What is the key integration technique needed for this double integral?

In summary, the integral is difficult to solve and the area around the purple dot is represented by a rectangle.
  • #1
docnet
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Homework Statement
Evaluate ∫∫ -4 / (1+ x^2 + y^2)^4 dx dy
Relevant Equations
∫∫ -4 / (1+ x^2 + y^2)^4 dx dy
Dear all,

Last semester on the final exam, our professor gave us an integral that seems difficult to solve.

The integral came at the end of a lengthy problem, where we were asked to find the net Gauss curvature of Enneper's surface.

The integral that emerged is the following.

Screen Shot 2021-01-05 at 12.11.30 AM.png


We tried substituting x = sinθ, y = cosΘ and ended up with

Screen Shot 2021-01-05 at 12.20.41 AM.png


which does not help.

We tried evaluating the following single integral to see if it simplifies.

Screen Shot 2021-01-05 at 12.11.39 AM.png
Screen Shot 2021-01-05 at 12.11.47 AM.png


Screen Shot 2021-01-05 at 12.11.58 AM.png


...And the calculations continue for another 3 pages and become more complicated.The evaluated integral is the following.

Screen Shot 2021-01-05 at 12.12.28 AM.png
We could integrate this expression with respect to y and evaluate the result from -∞ to ∞ .

The answer for the net curvature is - 4π in differential geometry textbooks.

Is there a key integration technique we are missing?

This integral seems difficult to solve without using technology.

Thank you.
 
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  • #2
docnet said:
Homework Statement:: Evaluate ∫∫ -4 / (1+ x^2 + y^2)^4 dx dy
Relevant Equations:: ∫∫ -4 / (1+ x^2 + y^2)^4 dx dy

Change to polar coordinates.
 
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  • #3
docnet said:
We tried substituting x = sinθ, y = cosΘ
George Jones said:
Change to polar coordinates.
I agree with George. You almost had it, @docnet.
 
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  • #4
Dear @Mark44 and @George Jones,

Thank you for your suggestions.

We substituted x = r sinΘ, y = r cosΘ and used the chain rule to determine dx and dy.

Screen Shot 2021-01-05 at 1.05.50 AM.png


The new boundaries of integration in polar coordinates are determined by the sets {0 ≤ r ≤ ∞} and {0 ≤ Θ ≤ 2π}.

We then expanded the numerator to obtain the following integrals.

Screen Shot 2021-01-05 at 1.22.42 AM.png

Screen Shot 2021-01-05 at 1.23.12 AM.png

Screen Shot 2021-01-05 at 1.24.33 AM.png

Screen Shot 2021-01-05 at 1.24.47 AM.png
Evaluating the first integral

We try substituting r = tan2u to simplify the process, and treat sin2Θ as a constant. The new boundaries of integration are determined by the set {0 ≤ u ≤ .5π} and we have

Screen Shot 2021-01-05 at 1.38.24 AM.png


to be continued.
 
  • #5
When you convert from rectangular to polar, dxdy becomes ##rdrd\theta##, so the numerator will become ##-4rdrd\theta##.
 
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  • #6
@docnet , what is the area bounded by the following four curves:
a circle about the origin of radius ##r##;
a circle about the origin of radius ##r + dr##;
a line from the origin at angle ##\theta##;
a line from the origin at angle ##\theta + d\theta##.

Draw a diagram to see.
 
  • #7
Mark44 said:
When you convert from rectangular to polar, dxdy becomes ##rdrd\theta##, so the numerator will become ##-4rdrd\theta##.

@Mark44
Thank you for pointing out my error. This would have been a time sink.

So we have
Screen Shot 2021-01-05 at 1.59.13 AM.png


and
Screen Shot 2021-01-05 at 2.12.13 AM.png


and

Screen Shot 2021-01-05 at 2.03.33 AM.png


Which is our net Gaussian curvature of Enneper's Surface.
 
Last edited:
  • #8
George Jones said:
@docnet , what is the area bounded by the following four curves:
a circle about the origin of radius ##r##;
a circle about the origin of radius ##r + dr##;
a line from the origin at angle ##\theta##;
a line from the origin at angle ##\theta + d\theta##.

Draw a diagram to see.

@ George Jones

The area bounded by the four curves is represented in the following diagram, the area around the purple dot. The distance between the two red circles is dr and the angle created by the two lines is dΘ.

Screen Shot 2021-01-05 at 2.18.11 AM.png
 
  • #9
docnet said:
The area bounded by the four curves is represented in the following diagram, the area around the purple dot.

