Emmy Noether

Amalie Emmy Noether (German: [?nø?t?]; 23 March 1882 – 14 April 1935) was an influentialGerman mathematician known for her groundbreaking contributions to abstract algebra andtheoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics,[1][2] she revolutionized the theories of rings,fields, and algebras. … Continue reading

Amalie Emmy Noether (German: [?nø?t?]; 23 March 1882 – 14 April 1935) was an influentialGerman mathematician known for her groundbreaking contributions to abstract algebra andtheoretical physics. Described by David HilbertAlbert Einstein and others as the most important woman in the history of mathematics,[1][2] she revolutionized the theories of rings,fields, and algebras. In physics, Noether’s theorem explains the fundamental connection between symmetry and conservation laws.[3]

http://en.wikipedia.org/wiki/Emmy_Noether

http://fora.tv/2010/11/07/Wonderfest_2010_Emmy_Noether_Mistress_of_Natures_Laws

http://fora.tv/embedded_player

Deriving the Dot Product

Some books actually use as the definition of the dot product. Another definition is . The idea is to find the angle between two vectors. At way to do this is to look at the angles made with the x-axis. We … Continue reading

Some books actually use |U||V|\cos(\theta) as the definition of the dot product. Another definition is u \cdot v =x_u x _v +y_u y_v.

The idea is to find the angle between two vectors. At way to do this is to look at the angles made with the x-axis. We want to know the difference between the two angles, which I’ll call \theta_u, \theta_v. Similarly I’ll let the vector u have two components (x_u, y_u) and v be (x_v, y_v) .

I want to find cos(x), which is:

\displaystyle \cos(\theta_u -\theta_v)  = \cos \theta_u \cos \theta_v + \sin \theta_u \sin \theta_v  = (\frac{x_u }{ |u|}) (\frac{x_v }{ |v|}) + (\frac{y_u} {|u|}) (\frac{y_v }{ |v|})

Very simple, it turns out, when you look at it the right way.

Make sense? Now, why don’t you try to derive the same result for
3-dimensional vectors? If you’re slick, you can actually use the
2-dimensional result (hint: there’s a plane that contains the two
vectors and the origin).

http://mathforum.org/library/drmath/view/53928.html

http://www.physics.orst.edu/bridge/mathml/dot+cross.xhtml

http://en.wikipedia.org/wiki/Cross_product