3D surface on a weather map.

Weather Maps as Two-Variable Surfaces | 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

Weather Maps as Two-Variable Surfaces

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