Two quick questions on subspaces (intro to tensor calc)

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SUMMARY

The discussion focuses on deriving a single equation for a hypersurface defined by parametric equations in a multi-dimensional space, specifically VN. The equations utilize trigonometric functions to express the coordinates in terms of a constant 'a' and parameters u1 to uN-1. To determine the relative positions of the points (1/2a, 0, 0, ..., 0) and (0, 0, 0, ..., 0, 2a) concerning the hypersurface, one must analyze the sum of the squares of the coordinates, which leads to insights about their placement in relation to the surface.

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  • Understanding of parametric equations in multi-dimensional geometry
  • Familiarity with trigonometric functions and their applications
  • Basic knowledge of hypersurfaces in tensor calculus
  • Concept of coordinate systems in higher dimensions
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  • Research the derivation of hypersurface equations in VN using parametric forms
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Students and professionals in mathematics, particularly those studying geometry and tensor calculus, as well as researchers interested in multi-dimensional analysis and hypersurfaces.

Ed Quanta
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So the parametric equations of a hypersurface in VN
are

x^1=acos(u^1)
x^2=asin(u^1)cos(u^2)
x^3=asin(u^1)sin(u^2)cos(u^3)
...
x^(N-1)=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))cos(u^(N-1))
x^N=asin(u^1)sin(u^2)sin(u^3)...sin(u^(N-2))sin(u^(N-1))

where a is a constant. How do I find the single equation of the hypersurface? And then from this how can it be determined whether the points (1/2a,0,0,...,0) and (0,0,0,...0,2a) are on the same or opposite sides of the surface?

Any insight will be appreciated. Thanks. I ask another question when this thing is resolved in my mind.
 
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Hint: What is the sum of the squares of all the coordinates?

Regards,
George
 

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