course Mth 275
I tried something different to keep the formatting when I send my work--I saved it in rich text format and then copied/pasted it, rather than copying directly from a MS Word document. Hopefully, this will show up correctly on your end.
Section 11.2:I had no difficulty understanding the topics in the section; it all seems to be reasonably extended from prior material. The problems were all straightforward, except for one. I was able to work out the “C” level proofs in #51, 54, and 57, but #60 stumped me. It states, “If G = F dot (F’ X F”), what is G’?” I see that since G is a scalar, so must be G’. I tried letting F = ai + bj + ck, and then doing the math, but all I get is a collection of 18 products of various combinations of a, b, c, a’, b’, c’, a”, b”, c”, a’”, b’”, and c’”—which led me nowhere.
The product rule generalizes to the dot and cross products, which is not difficult to prove. So
(F dot (F ' x F '') ) ' = F ' dot (F ' x F '') + F dot (F ' + F '') '.
(F ' x F '') ' = F '' x F '' + F ' x F ''' , so
( F dot (F ' x F '') ) ' = F ' dot (F ' x F '') + F dot (F '' x F '') + F dot (F ' x F ''') = F ' dot (F ' x F '') + F dot ( F ' x F ''').
The very last step occurs because F '' x F '' = 0 (the 0 vector).
This doesn't completely answer the question 'what is ...', but it gives a formula for computing the quantity.
If F(t) is regarded as the position of a particle, then F ' (t) is its velocity and F ''(t) its acceleration.
If the particle is moving in a straight line, then F ' and F '' will be parallel and the cross product will be 0.
If the particle is not moving in a straight line, then F '' will have a component which is not parallel to F '. The cross product will be perpendicular to F ' and to F '' (i.e., perpendicular to the velocity and to the acceleration), and hence perpendicular to what we might call the instantaneous plane of motion of the particle ('instantaneous' because the particle might well change its 'plane of motion' from instant to instant--e.g. if it moves along a helical path).
The full interpretation builds on this, and on the fact that F ''' is the rate of change of the acceleration, a quantity sometimes called 'jerk' which gives the magnitude and direction of the 'jerk' we feel when acceleration changes (e.g., when you hit the brakes or suddenly lift your foot off the brakes of a moving car). F ' x F ''' would be a vector quantity whose magnitude is proportional to the product of how fast you are moving and how great the 'jerk' you experience, and to the degree to which the velocity and the 'jerk' are perpendicular, and whose direction is perpendicular to these two quantities. If velocity and 'jerk' are in the same direction, this cross product would be zero. For uniformly accelerating motion F '' would be constant and F ''' would be zero. For uniform circular moton F ' would be parallel to F ''' and this term would be zero.
Many more examples could be considered to illustrate just what this all means. If you want to think more about these meanings let me know.
Regarding formatting:
Rich text can help somewhat, but it the text format of the form is still very limited.
If the formatting is really important you can save the document as a Web Page or HTML file. Send me a text copy, copied from your word processor.
Then open the file in your Internet browser. Open the source code (in Internet Explorer you can just click on View > Source and the code opens in a text box; or you can right-click within the document and the option to view the source should appear). You can then send me a copy of that document, which I can easily edit in frontPage and post to your access page.