Intereting Posts

Let $f$ be differentiable on $(0,\infty)$.Show that $\lim\limits_{x\to \infty}(f(x)+f'(x))=0$,then $\lim\limits_{x\to \infty}f(x)=0$
Unnecessary property in definition of topological space
Square-root for nonnegative (and not necessarily self-adjoint) operators
Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? (“Reciprocal addition” common for parallel resistors)
Statements with rare counter-examples
Cross product of the reals question
How to make a smart guess for this ODE
Quotient ring of a localization of a ring
Monad = Reflective Subcategory?
Can a Mersenne number be a power (with exponent > 1) of a prime?
If $P(x)=2013x^{2012}-2012x^{2011}-16x+8$,then $P(x)=0$ for $x\in\left$ has
Mutual set of representatives for left and right cosets: what about infinite groups?
Distance between a point to a set in metric spaces
If $R$ is commutative, and $J\lhd I\lhd R,$ does it follow that $J\lhd R?$
The derivative of a function of a variable with respect to a function of the same variable

In my notes, the lecturer considers a smooth vector field $v: TM\to T(TM)$, with $M$ a smooth manifold. Let’s write

$$v(u,e)=((u,e), (a(u,e),b(u,e)).$$

It is said that $v$ is a second order equation if $T\pi_M\circ v=\text{id}$. It implies that $a(u,e)=e$, i.e.

- A question regarding Frobenius method in ODE
- What is the physical meaning of fractional calculus?
- Regarding Ladder Operators and Quantum Harmonic Oscillators
- Solving a non-homogeneous differential equation via series solution
- Differentials Definition
- Differential Equations- Wronskian Fails?

$$v(u,e)=((u,e),(e,b(u,e))).$$

Here, my notes claim that if $c(t)=(u(t),e(t))$ is a curve on $TM$ satisfying the above, then

$$\frac{du}{dt}=e(t), \ \frac{de}{dt}=b(t)$$

This is probably very stupid but I can get why…

- When do Harmonic polynomials constitute the kernel of a differential operator?
- What went wrong?
- How to solve $y''' - y = 2\sin(x)$
- Questions about definition of Tangent Space
- Verify for $f(x,y)$, homogeneous of degree $n$: $xf_x+yf_y=nf$
- What does it mean for a function to be a solution of a differential equation?
- How do you find the Maximal interval of existence of a differential equation?
- Solve a second order DEQ using Euler's method in MATLAB
- Solving $-u''(x) = \delta(x)$
- Solving an ODE from a PDE

The requirement for v(u,e) that a(u,e) = e is called “obeying the canonical flip on the double tangent bundle”; it is precisely the vector fields on the double tangent bundle that satisfy or obey the canonical flip that are called second-order ODEs on a closed, connected, smooth manifold M.

For a second-order ODE on a manifold v(u,e), if c(t) = (u(t),e(t)) is a solution, then u(t) is called a “base curve” on the manifold M and its derivative c(t) is the solution to the second-order ODE; this is one of the reasons that v(u,e) is called a second-order ODE.

For more information on the canonical flip, try *Transveral Mappings and Flows* by Abraham and Robbin; for more information second-order ODEs on manifolds, try *Foundations of Mechanics* by Abraham and Marsen

- What is the geometric meaning of the inner product of two functions?
- What function could describe this GIF animation?
- When do we have $\liminf_{n\to\infty}(a_n+b_n)=\liminf_{n\to\infty}(a_n)+\liminf_{n\to\infty}(b_n)$?
- Euler's summation by parts formula
- Rigorous Proof: Circle cannot be embedded into the the real line!
- Complex valued trial solution
- Intuituive reason why Fermats last theorem holds
- If $\sum a_n$ converges, then $\sum \sqrt{a_na_{n+1}}$ converges
- how many distinct values does it have?
- Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
- Prove that the identity map $(C,d_1) \rightarrow (C,d_\infty)$ is not continuous
- Convergence in mean for the sequence of positive random variables
- Demystifying modular forms
- A problem about generalization of Bezout equation to entire functions
- Evaluate the integral $\int_0^{\infty} \left(\frac{\log x \arctan x}{x}\right)^2 \ dx$