How to Solve Integrals with Change of Variables and Jacobians?

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In summary: The u and v were given changes, so I need to find out how to solve this integral with the indicated changes/transformations.
  • #1
robierob
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I need to find out how to solve this integral with the indicated changes/transformations.

int.[0 to 2/3]int[y to 2-2y]
(x+2y)e^(y-x)
dxdy

u=x+2y v=x-y

I know that the xy region is x=y y=0 and y=1- (x/2)
which is a triangle

so I created systems with U and V but can't get a new bounded region...

I just get u=v and two other lines that intersect at the origin.
If anyone can tell me what's up with this it would be helpfull.

Also, does it seem weird that my given v doesn't match the problem exactly?
the u and v were given changes.
 
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  • #2
robierob said:
I need to find out how to solve this integral with the indicated changes/transformations.

int.[0 to 2/3]int[y to 2-2y]
(x+2y)e^(y-x)
dxdy

u=x+2y v=x-y

I know that the xy region is x=y y=0 and y=1- (x/2)
which is a triangle

so I created systems with U and V but can't get a new bounded region...

I just get u=v and two other lines that intersect at the origin.
If anyone can tell me what's up with this it would be helpfull.

Also, does it seem weird that my given v doesn't match the problem exactly?
the u and v were given changes.
What does the region look like in the xy-coordinate system?

One line is y= x. If v= x- y, what is v on that line? Another line is y= 1- (1/2)x or 2y= 2- x so x+ 2y= 2. What is u on that line? Finally, the third line is y= 0. In that case, u= x and v= x so y= 0 corresponds to the line u= v. What does that look like in the uv-coordinate system?
 
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  • #3
Thanks

Thanks, that put me back on track!
 

What is a change of variables?

A change of variables refers to the process of transforming a set of variables in a mathematical equation or problem into a new set of variables. This can often simplify the problem or make it more easily solvable.

What is a Jacobian?

A Jacobian is a mathematical matrix that is used to describe the relationship between two sets of variables in a change of variables. It is often used to calculate the transformation of integrals between different coordinate systems.

Why is a Jacobian important in change of variables?

The Jacobian is important because it allows us to use the change of variables technique to solve complex mathematical problems and make them more manageable. Without the Jacobian, it would be difficult to perform these transformations accurately.

What is the relationship between the Jacobian and the determinant?

The Jacobian is equal to the determinant of the transformation matrix. This means that the determinant can be used to calculate the Jacobian, and vice versa. The determinant is important in change of variables as it helps determine the scaling factor of the transformation.

How is the Jacobian used in real-world applications?

The Jacobian is used in various fields such as physics, engineering, and economics to solve problems involving multiple variables and coordinate systems. It is also used in computer graphics and machine learning to transform data and images into different coordinate systems for analysis and processing.

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