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

[Oggy]

I understand double and triple integrals and all, but I'm just wondering why is
$$dxdy=|J|dudv\ x=f(u,v)\ y=g(u,v)$$ Where does that derive from? Why is it? (and also for triple integrals)

 

[matt grime]

It comes from considering the area change of an infinitesimal area moving from the xy plane to uv plane

 

[M98Ranger]

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.