Away from certain kinds of points, you can always do it to make it what you said, but that requires a real differential equations theorem (proved in appendix of my notes if you care).
Hey @Ted, could you give me a hint on this problem? Given the triangle $ABC$, with a point $D$ $\frac{2}{3}$ down $\overrightarrow{AB}$, and point $E$ $\frac{2}{5}$ down $\overrightarrow{CB}$, where the point $Q$ bisects $\overline{CD}$. Show that $||\overrightarrow{AQ}|| = c||\overrightarrow{AE}||$. Find the value $||\overrightarrow{AQ}|| / ||\overrightarrow{AE}||$. What is the value for which $\overline{CD}$ divides $\overline{AE}$?
With all problems like that, @Cookie, as I said in the lectures, write $\overrightarrow{AB} =\vec x$ and $\overrightarrow{AC} = \vec y$ and write everything under the sun in terms of $\vec x$ and $\vec y$.
The big man that comes up from this computation is the 2nd fundamental form $\Bbb{II}(v, w) = v \cdot S_p(w)$. This theorem says it's a billinear form on $T_p M$
Those points are called umbilics, DogAteMy (you can explain why). Those are the bad points to which I referred earlier where you cannot construct such a parametrization.
@Ted can I get one more clarification? Let $\vec{u},\vec{v}$ $\in$ $\mathbb{R}^{2}$. Describe the vectors $\vec{x} = s\vec{u} + t\vec{v}$, where $s + t = 1$.
@gian this is a vague analogy, but ithere are some formal analogies to adjoint operators in linear algebra if you think about $\operatorname{Hom}(-,-)$ in analogy to the canonical pairing $V^* \times V \to K$ or as a "scalar product" (I think that's where the name comes from)
@gian I'd say that I have an intuition about adjunctions, but it's difficult to put it in words. My intuition comes from seeing/recognizing lots of examples and perceiving some sense of analogy/similarity between them.
So the formula $\frac{dE_n}{dx}=-\left(\frac{1}{R_1}+\frac{1}{R_2}\right)E_n$ definitely seems like it should boil down to a statement of geometry alone
@EricSilva I want to say it's something like: If I move away from the surface of the conductor, $\mathbf{E}\cdot d\mathbf{A}=E_n dA$ should remain the same since the charge is confined to the conductor's surface
if there were a closed minimal surface you do this stuff on it and whatnot and you would have $\partial E/ \partial n = 0$ so the normal components of these dudes are constant but closed surfaces dont work like that cause height functions so baddabing baddaboom impossible
Right, DogAteMy. Writing the forms in terms of basis $1$-forms, we have $\omega_{13} = h_{11}\omega_1 + h_{12}\omega_2$ and $\omega_{23} = h_{21}\omega_1 + h_{22}\omega_2$ for some functions $h_{ij}$.
I should note that this is an integration algorithm and therefore intermediate steps DO appear to be unjustified. The purpose of this question is to justify or reject this algorithm as always giving correct solutions.
Suppose we have a function $f(x)$ that has no vertical asymptotes and can be w...
i more mean something along the lines of: If I transfer a finite amount of heat to some point on the surface of a sphere and allow it to diffuse according to the heat equation, I expect it to eventually look uniform
@ericsilva I could be talking nonsense, though, since the fact that you were talking about cohomology makes me think I'm taking the analogy too literally
like formally it's hard to work with $\Delta$ on a manifold so you want to look at the spectral decomposition (if you live in a compact world that is, closed surfaces and whatnot) and as it turns out that spectrum is loaded with curvature
Four dogs stand in four corners of a square . The side of the square is $1$ km . Now closing eyes, each dog runs at the same velocity to the dog residing to the right . By this, they cover half distance . After opening eyes , each dog runs at the same velocity to dog in the right and covers the s...