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The "coordinate axes" in a [itex]\tau[/itex], [itex]\sigma[/itex] coordinate system are the lines [itex]\tau= 0[/itex] and [itex]\sigma= 0[/itex] which mean [itex]t_1- t_2= 0[/itex] and lines parallel to that.inviziblesoul said:Thank you very much for your excellent efforts and this great explanation. However, I am not clear at certain points.
You have rightly pointed out: the aim here is to express the double integral in terms of a single integral. Furthermore, C is a function of the difference [itex]\tau=t_1−t_2[/itex].
I did not understand your phrase <<That is a rectangle in the τ, σ plane with its digonals parallel to the axes.>> How do you know that its a rectangle and its diagonals are parallel to the axes (the [itex]\tau, \sigma[/itex] axes ?).
The lines [itex]\tau= t_1+ t_2= constant[/itex] is the same as [itex]t_2= -t_1+ constant[/itex] have slope -1. The lines [itex]\sigma= t_1- t_2= constant[/itex] or [itex]t_2= t_1- constant[/itex] have slope 1. They are perpendicular so we still have an "orthogonal" coordinate system.and how did you choose [itex]\sigma = t_1 + t_2?[/itex] why not some other function?
I will greatly appreciate if you can kindly refer me some reading on this topic.
I have attached my solution as well. I have not introduced a new variable, however, I have used [itex]\tau[/itex] and [itex]t_1[/itex].
Thank you for your time.
A double integral is a type of mathematical calculation that involves evaluating a function over a two-dimensional region. It can be thought of as finding the volume under a surface in three-dimensional space.
A double integral is typically computed by first determining the limits of integration for both the x and y variables, then breaking the region into small, rectangular elements. The function is then evaluated at each element and the results are summed together to find the total volume.
Double integrals are used in many areas of science and engineering to calculate properties such as area, volume, and mass. They are particularly useful in physics and engineering for calculating the work done by a force over a given region.
A single integral computes the area under a curve in one dimension, while a double integral computes the volume under a surface in two dimensions. Essentially, a double integral involves performing a single integral over a range of values for a second variable.
Yes, a double integral can be applied to any continuous function over a two-dimensional region. However, the computation may become more complex for certain functions, and may require advanced techniques such as change of variables or integration by parts.