Intereting Posts

Is there a pure geometric solution to this problem?
Combinatorial Argument for Recursive Formula
Cross product of cohomology classes: intuition
Chuck Norris' Coupling of Markov Chains: An Invariant Distribution
Solving the recurrence relation $a_n=\frac{a_{n-1}^2+a_{n-2}^2}{a_{n-1}+a_{n-2}}$
Is the adjoint representation of SO(4) self-dual?
Compass-and-straightedge construction of the square root of a given line?
RSA in plain English
Can I say $f$ is differentiable at $c$ if $D_u(c) = \nabla f(c) \cdot u$ for all unit vectors $u$?
Finding expected number of distinct values selected from a set of integers
If $a$ and $b$ are positive real numbers such that $a+b=1$, prove that $(a+1/a)^2+(b+1/b)^2\ge 25/2$
What is the proof that covariance matrices are always semi-definite?
Hermitian positive semi-definite-Square root
covering map with finite fibres and preimage of a compact set
Finding the rate of change in direction

So consider the heat equation on a rod of length $L$,

$u_t (x,t) = c^2 u_{xx} (x,t)$, $\forall (x,t) \in [0,L]$ x $\mathbb{R}^+ $,

and the energy at time $t$ defined as,

- Find a power series representation for the function.
- Proving that if $f: \Bbb{R}\rightarrow\Bbb{R}$ is continuous in $“0”$ and fulfills $f(x)=f(2x)$ for each $x$ $ \in\Bbb{R}$ then $f$ is constant.
- Dirac Delta Function of a Function
- A young limit $\lim_{n\to\infty} \frac{{(n+1)}^{n+1}}{n^n} - \frac{{n}^{n}}{{(n-1)}^{n-1}} =e$
- Limit of derivative is zero. Does it imply a limit for f(x)?
- Calculus in ordered fields

$$E(t)=\frac{1}{2}\int_{0}^{L} u(x,t)^2 dx.$$

How would I show that $E(t) \geq 0$ for every $t \in \mathbb{R}^+$, and that

$$

E'(t) = -c^2 \int_{0}^{L} (u_x (x,t))^2 dx + c^2 \big(u(L,t)u_x(L,t) – u(0,t)u_x(0,t)\big)?

$$

Here’s my attempt:

$E'(t) = \frac{d}{dt} \int_{0}^{L} \frac{u^2}{2} dx = \int_{0}^{L} \frac{1}{2} (u^2) dx = \int_{0}^{L} uu_t dx$

and if $u_t(x,t) = c^2 u_{xx}(x,t)$, then,

$E'(t) = c^2 \int_{0}^{L} u u_{xx} dx = \int_{0}^{L} uu_t dx$

But I don’t really know where to go from here.

- Conjectured closed form for $\int_0^1x^{2\,q-1}\,K(x)^2dx$ where $K(x)$ is the complete elliptic integral of the 1ˢᵗ kind
- Understanding differentials
- Evaluate $\int_0^\infty\frac{\ln x}{1+x^2}dx$
- Unconventional (but instructive) proofs of basic theorems of calculus
- Example of distinctions between multiple integral and iterated integrals.
- Proving that the given two integrals are equal
- Prove $\int_a^b f(x)dx \leq \frac{e^{2L\beta}-1}{2L\alpha}\int_c^d f(x)dx$
- Partial Fraction Decomposition?
- Limit-Fundamental Concept?
- Intuitive explanation for integration

Hint. Using integration by parts we obtain that

$$

\int_0^L u(x,t)u_{xx}(x,t)\,dx=u(x,t)u_{x}(x,t)\,\big|_0^L-\int_0^L u_{x}^2(x,t)\,dx.

$$

- Chain Rule Intuition
- Show that $(0,1)$ is completely metrizable
- A function takes every function value twice – proof it is not continuous
- Fractional Powers
- If $N = q^k n^2$ is an odd perfect number with Euler prime $q$, and $k=1$, does it follow that $\frac{\sigma(n^2)}{n^2} \geq 2 – \frac{5}{3q}$?
- Penrose tilings as a cross section of a $5$-dimensional regular tiling
- Defining cardinality in the absence of choice
- Projective Nullstellensatz
- What is known about the 'Double log Eulers constant', $\lim_{n \to \infty}{\sum_{k=2}^n\frac{1}{k\ln{k}}-\ln\ln{n}}$?
- Prove that $(X\times Y)\setminus (A\times B)$ is connected
- What does proving the Riemann Hypothesis accomplish?
- $L_p$ complete for $p<1$
- Local homeomorphisms which are not covering map?
- Prove that: the center of any group is characteristic subgroup .
- Seeking a layman's guide to Measure Theory