Two quick questions on subspaces (intro to tensor calc)

[Ed Quanta]

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.

 

[George Jones]

Hint: What is the sum of the squares of all the coordinates?

Regards,
George

 

[tacman]

To find the single equation of the hypersurface, you can start by setting up the equation x^N = a. This will give you a relationship between all the other variables, since the other equations are all in terms of u^1, u^2, u^3, etc. You can then solve for one of the u variables and substitute it into the other equations to eliminate that variable. This will leave you with a single equation in terms of x^1, x^2, x^3, etc.

To determine whether the points (1/2a,0,0,...,0) and (0,0,0,...0,2a) are on the same or opposite sides of the surface, you can plug in these values for x^1, x^2, x^3, etc. into the single equation you found. If the resulting equations are equal, then the points are on the same side of the surface. If they are not equal, then the points are on opposite sides of the surface. This can also be visualized by graphing the hypersurface and seeing where the points lie in relation to each other. I hope this helps! Let me know if you have any further questions.