What does it mean for a change of variables to be UNITARY?

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A change of variables is considered unitary if the Jacobian matrix of the transformation is unitary, meaning it preserves complex norms. To verify this, one must differentiate the new variables with respect to the original variables, form a matrix from these derivatives, and check if the product of the matrix and its Hermitian conjugate equals the identity matrix. Unitary transformations maintain the volume of the space, ensuring that the integral of the transformed variables remains invariant. This concept parallels orthonormal transformations in real-valued functions. Understanding these properties is crucial for applications in quantum mechanics and other fields involving complex variables.
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So if I'm changing from variables x,y to variables \alpha = f(x,y), \beta = g(x,y), what exactly does it mean to stay this change of variables is unitary, and how can I tell if it is or if it isn't?
 
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I never encountered it, but probably it means that the jacobian matrix of the change of variables is a unitary matrix. In other words: you differentiate f and g with respect to x and y, put this four functions into a matrix M, and verify that U multiplied by the hermitian conjugate of U is the identity 2 x 2 matrix. To find the hermitian conjugate of a matrix you transpose it and then you take the complex conjugate.

I'm not really sure that all this is true though.
 
A unitary transformation is one that preserves complex norm, ie complex numbers (or functions) of unit absolute value transforms to complex numbers (or functions) of unit absolute value. If you consider real valued functions of real variables this is the same as orthogonal (or more correctly orthonormal) transformations. For a transformation to be unitary the Jacobian must be a unitary matrix as stated above. This will preserve the volume V = \int_{V}dx_{1}...dx_{n}.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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