This is a problem that I am having trouble figuring out in my Differential Geometry course. Here it is below:

Demonstrate that the curve
r(t) = <a(sint)^2, asin(t)cos(t), acos(t)>
lies on a sphere and that all normal planes pass through the origin.

I think r(t) is a vector, but my teach really didn't give too much information on it and I am curious. Could someone help me with this problem? Try to show it and detail so that I can understand it, since this is a relatively new topic for me. Thanks.

2. Originally Posted by Dream
This is a problem that I am having trouble figuring out in my Differential Geometry course. Here it is below:

Demonstrate that the curve
r(t) = <a(sint)^2, asin(t)cos(t), acos(t)>
lies on a sphere and that all normal planes pass through the origin.

I think r(t) is a vector, but my teach really didn't give too much information on it and I am curious. Could someone help me with this problem? Try to show it and detail so that I can understand it, since this is a relatively new topic for me. Thanks.
well the defintion of a sphere is

$\displaystyle x^2+y^2+z^2=r^2$

the vector eqation gives

$\displaystyle x=a\sin^2(t),y=a\sin(t)\cos(t),z=a\cos(t)$

so

$\displaystyle x^2+y^2+z^2=a^2\sin^4(t)+a^2\sin^2(t)\cos^2(t)+\co s^2(t)=$
factoring gives
$\displaystyle a^2[\sin^2(t)\left[ \sin^2(t)+\cos^2(t)\right] +\cos^2(t)]=a^2[\sin^2(t)[1]+\cos^2(t)]=a^2(1)=a^2$

so this is and eqation of a sphere centered at the origin with radius a

$\displaystyle x^2+y^2+z^2=a^2$

taking the gradient gives the normal vector to the surface

$\displaystyle F(x,y,z)=x^2+y^2+z^2-a^2$

$\displaystyle \nabla F=2x \vec i + 2y \vec j +2z \vec k$

I hope this helps. Good luck

3. Thank you.