Let $0$ be a regular value of the smooth function $g:\mathbb{R}^n\rightarrow\mathbb{R}^p$ and $M=g^{-1}(0)$. Define the preimage orientaton of $M$ as such that $$[\triangledown g_1,\ldots,\triangledown g_p]\oplus[M]=[\mathbb{R}^n].$$
I've been asked for the following:
- Compare the preimage orientation of $\mathbb{S}^{n-1}=f^{-1}(0)$ ($f:\mathbb{R}^n\rightarrow\mathbb{R}$ given by $f(x)=\Vert x\Vert ^2 -1$) with the induced orientation on boundary of the closed unit ball, $\mathbb{S}^{n-1}=\partial B^n$ ($B^n$ with induced orientation of $\mathbb{R}^n$).
- Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function and $F:\mathbb{R}^3\rightarrow\mathbb{R}$ defined by $F(x,y,z)=z-f(x,y)$. Find positive basis for the tangent spaces to the graph of $f$, viewed with the preimage orientation given by $F$.
I don't know how to even start, so any detailed solution will be helpful to understand.
Here's a starting point: to compare these orientations, you only need to compare them at a single point. [Important question: Do you see why this is true?]
Let's look at the first problem, with $n = 3$. Then at the point $(1, 0, 0)$, the gradient of $f$ is something like $(2, 0, 0)$, right? And you need a basis for the tangent space with the property that $$ [(2,0,0), b_1, b_2] $$ is the positive orientation for $3$-space. I'm going to say that $e_2, e_3$ looks like a pretty good choice. (After all, the first vector is just $2 e_1$!)
Now you need to use the OTHER technique for finding a basis (the Spivak/Munkres one) at $(1,0,0)$, and see whether they agree or disagree.
And then you can try it again for $n = 2,$ and $n = 4$, and then you should be able to generalize pretty easily.