Looking for a rectifying chart

  • Thread starter mihyaeru
  • Start date
In summary, we have found a solution for the differential equations given in the problem by changing the variables and using the Pythagorean identity. We can now plot this solution on a rectifying chart at (0,0) of the vector field X: R^2 -> TR^2 , (x,y) -> X(x,y) with X(x,y):= (cos^2(x+y)+sin^2(x-y))d/dx +
  • #1
mihyaeru
4
0
given task
Find a rectifying chart at (0,0) of the vector field
X: R^2 -> TR^2 , (x,y) -> X(x,y)
with X(x,y):= (cos^2(x+y)+sin^2(x-y))d/dx + (cos^2(x+y)-sin^2(x-y))d/dy .

attempt of solution
We know that there exists a rectifying chart since X(0,0)=(1,1).
A transversal direction would be (1,-1).

Therefore we get the differential equations:
dx/dt= cos^2(x+y)+sin^2(x-y) (1)
and
dy/dt= cos^2(x+y)-sin^2(x-y) (2)

Next we have to find theire solutions. Change of variables and integration lead me to the general solution of (1):

t+const = sec(2y)*tan^-1(sin(x-y)*sec(x+y))

=> tan(t*sin(2y))+const = sin(x-y)/sin(x+y)
where I got stuck... //Recall we have to solve this equation for x and (2) for y.

Maybe I should change my coordinate system at the very beginning, such that x+y=a and b=x-y? What do you think? Any ideas?

Kind regards,
mihyaeru
 
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  • #2



Dear mihyaeru,

Thank you for sharing your attempt at solving this problem. It seems like you are on the right track, but as you mentioned, you have reached a point where you are stuck. Let's take a closer look at your equations and see if we can find a solution.

First, let's change the variables as you suggested, such that x+y=a and b=x-y. This will give us the following equations:

dx/dt= cos^2(a)+sin^2(b) (1)
dy/dt= cos^2(a)-sin^2(b) (2)

Now, we can use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to rewrite these equations as:

dx/dt= 1 (1a)
dy/dt= cos(2a) (2a)

Integrating both equations with respect to t, we get:

x= t + const (1b)
y= 1/2sin(2a)t + const (2b)

Now, we can use these equations to eliminate t and obtain a solution for x and y in terms of a and b. Substituting (1b) into (1a), we get:

dx/dt= 1 = cos^2(a)+sin^2(b)
=> t= cos^2(a)+sin^2(b) - const

Similarly, substituting (2b) into (2a), we get:

dy/dt= cos(2a) = cos^2(a)-sin^2(b)
=> t= 1/2sin(2a) + const

Since both t's are equal, we can equate the right sides of these equations and solve for a and b:

cos^2(a)+sin^2(b) - const = 1/2sin(2a) + const
=> cos^2(a)-1/2sin(2a) + const = 0

This is a quadratic equation in terms of cos(a) with solutions:

cos(a)= 1/4 +/- sqrt(1/16-const)
=> a= arccos(1/4 +/- sqrt(1/16-const))

Now, we can use this solution for a in (1b) and (2b) to get the solutions for x and y:

x= t + const =
 

FAQ: Looking for a rectifying chart

1. What is a rectifying chart?

A rectifying chart is a type of chart used in scientific experiments to compare the experimental data with a standard or reference data set. It helps to identify any discrepancies or errors in the experimental data and make necessary adjustments to ensure accurate results.

2. How is a rectifying chart created?

A rectifying chart is created by plotting the experimental data on a graph and overlaying it with the reference data set. The data points are then compared and any differences are noted. The rectifying chart can be created manually or with the help of software programs.

3. What is the purpose of using a rectifying chart in scientific experiments?

The purpose of using a rectifying chart is to ensure the accuracy and reliability of the experimental data. It helps to identify any systematic errors or inconsistencies in the data and make necessary adjustments to eliminate them. This can help to improve the validity of the results and increase the credibility of the experiment.

4. Can a rectifying chart be used in all types of scientific experiments?

Yes, a rectifying chart can be used in various fields of science, including biology, chemistry, physics, and engineering. It is particularly useful in experiments where precise measurements are crucial, and the results need to be compared with a standard or reference data set.

5. What are the limitations of using a rectifying chart?

One limitation of using a rectifying chart is that it relies on the accuracy and reliability of the reference data set. If the reference data set is flawed or outdated, it can lead to incorrect conclusions. Additionally, a rectifying chart may not be effective in identifying random errors or errors caused by external factors that are not included in the reference data set.

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