Why Is the Jacobian Determinant Used in Double and Triple Integrals?

In summary, a double integral is a mathematical operation used to calculate the volume of a three-dimensional shape over a specific region in two-dimensional space. It involves integrating a function with respect to two variables, typically x and y, over a given region. The main difference between a double integral and a triple integral is that a triple integral involves integrating a function with respect to three variables, making it a three-dimensional operation. Double and triple integrals have various applications in physics, engineering, economics, and probability and statistics. To set up a double integral, the limits of integration for each variable must be determined, and the double integral formula can be used to integrate the function over the given region. Some common techniques for evaluating double and triple integrals include
  • #1
Oggy
26
0
I understand double and triple integrals and all, but I'm just wondering why is
[tex]dxdy=|J|dudv\ x=f(u,v)\ y=g(u,v)[/tex] Where does that derive from? Why is it? (and also for triple integrals)
 
Last edited:
Physics news on Phys.org
  • #2
It comes from considering the area change of an infinitesimal area moving from the xy plane to uv plane
 
  • #3


The expression dxdy=|J|dudv is known as the Jacobian determinant, and it is used to convert integrals from one coordinate system to another. In this case, it is being used to convert from Cartesian coordinates (x and y) to parametric coordinates (u and v).

The Jacobian determinant is derived from the chain rule of multivariable calculus. It represents the change in area between two coordinate systems. In the case of double integrals, the Jacobian determinant is a 2x2 matrix and represents the change in area between a small rectangle in the u-v plane and its corresponding rectangle in the x-y plane.

The reason for using the Jacobian determinant in double and triple integrals is to make the integration process simpler and more efficient. By converting to parametric coordinates, we can often simplify the integrand and make the limits of integration easier to work with.

In the case of triple integrals, the Jacobian determinant is a 3x3 matrix and represents the change in volume between a small rectangular prism in the u-v-w space and its corresponding prism in the x-y-z space.

Overall, the use of the Jacobian determinant in double and triple integrals is a powerful tool that allows us to solve complicated integrals in a more manageable way.
 

Related to Why Is the Jacobian Determinant Used in Double and Triple Integrals?

What is a double integral?

A double integral is a type of mathematical operation used to calculate the volume of a three-dimensional shape over a specific region in two-dimensional space. It involves integrating a function with respect to two variables, typically x and y, over a given region.

What is the difference between a double integral and a triple integral?

A double integral involves integrating a function with respect to two variables, while a triple integral involves integrating a function with respect to three variables. This means that the region in which the function is being integrated is three-dimensional instead of two-dimensional.

What are the applications of double and triple integrals?

Double and triple integrals are commonly used in physics, engineering, and economics to calculate volume, surface area, mass, and other quantities. They are also used in probability and statistics to calculate the probability of an event occurring within a given region.

How do you set up a double integral?

To set up a double integral, you first need to determine the limits of integration for each variable. This involves identifying the boundaries of the region in which the function is being integrated. Then, you can use the double integral formula to integrate the function over the given region.

What are some common techniques for evaluating double and triple integrals?

Some common techniques for evaluating double and triple integrals include using the properties of integrals, changing the order of integration, and using symmetry to simplify the integral. Other techniques include using substitution, integration by parts, and partial fraction decomposition.

Similar threads

Replies
5
Views
2K
Replies
2
Views
1K
Replies
4
Views
2K
  • Calculus
Replies
5
Views
1K
  • Calculus
Replies
24
Views
3K
  • Calculus
Replies
6
Views
2K
  • Introductory Physics Homework Help
Replies
5
Views
677
Replies
1
Views
863
  • Calculus
Replies
1
Views
1K
Back
Top