Owen's Math Portal

To prove that the space of all invetible matricies is dense in the space of all matricies, it will suffice to show that the space of all invertible matricies is dense the space of matricies with distinct eigenvalues, as the space of matricies with distinct eigenvalues is dense in the space of all matricies. Denote \(E(n, \mathbb{R}) \) as the space of matricies with n distinct eigenvalues. Formally, $$ E(n, ℝ) = { \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \\ \end{bmatrix} : a_{ij} \in ℝ, \hspace{0.

In this article, we will examine the nature and utility of the first fundamental form, a quadratic form on a surface. Given a curve parametrized by \(u=u(t)\) and \(v=v(t)\) which lies on a surface \(X\) parametrized by \(X=X(u,v)\) we have that $$ ds = |\frac{dX}{dt}| dt = |X_u \frac{du}{dt} + X_v \frac{dv}{dt}| dt = \sqrt{(X_u \dot{u} + X_v \dot{v}) \cdot (X_u \dot{u} + X_v \dot{v})} $$ Now defining the coefficients, called the First Fundamental Form coefficients E, F and G, we have

The Gaussian Integral is given by $$ \int_{-\infty}^{\infty} e^{-x^2} dx $$ This is a beautiful integral with many important applications to probability and statistics, namely normal distributions. Here, we will evaluate this integral using techniques from Calculus 3. First, if we define \(I=\int_{-\infty}^{\infty} e^{-x^2} dx\) we have that $$ I^2=\int_{-\infty}^{\infty} e^{-x^2} dx \int_{-\infty}^{\infty} e^{-y^2} dy= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-x^2-y^2} dy dx= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)} dy dx $$ From here, we will preform a change into polar coordinates, with \((x,y) \rightarrow (t, \theta)\) and \(x=r\cos\theta\) , \(y=r\sin\theta\) .