Irid said:Homework Statement
Consider a function which depends only on a difference between two variables, and integrate it with respect to both:
[tex]
\int_a^b \int_a^b f(x-y)\, dxdy
[/tex]
Is there any way to simplify this expression, like reducing it into a 1-D integral?
Irid said:This gives me
[tex] dxdy = (dv^2-du^2)/4 [/tex]
and I don't see how this makes the integral any easier
An integral depending on coordinate differences is a mathematical concept used in the field of multivariate calculus. It involves calculating the area under a curve or the volume under a surface by considering the differences in coordinates between two points.
A regular integral calculates the area or volume by considering the entire region under the curve or surface. An integral depending on coordinate differences only takes into account the differences in coordinates between two points, making it a more precise and localized calculation.
Integrals depending on coordinate differences are commonly used in fields such as physics, engineering, and economics to calculate quantities such as work, force, and profit. They are also used in optimization problems and in the study of fluid dynamics.
The basic steps for solving an integral depending on coordinate differences are: 1) determining the limits of integration, 2) setting up the integral using the appropriate formula, 3) evaluating the integral using techniques such as substitution or integration by parts, and 4) interpreting the result in the context of the problem.
One limitation of using integrals depending on coordinate differences is that they can only be applied to functions that are continuous and differentiable on the given interval. Additionally, they may not always provide an accurate representation of the entire region under the curve or surface, as they only consider the differences between two points.