Re: [Phys-L] f'(t) / g'(t) question

[John Denker <jsd@av8n.com>]

Let's make yet more connections.  As pointed out by James (1898), we
should not think of learning in terms of the ideas, but rather in the
/connections/ between ideas.

1) Whenever you see a slope y'/x' you should consider replacing
 it with the angular direction θ such that y'/x' = tan θ

 A lot of things that are nonlinear functions of slope are
 linear functions of angle, in which case this is a tremendous
 simplification.

 The same idea applies to relativity:  A lot of things that are
 nonlinear functions of velocity are linear functions of rapidity.

2) Whenever you see an arctangent, especially the arctangent of
 a ratio such as atan(y'/x'), you should very strongly consider
 replacing it by atan2(y', x').

 The point is, if you flip the sign of y' and x' the ratio y'/x'
 is unchanged, whereas θ = atan2(y', x') changes by 180 degrees
 as it should.  This is a big deal.  Train yourself to never use
 the unadorned single-argument atan function.

3a) This is related to the previous message about thermodynamics,
 where you need exterior derivatives to make sense of things like
     dE = - P dV + T dS

3b) It is also related to the recent discussion of geodesics,
 where you need exterior derivatives to make sense of things like
    ds^2 = cos^2(φ) dθ^2  +  dφ^2

 Train yourself to not refer to the x direction or the y direction.
 It is much better to refer to the dx and dy directions.  This is
 particularly clear in (r, θ) polar coordinates:  Plain θ is a 
 coordinate, whereas at each point dθ is a vector with a direction.

4) Previously I said direction and magnitude are complimentary
 ideas.  More specifically, we can say that they are /perpendicular/
 in the following sense:

   rectangular coordinates (x, y) : dx is perpendicular to dy
   polar coordinates (r, θ)       : dr is perpendicular to dθ

5) Not coincidentally, we can think of direction and magnitude
 as the real part and the imaginary part (if we restrict
 attention to two dimensions).

 Given a vector r in the plane, we can treat it as a complex
 number.  Write r = |r| exp(i θ).  Then take the logarithm.
 The real part is log(|r|) and the imaginary part is θ.

 This shows up all over the place, including in electronics.
 A Bode plot shows log(|r|) versus log frequency /and/ θ
 versus log frequency.  Sometimes students are slow to realize
 just how intimate is the relationship between log(|r|) and θ.
 They are the real and imaginary parts of the same thing.

 In other words, the arctangent is the imaginary part of the
 logarithm.

 Beware that θ is measured CCW from east, whereas in a previous
 message I measured heading CW from north.  The conventions for
 math and the conventions for real-world navigation are inconsistent.

On 04/08/2016 09:14 AM, Jeffrey Schnick wrote:
> For motion of a particle confined to the xy plane, a graph of v_y vs.
> v_x is called a hodograph.
[...]
> The acceleration of a particle [...]

Such a graph exists, and sometimes it's called a hodograph, and
dictionaries agree with this usage:
  http://www.dictionary.com/browse/hodograph

However, the same word is sometimes used with a different meaning,
namely a plot of vx and vy for different parts of the body ... in 
which case there is no depiction of acceleration.
  https://en.wikipedia.org/wiki/Hodograph

I'm not arguing that either one is right or wrong;  I'm just
pointing out the possibility of confusion.  This is example #437
of weird inconsistent terminology.