Jacobian Calculation for Transformation (x, y) to (u, v)

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In summary, to calculate the Jacobian d(x,y)/d(u,v) for the given transformation, you can either solve for x and y as functions of u and v, use implicit differentiation to find the partial derivatives, or take the reciprocal of d(u,v)/d(x,y).
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franky2727
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calculate the jacobian d(x,y)/d(u,v) of the transformation u=x2+y2
v=x+y

for this do i first have to calculate the jacobian d(u,v)/d(x,y) then do 1over the answer? because i would assume the matrix to be det|{(dudx,dudy)(dvdx,dvdy)} but with (u,v) on top i cannot get this
 
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franky2727 said:
calculate the jacobian d(x,y)/d(u,v) of the transformation u=x2+y2
v=x+y

for this do i first have to calculate the jacobian d(u,v)/d(x,y) then do 1over the answer? because i would assume the matrix to be det|{(dudx,dudy)(dvdx,dvdy)} but with (u,v) on top i cannot get this

You can do any of three things:
1) Solve for x and y as functions of u and v.
2) Use implicit differentiation to find [itex]\partial x/\partial u[/itex], [itex]\partial y/\partial u[/itex], [itex]\partial x/\partial v[/itex], and [itex]\partial y/\partial v[/itex].
3) Take the reciprocal of d(u,v)/d(x,y).
 

1. What is the purpose of calculating the jacobian?

The jacobian is a mathematical tool used in multivariable calculus to determine the relationships between different variables in a system. It is commonly used in fields such as physics, engineering, and economics to analyze and model complex systems.

2. How is the jacobian calculated?

The jacobian is calculated by taking the partial derivatives of a set of equations with respect to each variable in the system. These partial derivatives are then arranged in a matrix, known as the jacobian matrix, which represents the relationships between the variables in the system.

3. What is the significance of the jacobian determinant?

The jacobian determinant, also known as the determinant of the jacobian matrix, is used to determine the change in volume of a system when the variables are transformed. It is an important concept in integration and is used in various applications, such as calculating the volume of a solid with irregular shape.

4. Can the jacobian be used for non-linear systems?

Yes, the jacobian can be used for non-linear systems. However, the calculations may be more complex and require numerical methods to approximate the jacobian matrix. In these cases, the jacobian may also vary at different points in the system.

5. What are some practical applications of the jacobian?

The jacobian has various practical applications, including optimization, control systems, and modeling physical systems. It is also used in machine learning and computer graphics to analyze and simulate complex systems. Additionally, the jacobian is used in differential equations to study the stability and behavior of dynamic systems.

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