[Math] Jacobian of parametrized ellipsoid with respect to parametrized sphere

differential-geometrymultivariable-calculusparametric

I'm not even sure how best to phrase this question, but here goes. Given $\theta$ (elevation) and $\phi$ (azimuth), the unit sphere can be parametrized as

$
x = \cos(\theta)\sin(\phi) \\
y = \cos(\theta)\cos(\phi) \\
z = \sin(\theta).
$.

A general ellipsoid can then be written as $X = ax$, $Y = by$, $z = cz$.

I'm trying to find the Jacobian that tells you how the sphere was transformed to the ellipsoid. In my mind, this involved computing the following matrix

$
\frac{\partial X}{\partial x},\frac{\partial X}{\partial y}, \frac{\partial X}{\partial z} \\ \frac{\partial Y}{\partial x}, \frac{\partial Y}{\partial y}, \frac{\partial Y}{\partial z} \\
\frac{\partial Z}{\partial x},\frac{\partial Z}{\partial y}, \frac{\partial Z}{\partial z}
$

  1. Is this correct, or should the "matrix" only have the diagonal entries?

  2. If it is not correct, would this idea be correct for an implicit surface?

  3. If it is correct, how do I do the differentiation for the off-diagonal entries? Could you work out a single example for differentiating a function with respect to $y$ and $z$?

I've been cracking my head on this one, though it seems like it should be extremely simple.

Best Answer

The map $(x,y,z)\mapsto (ax,by,cz) = (X,Y,Z)$ takes three variables to three variables, rather than the two variables of your parametrization. The Jacobian matrix of this three-dimensional transformation is $$J = \begin{bmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{bmatrix}.$$ (So the answer to your first question is, "both." The off-diagonal entries are $0$.)

What maybe is confusing you is that this is a three-dimensional transformation and makes no reference to $\theta$ or $\phi$. (Is this why you expect there to be off-diagonal entries?) To see what's going on with the ellipsoid, in particular, you need to find the Jacobian of your parametrization (which tells you how the $\theta$ and $\phi$ directions are distorted by embedding them in three-space) and then compose it with the matrix $J$. This will be equivalent to differentiating the composition $(X(\theta,\phi), Y(\theta,\phi), Z(\theta,\phi))$.

Just for kicks, here's another attack that doesn't make any explicit use of calculus. Since the unit sphere is defined implicitly by $x^2 + y^2 + z^2 = 1$, the tangent space to the sphere at the point $p$, is all the vectors perpendicular to $(x,y,z)$. Multiply these vectors by $J$ and you have the vectors to the ellipsoid at $(X,Y,Z)$.

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