Physics 1600 2 Quantitative Methods

from Physics 1600 Review Plan

Physics 1600 2 Quantitative Methods

• 2.1 Physical quantities, units and dimensions, notation … … . 22
• 2.1.1 Types of physical quantities … … … … … . 22
• 2.1.2 Coordinates … … … … … … … … . 22
• 2.1.3 Notation … … … … … … … … … 23
  • notation
    • $v^{\prime }$ generally refers to a variant (not a derivative).
• 2.1.4 Dimensions … … … … … … … … . 28
  • dimension analysis
    • Non-dimensional quantities like [θ] become 1.
• 2.1.5 Numerical values of physical quantities … … … 30
• 2.2 Mathematical functions, time derivatives and integrals … . . 31
• 2.2.1 Functions … … … … … … … … … 31
• 2.2.2 Behaviors of commonly-used functions … … … . 35
  • function, it is convenient in physics for a function to have only one output
    • If a solution to describe a phenomenon has two outputs, it becomes ambiguous which one to choose.

  • mathematical functions usually take dimensionless arguments and yield dimensionless values.

    • mathematical functions: like sin, cos, exponential
      • Not arbitrary defined functions like x(t)
        • Obviously, it takes t and returns x
      • When formulas like e^n or cos(n) appear, n is dimensionless
        • For example, $f(x)=x_0{e}^{t/\tau }$
          • (t/τ) is dimensionless, τ is time constant
        • $f(x)=x_0\cos (\omega t)$
          • ωt is dimensionless, ω is angular frequency
    • There are three exceptions
      • Functions like $f(t)=t^2+t$ naturally take dimensionful arguments
        • However, t usually only takes the form of t^2 or t
      • Quadratic function $f(t)=\sqrt{t^2+d^2}$
        • Although it takes t as an argument, it can be transformed into a dimensionless argument
        • It can be transformed into $f(t)=d\sqrt{1+(\frac{t}{d}{)}^2}$, where the argument is offset by d (blu3mo)(blu3mo)
        • So, $[f(t]=[d]=[\frac{1}{t}]$ I guess
      • Log function
        • $\log Q_1-\log Q_2$ may seem to have dimensions, but it can be written as $\log \frac{Q_1}{Q_2}$, which is dimensionless
          • In reality, the subtraction of logarithms is a problem of the ratio between Q1 and Q2, so there is no unit involved
• 2.2.3 Derivatives … … … … … … … … . . 41
  • $\frac{df}{dx}$ is not a fraction, but more like a function on its own ($deri{v}_x(f)$)
    • So, canceling dx as a fraction is not valid
• 2.2.4 Infinitesimal limits and differentials … … … . . 44
  • infinitesimal = extremely small
    • It’s easy to forget that there are no small elements
• 2.2.5 Second- and higher-order derivatives … … … . . 45
• 2.2.6 Integrals … … … … … … … … … 47
  • integration
    • It is important to differentiate the variables inside the integral and use ’ to distinguish position/coordinate (p50)
  • In the Fundamental Theorem of Calculus, it is crucial to calculate the lower limit as well
    • Even if t_0 is assumed to be 0, f(t_0) may not be 0
    • Conversely, using C in indefinite integrals is basically impossible
      • If t_0 is calculated properly, it becomes C
• 2.3 Functions of time … … … … … … … … . . 49
• 2.3.1 Time derivatives … … … … … … … . . 49
• 2.3.2 Time integration … … … … … … … . . 50
• 2.3.3 Time averages … … … … … … … … 54
• 2.4 Integrating first-order differential equations … … … . . 55
  • first order differential equation
    • Type where the function itself is included on the right side, like dQ/dt=-Q/t
      • It can be solved by moving terms around
• 2.4.1 Method 1 … … … … … … … … … 55
  • TODO: If there is time, I want to explore the difference between this and 2
• 2.4.2 Method 2 … … … … … … … … … 57
• 2.5 Approximations … … … … … … … … … 58
  • approx
    • This is the first time we are looking at specific examples
      • Consistency in dropping powers is important.- 2.5.1 Binomial approximation … … … … … … 58
  • p63 I want to ask a question (blu3mo)
  • $(1+x{)}^a\approx 1+ax$ should be intuitive and easy to remember
• 2.5.2 Small-angle approximations … … … … … . 63
  • TODO: I want to come up with this proof on my own (blu3mo)
  • ==Once you use one (for example, sinΘ=Θ), you need to use the others (cos, tan) as well==
  • θ << 1 means it is not the limit of θ approaching 0
    • So even if you can use the small angle approximation, it is important to be aware that it is just an approximation and not exact.
• 2.5.3 Taylor expansion and approximations … … … . 64
  • 
  • The n in $1/n!$, $f(x_0{)}^{(n)}$, and $(x-x_0{)}^n$ should match (blu3mo)(blu3mo)
    • It is important to understand that f(x_0) is a constant.
  • I want to understand the error
    • 
    • Take it at k+1
      • Since the expansion is estimated lower than it should be, the error is negative.
• 2.5.4 Example 1: binomial … … … … … … . . 65
  • Only terms like (a)(a-1) are added up
• 2.5.5 Example 2: exponential function … … … … . 66
  • ==e^x (x=0) is always 1 no matter how many times you differentiate it, so only the other parts are added up==
• 2.5.6 Example 3: sin θ and cos θ … … … … … . . 67
  • I understand the meaning of the sin/cos expansion in the data booklet.
  • ==When θ=0, sinθ=0, so terms that become sinθ or -sinθ when differentiated disappear==
  • So it becomes something like 1/1!-x^3/3!+x^5/5!-x^7/7!
  • If you truncate at the first term, sinθ becomes θ and cosθ becomes 1.
    • I see (blu3mo)(blu3mo)(blu3mo)