/sci/ - Science & Math

[Anonymous No.4412737]

hey sci

Suppose <span class="math">A=(a_{ij})_{i,j=1}^{n}[/spoiler] is a real matrix. Prove that
<div class="math">\int_{|x|<1} (x,Ax)dx=\frac{\omega_{n}}{n(n+2)} trace~A</div>
where \omega_{n} is the area of the unit sphere in <span class="math">\mathbb{R}^{n}[/spoiler] and <span class="math">(\cdot,\cdot)[/spoiler] is the dot product of vectors in <span class="math">\mathbb{R}^{n}[/spoiler].

my brain is literally dead can you please help

in return have a Calabi-Yau Manifold

[Anonymous No.4412744]

reported for homework

[Anonymous No.4412748]

>>4412744
It's not homework. Please tell me how can I start the proof then I'll leave?

[Anonymous No.4412790]

Anyone?

[theta !vacuaB.oEw No.4412865]

>>4412790
one sec. i'm pretty confident i got this.

[Anonymous No.4412872]

>>4412737

>Calabi-Yau Manifold

I hear this and think of Half-Life.

[theta !vacuaB.oEw No.4412870]

>>4412865
you should see that
<div class="math">\int_{|x|<1} a_{ij} x_{i} x_{j} dx=0</div>
and
<div class="math">\int_{|x|<1} x_{1}^{2}dx = \cdots = \int_{|x|<1} x_{n}^{2} dx</div>
so
<div class="math">\int_{|x|<1} (x, Ax) dx = \int_{|x|<1} \sum_{i=1}^{n} a_{ii} x_{i}^{2} dx = \frac{1}{n} trace A \int_{|x| < 1} |x|^{2} dx =\frac{\omega_{n}}{n(n+2)}~trace~A</div>

[Anonymous No.4412890]

>>4412872

Get out.