I'm looking at the solution for finding maximum and minimum of $f(x,y) = xye^{-x² -y^2}$ To do so, it first finds the critical points using the secon...

I have this function of a matrix and two vectors: $$f(x, V, y) = x^T V y$$ where $$V \in R^{d \times d}, x \in R^{d}, y \in R^{d}$$ Basically I need to op...

From Weyl's theorem, i.e.: Let $A$ and $E$ be $n\times n$ real symmetric matrices. Let $\alpha_1\geq\ldots\geq\alpha_n$ be the eigenvalues of $A$ and...

$x^2 = y^2 + xy + 5$, where $x$ and $y$ are natural numbers. Here is what I have so far: $x \neq y$ (from the equation). $x$ is always odd (using the equa...

Problem: Let T: $\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear map given by $$T\left[ \begin{matrix} x\\y\\ z\end{matrix} \right]= \left[ \begin{matrix...

What happens when the series is alternating and the tests that I know are inconclusive? $\sum_{n=1}^\infty (-1)^n (n/(n+5))$. The $b_n$ series is not decr...

"Let $P$ be the change-of-basis matrix from a basis $S$ to a basis $S'$ in a vector space $V$. Then, for any vector $v \in V$, we have $$P[v]_{S'}...

I am asked to show that the pseudosphere has Gaussian Curvature $-1$ at all points. So I parametrized the tractrix as: \begin{equation} \alpha(t) = (\sin{...

Find the limit of: $\lim_\limits{x \to 3} \frac{\ln x - \ln 3}{x - 3} $ For starters, I substitute $y = x - 3$. So: $\lim_\limits{x \to 3} \frac{\ln x - \...

From a bridge deck of $52$ cards, we draw $13$. What is the probability that we have $5$ spades in our hand? I think that there are $\dfrac{52!}{13! \cdot...