# Tagged Questions

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### contraction identity on $k$-forms

$i_\mathbb{X} \omega$ is the contraction of $\omega$ with respect to $\mathbb{X}$. In my notes it is stated that $i_\hat{\mathbb{X}} dx = dx(\hat{\mathbb{X_t}})$. I cannot see how this fits the ...
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### What's the Differential of this Map $f:S^3\rightarrow \mathbb{R}$

$f:S^3 \rightarrow \mathbb{R}$ is defined as $f(x,y,z,w)=x+zw$, where $S^3= \{(x,y,z,w) | x^2 +y^2 +z^2 +w^2 =1\}$ I tried using a stereographic chart but that got ugly. The function is so simple I ...
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### maximum area of a rectangle inscribed in a semi - circle with radius r.

A rectangle is inscribed in a semi circle with radius $r$ with one of its sides at the diameter of the semi circle. Find the dimensions of the rectangle so that its area is a maximum. My Try: ...
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### Differentiability of “positive part” function

Let the positive part function be defined as $\max(0,x)$; this function is obviously not differentiable in $x=0$. But what about the (more smooth) function $\big( \max(0,x) \big)^{2}$. I suspect the ...
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### The diffential of commutator map in a Lie group

Leb $G$ be a Lie group and $f:G\times G\rightarrow G$ be the commutator map $:(x,y)\mapsto xyx^{-1}y^{-1}$. How to obtain the Lie bracket in the associated Lie algebra of $G$ from the derivatives of ...
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### A question on self-adjointness of the shape operator

In the book Elementary Differential Geometry of Christian Bar, CUP, on the page 108, of the proof of theorem 3.5.5, the author wrote: Theorem: Let $S ⊂ \mathbb{R}^3$ be an orientable regular surface ...
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### Understanding QEF's and assorted techniques for dual marching cubes

I'm attempting to implement dual marching cubes in a 3d engine and I'm getting lost in the math portion. http://www.cs.rice.edu/~jwarren/papers/dmc.pdf is the link to the original white paper. Part ...
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### Chain rule for tensor of family of tensor fields

Let $f_\tau$ be a $\mathbb R$-family (parameter $\tau$) of diffeomorphisms that map from $\mathbb R^4$ to $\mathbb R^4$. $f^*_\tau$ is the corresponding pullback (I think that is the correct term). ...
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### Derivative of Normal Vector Field

This is an example from Do Carmo (Example 4, page 139). Consider the saddle point $p=(0, 0, 0)$ of the hyperbolic paraboloid $z=y^2-x^2$ with parameterization $\mathbf x(u, v)=(u, v, v^2-u^2)$. It is ...
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### Showing that a map $h:S^2\rightarrow \mathbb{R}^4$ is an immersion

The Problem Let $h:S^2\rightarrow \mathbb{R}^4$ be a smooth map of the form $$h(x,y,z)=(zy,yz,zx,ax^2+by^2).$$ Show that $h$ is an immersion for any $a,b\in \mathbb{R},a,b\neq 0,ab<0$. Attempt ...
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### Finding Arc Length using First Fundamental Form

Let $v=ln(u+\sqrt{u^2+1})+C$ be the curve given on the right helicoid $x=u\cos(v),y=u\sin(v),z=2v$. Calculate the arc lengths of this curve between the points $M_1(0,0)$ and $M_2(1,ln(1+\sqrt{2}))$ ...
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### Show that the Lie derivative is equal to the commutator

Let $\Omega \subseteq \mathbb{R}^d$ be open. Let $\epsilon > 0$. Let $(\phi_t)_{t \in (-\epsilon , \epsilon)}$ be a family in $\mathrm{Diff}(\Omega)$ such that $\phi_0 = id_{\Omega}$ and ...
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### shortest distant of curve from origin

what'd be shortest distance of curve from origin(0,0) function is $$y=\frac{e^x+ e^{-x}}{2}$$ I tried taking some x and y points on curve then using distance formula finding distant then i found ...
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### Finding x when slope = 1

I've been working out some problems relating to slope on the points of a curve. I'm having issues with this one: In the curve to which the equation is... $$x^2 + y^2 = 4$$ find the value of $x$ at ...
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### Is this really a typo?

Let $U \subseteq \mathbb R^n$ and $F: U \to \mathbb R^m$ a function with coordinate functions $f_i$. My notes say that: If $F$ is differentiable on $U$ the Jacobian of $F$ is defined at each point in ...
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### connection along a map

I'm interested in connection along a map between manifolds, i.e. the pullback construction. Let $f: M \rightarrow N$ be a map and $(V,\nabla^V) \rightarrow N$ be a vectorbundle with connection. Then ...
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### Does nullity at one point implies nullity everywhere?

Consider the following definition of the derivation. Now, consider a derivation $\delta_{p} :C^{r}(R^{n})\rightarrow R$ , i.e. it is defined on r-times differentiable functions defined on the ...
### Is $H^{\alpha}_{ij,k}=H^{\alpha}_{ik,j}$ for submanifolds in $R^n$?
Consider $\Sigma^n\subset {\bf R}^{n+p}$ a submanifold and $\{e_i, e_\alpha\}$ an orthonormal frame where the Greek indexes stand for the normal vectors. Then define \$H^{\alpha}_{ij,k}=e_k\langle ...