Surface Integral problem. May be some miss conception. November 28

Surface Integral problem. May be some miss conception.
Evaluate $\int \int \vec A.\hat n dS$, where$\vec A = 18z\hat i - 12\hat j + 3y\hat k$ and $S$ is that part of the plane $S$ $2x+3y+6z = 12$ which is located in the first octant. The surface ...

Floor and Ceiling functions November 28

I have been trying to proof ⌊log_2(⌈n/k⌉)⌋ = ⌊log_2(n/k)⌋, but I never learned any rules with floor and ceiling functions. I am not sure if this theorem is true either. So my question is: Is it safe to say ⌊log_2(⌈n/k⌉)⌋ = ⌊log_2(n/k)⌋ ?

Prove $nn^2$ for $n3$ November 28    5

I'm aware that induction is necessary. I have been stuck on this problem for a few days now. I'm having a hard time understanding how to apply the inductive hypothesis to the inequality to arrive at the $P_{n+1}$ step. Base case clearly holds as $24

Real life coordinate geometry problem November 28    3

Real life coordinate geometry problem
To conduct a sport activities, in a rectangular shaped school ground $ABCD$, lines have been drawn with chalk powder at a distance of $1$ m each. $100$ flower pots have been placed at a dist ...

Prove $n^2 (n+1)$ for all integers $n \geq 2$ November 28    7

I understand that I need to use induction for this, that's not a problem. I get stuck after I try to invoke the inductive hypothesis.$P_n: n^2 > n+1$... and we want to prove $P_{n+1}: (n+1)^2 > (n+1)+1$ holds.Base: $P_2: 4 > 3$Suppose $P_n$ holds

$n (n/e)^{n}$ for all $n \geq 1$ by induction November 28

I am stuck on this problem at the last part of the $p = n+1$ step. For $n = 1$ we get $1! > 1/e$, which checks out. So assuming $n=k$ for $k \geq 1$ is true, I want to show that $ (k+1)! > ((k+1)/e)^{k+1} $. So to show this I begin by expanding the

Limit of zeta function in $x = 1$ November 28    1

How can I prove that $\lim_{x \rightarrow 1}{\sum_{n=1}^{\infty}{\frac{1}{n^x}}} = \infty$? My idea is to show that we can exchange the positions of limit and sum, obtaining the harmonic sum, that we know that diverges, but, are we able to do such ex

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