1) Look up the Fundamental Theorem of Space Curves in at least two different reliable sources. Print out or photocopy the theorem as stated in your sources (include a citation of the sources). Look up any words in the statement of the theorem you do not understand. Then in your own words explain what the theorem says. Describe any differences in the statement of the theorem from your sources.
2) Compare and contrast the paths and the nature of the motion along the paths for six different objects moving according to the following position functions, where t is a time parameter. Thoroughly justify your claims.
r1 (t) = cos t i + sin t j+ t k
r2 (t) = (2+ cos t) i+ (2 + sin t) j+ t k
r3 (t) = cos 5t i + sin St j + 5t k
r4 (t) = cos(100π - t) i + sin(100π - t) j + (100π - t) k
r5 (t) = cost i- sint j- t k
r6 (t) = cos t2 i + sin t2 j + t2 k
3) Prove that T and N are orthogonal. To do this, it might be helpful to use the following:
a. w · w = |w|2 for any vector w
b. |T| =1
c. There is a product rule for derivatives of dot products of vector-valued functions that works the way you should expect it to work.