See: Math Notebook

I'm writing up my solutions to math exercises that I have done to increase my understanding.

To Do

- Show that {$SU(2)$} is closed (and thus compact, because it is bounded).
- Show that {$SL(2,\mathbb{C})$} is an analytic manifold.
- Show that {$SU(2)$} is a differentiable manifold but not analytic.
- Show that {$SL(2,\mathbb{R})$} is a differentiable manifold but not analytic.
- Show that the complex solutions {$x$} and {$y$} to the equation {$x^2+y^2=1$} form a sphere with two holes.
- Learn how to calculate the hyperarea of an n-sphere and the hypervolume of the n-ball.

Need to Write Up

- {$SU(N)$} is bounded. Multiplying by the conjugate transpose shows that, for each row, and each column, the sum of squares equal 1, thus for each entry the magnitude is at most 1.
- {$SL(N)$} is not bounded. The determinant is 1, thus the volume is fixed, thus the vectors can be arbitrary large in any dimension.

Solved

- Combinatorial interpretation of {$e^{A} \cdot e^B = e^{A+B}$}
- Square root of identity matrix
- Interpret combinatorially det(A)det(B) = det(AB)
- Why symplectic manifolds are even dimensional
- Confirm that a harmonic oscillator preserves symplectic area
- What is the conjugate of a multidimensional number
- Relating definitions of {$e$}

Parsiųstas iš http://www.ms.lt/sodas/Book/MathExercises

Puslapis paskutinį kartą pakeistas 2020 vasario 27 d., 11:09