Jacobian Transformation.

In summary, the given integral \int\int(x-3y)DA is evaluated using the transformation x = 2u + v and y = u + 2v, where R is the triangular region with vertices (0,0), (2,1) and (1,2). The correct limits for u and v are 0<= u <= 1 and 0<= v <= 1-u. The Jacobian for this transformation is 3. The final answer for the integral is -3.
  • #1
tnutty
326
1

Homework Statement



Use the given transformation to evaluate the given integral.


[tex]\int[/tex][tex]\int(x-3y)DA[/tex]
R.

where R is the triangular region with vertices (0,0), (2,1) and (1,2) ; x = 2u + v , y = u + 2v



Trial :

Using the points given I came up with these equations for the triangle lines :

y = 2x;
y = 1/2x;
y = 3 - x;

Now just substituting x = 2u and y = u+2v I get this :

u = 0
v = 0
v = 3(1-u)

plotting this , I see that 0<= u <= 1 and 0<= v <= 3(1-u)

And from the original integral the integrand is x - 3y. Again substituting
for x and y with the given transformation coordinate for u and v I get :

(x-3y)DA = -(u+5v) dv*du

Now the integral becomes :

[tex]\int^{1}_{0} du[/tex][tex]\int^{3-3u}_{0} -u - 5v dvdu[/tex]

In which I get the answer -4/5 which is not correct. Any help?
 
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  • #2
ok so the variable transformation sounds alright, but where is the Jacobian...? (as in the title)

as do you the variable change, you are effectivley rotating & squeezes the axes, [tex] dA = dx dy \neq dudv [/tex]. So to preserve the original area you need to include the jacobian
 
Last edited:
  • #3
Since :

x = 2u + v , y = u + 2v

The jacobian is :

|2 1|
|1 2|
= 2*2 - 1*1 = 3 ;

The the answer is -4/5* 3 = -12/5 ?

I still did something wrong I guess.
 
  • #4
the limit for v is 0<y<1-u because v=1-u not 3(1-u) and you also need to include the jacobian in your equation, but you can just multiply the final answer you get by (3) instead of doing all your integrals after multiplying with the jacobian. And after doing this question I got -3 as the final answer.
 

What is a Jacobian Transformation?

A Jacobian Transformation is a mathematical tool used in multivariable calculus to transform coordinates from one coordinate system to another. It is used to calculate derivatives and integrals in different coordinate systems.

How is a Jacobian Transformation calculated?

To calculate a Jacobian Transformation, you first need to determine the partial derivatives of the original coordinate system with respect to the new coordinate system. These partial derivatives are then used to construct a matrix called the Jacobian matrix, which is then used to perform the transformation.

What is the significance of the Jacobian Transformation?

The Jacobian Transformation is significant because it allows for the calculation of derivatives and integrals in different coordinate systems, making it a powerful tool in many areas of science and engineering. It also helps to simplify complex mathematical problems by transforming them into a more manageable form.

What are some applications of the Jacobian Transformation?

The Jacobian Transformation has a wide range of applications in various fields such as physics, engineering, and economics. It is used in fluid dynamics, electromagnetics, robotics, and optimization problems, among others. It is also used in machine learning and computer graphics to transform data into different coordinate systems.

Are there any limitations to the use of Jacobian Transformation?

While the Jacobian Transformation is a powerful tool, it does have some limitations. It can only be applied to continuous functions, and the transformation may not be well-defined if the partial derivatives of the coordinate systems approach zero. Additionally, the calculation of the Jacobian matrix can be computationally intensive for complex systems.

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