Where is $-2R_{ij}\nabla_jf$ from December 2

Where is $-2R_{ij}\nabla_jf$ from
$M$ is a compact Riemannian manifold,$g_{ij}$ and $f(t)$ is defined as first picture.I want to compute (as equality with red line in second picture) $$ \int_M-\nabla_if\nabla_i(2\Delta f-|\n ...

random walk and calculating the probability of paths December 2

Consider a random walk $(X_n)_{n≥0}$ with $p = 0.7$, starting from $X_0 = 3$. Find the probability that $X_{10} = 5$, but $X_n ≥ 1$ for $n = 0, . . . , 10.$.Essentially what I got from the reflection principle is this. $P(x_0=1,\text{staying} \geq 1,

Set Theory (Counting Sets) December 2

When Amy and John go out for dinner it is either just the two of them, or the two of them together with one or both of their two closest friends. This month they have gone out with each of these friends a total of 7 times, and with both of them toget

Calculus- Find the maximum height of a function December 2

A ball is thrown into the air with an initial velocity of 16 ft per second. its height after t seconds is given by -t^2 . After how many seconds does the ball reach its maximum height?I can't remember back whether this fell under the derivative s

Definite integral, $\frac 1{\ln(x)}$ December 2    2

What is $$\int_0^{\pi^{2}}\frac 1{\ln(x)}dx$$I tried using complex residues and some identities, but no luck. Any suggestions?This type of integral is called a logarithmic integral function $\mathrm{li}(x)$. A logarithmic integral function is used in

deriving general form of rational log integral December 2    3

Is there a nice/clever method to derive a general closed form for:$$\displaystyle \int_0^1 \frac{\ln(1+x^a)}{1+x}dx, \;\ a>1\quad?$$I thought maybe start with differentiating w.r.t. $a$.This gives $\displaystyle \int_0^1 \frac{x^{a}\ln(x)}{(1+x^{a})(

Cyclic integration by parts trick December 2    2

I have seen cyclic integration by parts trick used for trignometric integrals. I got the impression that it is necessary for the trick to work that derivative of the function is cyclic. However it is not so, because the trick works for polynomials as

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