Math and stuff

Proposition Show that the function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by \[\begin{align*} f(x) &= \begin{cases} x^2\sin(1/x) & (x \ne 0), \\ 0 & (x = 0) \end{cases} \end{align*}\] is differentiable in $\mathbb{R}$, yet $f’$ is not even continuous at 0. Solution $f’(x) = 2x\sin(1/x) - \cos(1/x)$ when $x \ne 0$....

Proposition Evaluate $\int_{\gamma} \exp(3z) dz$ for each of the following paths: $\gamma$ is the line segment from 1 to $i$ $\gamma = C[0, 3]$ $\gamma$ is the arc of the parabola $y = x^2$ from $x = 0$ to $x = 1$. Solution 1 Let $\gamma(t) = 1 - t...

Proposition Prove that a $k$-dimensional (vector) subspace of $\mathbb{R}^n$ is a $k$-dimensional manifold. Solution Let $V$ denote the subspace and $v_1, \cdots, v_k$ denote a basis of $V$. We can extend it to a basis of $V$. Then $A = \begin{bmatrix} v_1 & \cdots & v_n \end{bmatrix}$ is a invertible...

Proposition If $T$ is an inner product on $V$, a linear transformation $f: V \rightarrow V$ is called self-adjoint (with respect to $T$) if $T(x, f(y)) = T(f(x), y)$ for $x, y \in V$. If $v_1, \cdots, v_n$ is an orthonormal basis and $A = (a_{ij})$ is the matrix of...

Proposition If $A[x]$ is Noetherian, is $A$ necessarily Noetherian? Solution Yes. Let $\phi: A[x] \rightarrow A$ be a ring homomorphism such that $a \mapsto a$ for all $a \in A$ and $x \mapsto 1$. This uniquely determines the homomorphism because every element in $A[x]$ is a polynomial in $x$ with...