Connection between a symmetric linear transformation and the unit sphere

 

There is an interesting correspondence among the quadratic form of a symmetric linear transformation T:V \mapsto Von a real Euclidean space,the extreme values of the sphere and the eigenvectors of T

Let Q(x)=(T(x),x) be the quadratic form associated with a symmetric linar transformation which maps V into itself,then the set of elements u in V satisfying \langle u,u  \rangle=1 is called the unit sphere of V

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The logarithmic spiral

The logarithmic spiral has some very interesting properties and Bernoulli was especially fascinated by it.I’ll prove it’s most important property(the angle between the curve and the radius at every angle is constant) and proceed with an example.

In polar co-ordinates,the equation of the spiral is given by:

r(\theta)=ae^{k\theta} where a,k are constants and a>0

Now,to prove that any line from the origin which intersects the curve does so by making a constant angle(say \phi) with the curve(direction of tangent line),we consider the derivatives of the parameter equations which correspond to r(\theta)

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