25 Sep, 2015

First Applied Math Class

Preliminaries:

  • Book: Mathematical Methods for Physics and Engineering, by Riley, Hobson and Bence

Derivatives:

  • Everything that starts with a $c$ will be negated
  • $\frac{d}{dx}(sin~x) = cos~x$
  • $\frac{d}{dx}(cos~x) = -sin~x$
  • $\frac{d}{dx}(tan~x) = sec^2x$
  • $\frac{d}{dx}(sec~x) = sec~x \cdot tan~x$
  • $\frac{d}{dx}(csc~x) = -csc~x \cdot cot~x$
  • $\frac{d}{dx}(cot~x) = -csc^2x$
  • $\frac{d}{dx}(sin^{-1}x) = \frac{1}{\sqrt{1-x^2}}$
  • $\frac{d}{dx}(cos^{-1}x) = -\frac{1}{\sqrt{1-x^2}}$
  • $\frac{d}{dx}(tan^{-1}x) = \frac{1}{1+x^2}$
  • $\frac{d}{dx}(sec^{-1}x) = \frac{1}{|x|\sqrt{x^2-1}}$
  • $\frac{d}{dx}(csc^{-1}x) = -\frac{1}{|x|\sqrt{x^2-1}}$
  • $\frac{d}{dx}(cot^{-1}x) = -\frac{1}{1+x^2}$

Exact Differentials:

If $df = \frac{\partial f}{\partial x} \partial x + \frac{\partial f}{\partial y} \partial y$ then $df$ is exact. So, to check if $df$ is exact, suppose exactness and check if $f_{xy} = f_{yx}$. (see HW 5.2a)

Homework Problems to Review before Midterm:

  • Chapter 5:
    • 4: Exact differntials - p.156 (Section 5.9)
      • 2: (additional practice)
    • 6: Partial derivatives (a ln trick to remember)
    • 9,12: Partial derivative (logic problema)
    • 14: min, max, saddle point
    • 16,18: Lagrange Multipliers!!! (p.168-169)
      • Maximize $f(x,y)$; subject to $g(x,y) = 0$
        • $f_x:= \lambda g_x$
        • $f_y:= \lambda g_y$
      • Maximize $f(\mathbf x_n$), subject to $g_j(\mathbf x_n) = 0$, for $j = 1,…,m$,
      • In other words, maximize $f(\mathbf x_n$), subject to $\mathbf g_m(\mathbf x_n) = \mathbf 0_m$
        • $f_{x_i}:= \mathbf g_{x_i} \mathbf\lambda$, for $i = 1,…,n$