Need help finding Jacobian matrix of diffeomorphism of spheres

Question

Let $S_a \subset \mathbb{R}^{n+1}$ and $S_b \subset \mathbb{R}^{n+1}$ be two spheres of radius $a$ and $b$ respectively. So $S_a$ are $n$-dimensional.

Let $F:S_a \to S_b$ be the diffeomorphism $F(s) = \frac{bs}{a}$.

I want to calculate $D_F$ the Jacobian of the determinant so that I can use substitution in an integral: $$\int_{S_b}u(b) = \int_{S_a}u(F(a))\text{det}(DF)$$ so my question is what is $DF$?

I know $DF$ is a matrix wrt. the orthogonal bases of the tangent space but I don't know what it means in reality.

Answer 1

I understand the question as follows: You are talking about $n$-dimensional spheres (not balls) $$S_a:=\bigl\{x\in{\mathbb R}^{n+1}\>\bigm|\> |x|=a\bigr\}$$ embedded in ${\mathbb R}^{n+1}$, and integration refers to the $n$-dimensional euclidean "area" measure on these spheres.

The Jacobian determinant is in force when maps $f: \>{\mathbb R}^m\to{\mathbb R}^m$ are involved, and $m$-dimensional volumes are at stake ($m=n+1$ in your case). But in the problem at hand we are talking about the lower-dimensional "area" measure of an embedded submanifold. For a formal proof of a corresponding formula one would have to look at so-called Gram determinants.

But in the special situation you describe you can trust your intuition! A linear stretching $$F_\lambda:\quad {\mathbb R}^m\to{\mathbb R}^m, \qquad x\mapsto y:=\lambda x$$ of ${\mathbb R}^m$ by a factor $\lambda>0$ multiplies the $d$-dimensional "area" of all nice (large or "infinitesimal") $d$-dimensional submanifolds by the same factor $\lambda^d$, and this factor $\lambda^d$ is the "determinant" you are after. In your case $\lambda={b\over a}$, and $d=n$, so that the resulting formula reads as follows: $$\int_{S_b} u(y)\ {\rm d}\omega(y)=\left({b\over a}\right)^n\ \int_{S_a}u\bigl(F(x)\bigr)\ {\rm d}\omega(x)\ .$$ Here $ {\rm d}\omega(y)$ and $ {\rm d}\omega(x)$ denote the $n$-dimensional euclidean "surface element" on $S_b\,$, resp. $S_a\,$.

Answer 2

Christian's answer is spot-on, but the thing you need -- the Jacobbian determinant -- is sitting right there inside it.

You've got the map $$F: R^3 \to R^3 : (x, y, z) \mapsto (cx, cy, cz),$$ where $c = b/a$. This map happens to be very nice: it stretches every vector by a factor of $c$.

Now consider a pair of orthogonal unit basis vectors $v, w$ for the tangent space of the first sphere at the point $p = (x, y, z)$. To be tangent vectors, they have to be perpendicular to the radius vector $[x, y, z]$. Now look at the second sphere, and the point $(cx, cy, cz)$. THe vectors $v$ and $w$ are still orthogonal to the radius vector for that point. So they once again form a basis for the tangent space.

Now: what does $DF$ do to each of those vectors?

$$ DF(p)(v) = cv + 0w \\ DP(p)(w) = 0v + cw. $$ So the matrix for the restriction of DF to the tangent space, with respect to the bases $\{v, w\}$ and $\{v, w\}$ for the tangent spaces fo the first and second sphere, respectively, is $$ \begin{bmatrix} c & 0 \\ 0 & c \end{bmatrix} $$ and the determinant of that is just $c^2$.

This is a very special case. It'd be tough to do a similar analysis for almost any other map. But in this case, it's possible, and I personally find it helpful to see all the details worked out.