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

Imagine standing on a gently curved hill and focusing on a single point underfoot. At a very small scale, the surface looks almost flat.This can be visualized as a flat board resting on the hill at that specific point. Mathematically, this is the tangent plane.To define this plane for a surface where height z depends on x and y, two slices of the surface are taken at the chosen point: one parallel to the xz-plane and the other parallel to the yz-plane.Each slice forms a curve on the surface, and each curve has a tangent line with a specific slope at that point. These slopes come from partial derivatives. The tangent plane is the unique flat surface that contains the tangent lines of both curves.The equation of this plane shows the change in height as a linear combination of changes in x and y. In this equation, the coefficients are the partial derivatives, representing the slopes of the two curves.Near the chosen point, this linear equation closely approximates the surface, making complex shapes easier to analyze.

tangent planes to surfaces

In multivariable calculus, the concept of a tangent plane plays a central role in approximating curved surfaces. When dealing with a surface defined by a function of two variables, such as z = f(x, y), the tangent plane at a given point provides the best linear approximation to the surface near that point. This local linearization allows complex, nonlinear geometries to be treated using simpler, planar models.The construction of the tangent plane involves taking vertical slices of the surface along planes where either x or y is held constant. These slices yield curves on the surface, and the tangent lines to these curves at a chosen point define the directional derivatives in the x and y directions. These directional slopes, given by the partial derivatives fx and fy, find the orientation of the tangent plane.The equation of the tangent plane at a point (x0, y0, z0) is given by:\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*}This equation provides a linear approximation of the surface near the chosen point, accurately describing its behavior over small distances.

Tangent Planes to Surfaces

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Partial Derivatives and Gradients

Derivadas Parciais e Gradientes

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

Planos tangentes às superfícies

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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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...

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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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