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Jun
4
answered A tangent vector to the unit tangent bundle $T_1S^n$ at $(x,\xi)$ can be written $(X,Z)$ with $X \cdot x = 0$ and $Z \cdot \xi = 0$.
Jun
3
awarded  Enlightened
Jun
3
awarded  Nice Answer
Jun
2
comment How can you return to studying/researching pure mathematics from degree level theoretical physics?
I worked on physics for my masters and transferred to math for PhD work in the USA. It was a good fit for me. However, I also did a double major in math and physics as an undergraduate. In any event, there are certainly topics in physics which yield themselves to the path you outline. Sorry I know nothing about the UK
Jun
2
comment Serret-Frenet for Non-unit “speed” space curves
welcome to the MSE. Nice answer.
Jun
1
comment Where does the chain rule come from for multivariable functions?
Since the derivative represents the best linear approximation to the change in a function the change in the composition of two functions best linear approximations turns out to be the composite of the linear approximations whose formula in turn is represented by the matrix multiplication of the Jacobian matrices for the functions... hence the rule. Well, hence all the chain rules.
Jun
1
revised Serret-Frenet for Non-unit “speed” space curves
added 1 character in body
Jun
1
answered Serret-Frenet for Non-unit “speed” space curves
Jun
1
comment Smooth parameterized surface/area.
The derivative is not with respect to "$x$". Rather, with respect to the parameters of the surface... I hope my answer makes sense. I did not attempt your 1.2
Jun
1
answered Smooth parameterized surface/area.
Jun
1
comment some vector calculus frequently occurs in the study of PDE
your question is not answerable in general. To answer which is "in" or "out" the context matters. Generally, the choice is just one of contextual convention. For example, for compact surfaces traditionally the outward pointing normal is taken to be positive. Here inward is the side of the surface which contains a finite volume.
Jun
1
comment Directional derivative expression
I think your last comment is reasonable, however, $\ddot{r}$ has both a normal and tangential component. Therefore, it is not fair to call $\ddot{r}$ the "normal" (generally)
Jun
1
comment Directional derivative expression
The real question is what you mean by $d/dn$. How would you define this in terms of a limiting process on functions ?
Jun
1
comment Directional derivative expression
I think what you wrote before is also true, in fact, modulo a little argument about the chain-rule and an agreement about notation, it is the definition of the directional derivative in the $A$-direction. We can write $\partial f /\partial a = (\nabla f) \cdot \hat{a}$ where $\hat{a} = A/\sqrt{A \cdot A}$.
May
30
comment inverse function theorem for analytic functions whose derivative might vanish
I see, and I'm tempted to follow your solution. Certainly we know that for such $t$ which give $f '(x(t))=0$ then $g'(y(t))=0$ as well (since $x'(t),y'(t)$ are nonzero and the chain rule tells us $f'(x(t))x'(t) = g'(y(t))y'(t)$). Perhaps that is useful?
May
30
comment inverse function theorem for analytic functions whose derivative might vanish
maybe I'm misreading something, but, take the bump function (smooth but, not analytic at zero) and add $t/2$. I think that makes an increasing function. Take $x(t)=y(t)$ the modified bump and let $f=g$ then certainly $f(x(t))=g(y(t))$ for all $t$ and it does not follow that $x$ or $y$ is analytic. Surely I misinterpret the spirit of the question.
May
30
comment Need some help understanding this exercise about injective plane curve
@user7530 agreed. The construction here is pretty adhoc, although, easy to set-up, so I see why it's done.
May
30
comment Next book in learning Differential Geometry
Nice list. I need to bookmark this. I would add one for the sake of physics. Many of my peers in theoretical physics studied Anomalies in Quantum Field Theory by Bertlemann. It has about 100 pages of pure math at the start and is one of the more lucid birds-eye views you'll find in the physics literature.
May
29
comment The Weierstrass-Enneper representation, the Gauss map
what is the question? Maybe add a sentence or two to summarize.
May
29
comment Need some help understanding this exercise about injective plane curve
Oh, I see your curve is just in $\mathbb{R}^2$. Perhaps ribbon is the wrong term, it's like taking a thick marker and either broadening the curve horizontally or vertically. If you had a path just stop at a point, then the marker strip I describe would abruptly end and I believe that would cause the mapping $\phi$ to fail to be locally invertible.