View attachment 275654
Yes. What does this area equal?
 
  • #10
George Jones said:
Yes. What does this area equal?

@George Jones,

Integrating 1 to determine the area, we have

Screen Shot 2021-01-05 at 2.25.45 AM.png


=

Screen Shot 2021-01-05 at 2.26.49 AM.png


=

Screen Shot 2021-01-05 at 2.27.16 AM.png


=

Screen Shot 2021-01-05 at 2.27.32 AM.png
 
  • #11
The textbook gives the net curvature of Enneper's surface as - 4π but frankly I don't know if that is equal to the net Gaussian curvature. I assume they are different.
 
  • #12
Let's do this a little more intuitively. Since ##dr## and ##d\theta## are "infinitesimally small", the bounded area is a "rectangle". What are the lengths of its sides?
 
  • #13
George Jones said:
Let's do this a little more intuitively. Since ##dr## and ##d\theta## are "infinitesimally small", the bounded area is a "rectangle". What are the lengths of its sides?

Since a length of an arc of a circle is the radius times the angle, we have a rectangle of side dr and side rdΘ. length rdΘ corresponds to the side perpendicular to the radial direction.

The corresponding area is r drdΘ, that is congruent to our infinitesimally small area element dydx in rectangular coordinates.
 
  • #15
@Delta2.

The description of the parallelogram in the wikipedia link feels reminiscent of the geometric representation of the determinant of a 2 x 2 matrix.

In differential geometry, we learned the area factor corresponding to the metric is determined by taking the square root of the determinant of the metric.

Screen Shot 2021-01-05 at 2.59.55 AM.png


I assume this is equivalent to the Jacobian method you mentioned?
 
  • #16
Ok, I frankly don't know the method you are saying, it uses the square root of the determinant of a different matrix (not the Jacobian matrix) but it gives the same result, at least for two variables.
 
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  • #17
oh ok. I think if we had time, we could show the Jacobian matrix and the matrix of the metric are at least related somehow.

The Jacobian matrix is

Screen Shot 2021-01-05 at 3.11.47 AM.png


The area element corresponding to the metric is determined by taking the partial derivatives of a map

x = r sinΘ
y = r cosΘ

dx = rcosΘdΘ + sinΘdr
dy = -rsinΘdΘ + cosΘdr


and computing dx2 + dy2 = dr2 +r22

The diagonal elements of the matrix corresponding to our metric are 1 and r2, and there are no off-diagonal elements since there are no cross terms in our metric.

Its determinant is simply r2 whose square root is r.

So r is the factor of integration whenever we switch from euclidian coordinates to polar coordinates. I assume dr and just follow.

We haven't learned about using the Jacobian to do change of variables in an integral. We only learned to use the Jacobian to give sufficient conditions in certain optimization problems in class.
 
  • #18
if ##J## is the jacobian matrix as you have it in post #17 then it is $$df_1df_2...df_n=|det(J)|dx_1dx_2...dx_n$$ when we change the variables from ##f_1,f_2,...,f_n## to ##x_1,x_2,...,x_n##.
 
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  • #19
Dear @Delta2,

so we have

x = r sinΘ
y = r cosΘ


whose Jacobian is

rcosΘ sinΘ
-rsinΘ cosΘ


and the determinant is

rcos2Θ + rsin2Θ = r

which gives our change of variables from xy space to polar coordinates.

dxdy = > rdrdΘ
 
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Related to What is the key integration technique needed for this double integral?

1. What is a double integral?

A double integral is a mathematical concept used to find the volume under a surface in three-dimensional space. It is essentially the integration of a function of two variables over a two-dimensional region.

2. How do you evaluate a double integral?

To evaluate a double integral, you first need to determine the limits of integration for both variables. Then, you can use various integration techniques, such as substitution or integration by parts, to solve the integral. Finally, you can evaluate the integral using the Fundamental Theorem of Calculus.

3. What is the purpose of evaluating a double integral?

Evaluating a double integral allows us to find the volume under a three-dimensional surface, which has many practical applications in fields such as physics, engineering, and economics. It also helps us understand and analyze the behavior of functions in multiple dimensions.

4. What are some common mistakes when evaluating a double integral?

Some common mistakes when evaluating a double integral include forgetting to change the limits of integration when using substitution, mixing up the order of integration, and not properly setting up the integral for the given region.

5. Are there any tools or techniques that can help with evaluating a double integral?

Yes, there are several tools and techniques that can make evaluating a double integral easier. These include using graphing calculators or computer software, using symmetry to simplify the integral, and breaking the integral into smaller, more manageable parts.

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