ive been asked to fine the jacobian matrix for an ellipsoid $$x^2/a^2 + y^2/b^2 + z^2 / c^2 = 1$$
ive been looking online for the parametric equations and i get two different answers
$$x=a\cos(u)\sin(v)\\ y=b\sin(u)\sin(v)\\ z=c\cos(v)$$
or
$$x=r\cos(u)\sin(v)\\ y=r\sin(u)\sin(v)\\ z=r\cos(v)$$ which is right?
i know the jacobian should equal $abc$. but these things have confused me. any help?
I don't quite know what your "Jacobian" is. But if you are looking for a parametrisation equation, then
1). Parametrisation is not unique. For the same object it's very likely that you have many different ones, each depending on the way you view it.
2). For the ellipsoid $(x^2/a^2+y^2/b^2+z^2/c^2\le 1)$, a common parametrisation is $$x=at\sin\theta\cos\phi, y=bt\sin\theta\sin\phi,z=ct\cos\theta,t\in[0,1],\theta\in[0,\pi],\phi\in[0,2\pi]. $$ If you are looking for a parametrisation for the elliptic surface, just let $t=1$ be fixed.