Name: using the jacobian to simplify integrals
Uploaded: 2026-06-08T09:38:39.155Z
Duration: 1 min 30 s
Channel: JoVE

In multivariable calculus, evaluating a multiple integral over a region, R, often seems difficult as the region, R, may have complex boundaries in x and y coordinates.To simplify this mathematical evaluation, a change of variables is made into a new coordinate system defined by the variables u and v through a specific set of transformation equations.This transformation completely reshapes the original region, R, from the xy-plane onto a new region, S, in the uv-plane, creating two different area elements, dxdy and dudv, in two different coordinate systems.To link these two area elements, a critical component, the Jacobian, is calculated. It measures how much the original region, R, is stretched or compressed during the transformation to the new region, S.The Jacobian is obtained by evaluating a determinant that contains the partial derivatives of the original variables x and y with respect to the new variables u and v.Finally, the newly substituted function is multiplied by the absolute value of the Jacobian. This step completes the transformation process and yields the fully rewritten integral over the new region, S.

change of variables in multiple integrals

Multiple integrals are often used to evaluate areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, however, the original region may have complicated curved boundaries when expressed in Cartesian coordinates. These complex boundaries can make the limits of integration difficult to describe and the overall calculation cumbersome. To simplify the evaluation process, a change of variables is introduced that transforms the original coordinates into a new coordinate system better suited to the geometry of the region.A change of variables replaces the original coordinates x and y with new variables u and v. Through this transformation, the original region R in the xy-plane is mapped onto a new region S in the uv-plane. The purpose of this transformation is to reshape the integration region into a simpler form, often converting curved boundaries into straight lines or rectangular regions.Because the coordinate system changes, the small area elements in the two systems are not identical. A small rectangular region in the uv-plane may become stretched, compressed, or skewed when mapped into the xy-plane. The factor that measures this local geometric distortion is called the Jacobian.The Jacobian determines how area changes under the transformation. It is obtained from the partial derivatives that describe how the original variables depend on the new variables. Geometrically, the Jacobian measures how much a small area element expands or contracts during the mapping process.When rewriting the integral in the new variables, the integrand must be multiplied by the absolute value of the Jacobian. This adjustment ensures that the transformed integral correctly preserves the area of the original region. By incorporating the Jacobian, complicated integrals over irregular regions can often be converted into much simpler integrals in the transformed coordinate system.

Change of Variables in Multiple Integrals

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JoVE Core

Calculus

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Multiple Integrals and Applications

KEY TERMS AND CONCEPTS

using the jacobian to simplify integrals

Multiple integrals are used to find areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, the original region has curved boundaries when written in Cartesian coordinates. Those boundaries can make the limits of integration hard to describe and the calculation more difficult.

A change of variables helps simplify the setup. It replaces the original coordinates x and y with new variables u and v. The original region R in the xy-plane is then mapped to a new region S in the uv-plane. This new coordinate system is often a better fit for the shape of the region and can turn curved boundaries into straight lines or rectangular regions.

The two coordinate systems do not use the same small area element. A small rectangle in the uv-plane may become stretched, compressed, or skewed when it is mapped into the xy-plane. The factor that measures this local distortion is called the Jacobian. It is found from the partial derivatives that describe how the original variables depend on the new variables, and it shows how much a small area element expands or contracts during the mapping.

When the integral is rewritten in the new variables, the integrand must be multiplied by the absolute value of the Jacobian. This adjustment makes sure the transformed integral correctly preserves the area of the original region. With the Jacobian included, complicated integrals over irregular regions can often be changed into much simpler integrals in the new coordinate system.

Using the Jacobian to Simplify Integrals

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key terms and concepts

Learn how change of variables and the Jacobian simplify multiple integrals by mapping curved regions into easier shapes for area calculations.

Using the Jacobian to Simplify Integrals | 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 the Jacobian to Simplify Integrals

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