Intereting Posts

Showing that a CCC with a zero object is the trivial category
The spectrum of normal operators in $C^*$-algebras
Closed sum of sets
An example of a function uniformly continuous on $\mathbb{R}$ but not Lipschitz continuous
Principal ideal domain not euclidean
What is the expected value of the number of die rolls necessary to get a specific number?
Is there a bijective map from $(0,1)$ to $\mathbb{R}$?
Derivatives of the commutator of flows (or, what are those higher derivatives doing in my tangent space?!)
What is the domain of $x^x$ as a real valued function?
Simplifying Relations in a Group
Compute integral of a Lebesgue measurable set
Finding the Dimension of a Matrix Polynomial: $W$ = { $p(B)$ : $p$ is a polynomial with real coefficients}
Possible distinct positive real $x,y,z \neq 1$ with $x^{(y^z)} = y^{(z^x)} = z^{(x^y)}$ in cyclic permutation?
Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$
$\mathbb{Q}/(X^2+Y^2-1)$ is integrally closed

The derivative of a map $F$ between manifolds $M$ and $N$ is defined by

$$F_*X(f)= X(f \circ F)$$

where $X \in T_P(M)$, the tanget space at the point $P$.

We know that $$\left\{\frac{\partial}{\partial x^i}\bigg|_P\right\}_i$$ is a basis for $T_P(M)$. How to show that $$\left\{dx^i\bigg|_P\right\}_i$$ is a basis for the cotangent space $(T_P(M))^*$?

First, by $dx^i$, I guess we mean the derivative of the map $x^i$ as defined above, right? Is this map $x^i$ just picking out the ith coordinate? Secondly, to show that it is a basis, we need to show that $dx^i\left(\frac{\partial}{\partial x^j}\bigg|_P\right) = \delta^i_j.$ Where to go from here:

$$\underbrace{(dx^i)_P}_{(\Phi_*)_P}\underbrace{\left(\frac{\partial}{\partial x^j}\bigg|_P\right)}_{X}f = \left(\frac{\partial}{\partial x^j}\bigg|_P\right)(f\circ x^i)?$$

- Curvature of the metric $ds^2=y^2dx^2+x^2dy^2$
- Differential Geometry-Wedge product
- An application of partitions of unity: integrating over open sets.
- Volume of neighborhood of the curve
- When is a $k$-form a $(p, q)$-form?
- On the smooth structure of $\mathbb{R}P^n$ in Milnor's book on characteristic classes.

I can use the chain rule but I am not sure exactly. Please help.

- How to identify surfaces of revolution
- The continuity of multivariable function
- About the second fundamental form
- What does it mean to say a boundary is $C^k$?
- Solution of eikonal equation is locally the distance from a hypersurface, up to a constant
- Car movement - differential geometry interpretation
- Calculating the differential of the inverse of matrix exp?
- Existence of Killing field
- Show that $SL(n, \mathbb{R})$ is a $(n^2 -1)$ smooth submanifold of $M(n,\mathbb{R})$
- Fermi - Walker coordinate system

We work locally in a chart $(U,\phi)$ of $p$ on $M.$ Let $x_i:M\to \Bbb R$ denote the $i^{\rm th}$ coordinate function. Then $dx_i:M_p\to\Bbb R_{p_i}$ where $M_p,\Bbb R_{p_i}$ denote the tangent spaces at $p=(p_1,\ldots,p_n)$ and $p_i.$

By definition, $dx_i(v)(f)=v(f\circ x_i)$ for any tangent vector $v\in M_p$ and $C^\infty$-function $f$ at $p_i.$ In particular, choosing $v={\partial\over\partial x_j}|_{p},$ we would get $0$ unless $j=i,$ in which case we get

$$dx_i({\partial\over\partial x_i}|_{p})(f)={\partial\over\partial x_i}|_{p}(f\circ x_i) \overset{\rm def}= {\partial(f\circ x_i\circ\phi^{-1})\over\partial r_i}|_{\phi(p)} = {\partial(f\circ (r_i\circ\phi)\circ\phi^{-1})\over\partial r_i}|_{\phi(p)} = {\partial(f\circ r_i)\over\partial r_i}|_{\phi(p)}$$

where $\phi: U\subseteq M\to \Bbb R^n$ is a chart of $p,$ and $r_i$ is a coordinate function on $\Bbb R^n.$

Now using calculus, ${\partial(f\circ r_i)\over\partial r_i}|_{\phi(p)} = {\partial r_i\over\partial r_i}(\phi(p))\times{\partial f\over\partial t}(p_i)={\partial f\over\partial t}(p_i),$ where $t$ is a coordinate on $\Bbb R.$ Thus, we have shown that $dx_i({\partial\over\partial x_i}|_{p}) = {\partial\over\partial t}|_{p_i}.$

- Integral $\int_0^1 \log \left(\Gamma\left(x+\alpha\right)\right)\,{\rm d}x=\frac{\log\left( 2 \pi\right)}{2}+\alpha \log\left(\alpha\right) -\alpha$
- What strategy do you use when solving vector equations involving $\nabla$?
- Prove that $n$ is a sum of two squares?
- Mean value theorem on Riemannian manifold?
- recurrence relation number of bacteria
- Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? (“Reciprocal addition” common for parallel resistors)
- Show that: $\lim \limits_{n \rightarrow+\infty} \int_{0}^{1}{f(x^n)dx}=f(0)$
- Topologies in a Riemannian Manifold
- The identity cannot be a commutator in a Banach algebra?
- Riemann Hypothesis and the prime counting function
- Extension of a group homomorphism
- Find the value of : $\lim_{n\to\infty}\prod_{k=1}^n\cos\left(\frac{ka}{n\sqrt{n}}\right)$
- Proof that this is independent
- Is $\sqrt{p+q\sqrt{3}}+\sqrt{p-q\sqrt{3}}=n$, $(p,q,n)\in\mathbb{N} ^3$ solvable?
- Probability of first actor winning a “first to roll seven with two dice” contest?