Morse index
for (0,0,-1)
det\left|\begin{array}{l[cr]}-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0
(2-\lambda )(-\lambda(2-\lambda)-144)=0
\lambda_1=2,\lambda_2=1-\sqrt{145},\lambda_3=1+\sqrt{145}
for (24,-144,-1)...
Please,check my solution.
Find critical points of the function f(x,y,z)=x^3+y^2+z^2+12xy+2z
and determine their types (degenerate or non-degenerate, Morse index for non-
degenerate).
Attempt
\frac{df}{dx}=3x^2+12y=0
\frac{df}{dy}=2y+12x=0
\frac{df}{dz}=2z+2=0
Critical points...
Then if we change the medium velocity will change,right?
Velocity of sound wave in the should be faster.
From V=fw
If velocity increases then wavelength also.Frequancy remains constant.
Is it Right?
Homework Statement
When you sing, the vocal folds in your throat act as a resonating cavity (one open end, one closed end) which allows only certain frequencies. Suppose you try to sing a pure tone which is detected by a microphone and measured to have a frequency 500.0 Hz. This is the...
Homework Statement
What's the speed and direction of the following wave(A,B and C are constants)
y(x,t)=Ae^{Bx^2+BC^2t^2-2BCxt}
The Attempt at a Solution
y(x,t)=Ae^{B(x-Ct)^2}
from (x-Ct)
v=C in the +x direction
It's easy question,but I don't know whether I solved it correctly.
Homework Statement
Calculate the length of the curve given by
r=a\sin^3 \frac{\theta}{3}
in polar coordinates. Here, a > 0 is some number.
Homework Equations
l=\int \sqrt{r^2(\theta)+(\frac{dr}{d\theta})^2}d\theta...
Moment of Inertia of the center of mass is
I_0=\frac{mL^2}{12}
If it is pivoted distance x from the centre
I=I_0+mx^2
T=2\pi\sqrt{\frac{I}{mgx}}=2\pi\sqrt{\frac{I_0+mx^2}{mgx}}
for T minimum we have
\frac{dT}{dx}=0