First, as the inequality is invariant under the transformations f \mapsto cf, x \mapsto -x , we may assume WLOG that f(0), \ f'(0) \ge 0, and ||f||_\infty = 1. Thus, we must prove
|f'(0)|^2 \le 4(1 + ||f''||_\infty) subject to ||f||_\infty = 1. Now, for all x \in (0, 1) we have
1 \ge f(x) \ge...
The waves book by A.P.French gives two characterizations of the reflection coefficient for a 1-D traveling wave encountering an interface between two media. On one hand, he writes
R = \frac{v_2 - v_1}{v_2 + v_1}
where v_i are the wave speeds in the two media. Later on, he writes the reflection...
I just stumbled upon this problem in Griffiths, and I, too, have the same difficulty as the OP. The problem seems to require the tangental derivatives \frac{\partial \vec{A}}{\partial x}, \ \frac{\partial \vec{A}}{\partial y} to be continuous across the current sheet. This link...
Can someone explain why the energy U of a closed system at constant entropy S and volume V reaches a minimum at equilibrium? Wikipedia has an article on principle of minimum energy, but I'm a little uncomfortable with the partial derivative manipulations in that article. Is it possible to argue...
So here's how I'm thinking about it. \mathbf{r} denotes two different things. On one hand, he write \mathbf{r} = \mathbf{r}(q_1, \dots, q_n, t) to denote the embedding of the configuration space in \mathbb{R}^3. The q_i's do not depend on time; the t-dependence signifies a possibly...
In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein arrives at the expression (equation 1.46) \mathbf{v}_i = \frac{d\mathbf{r}_i}{dt} = \sum_k \frac{\partial \mathbf{r}_i}{\partial q_k} \dot{q}_k + \frac{\partial \mathbf{r}_i}{\partial t}
where \mathbf{r}_i...