Exactly, ##1 = \sum_{k =1}^N (\frac{ \overrightarrow{v_k} \cdot \overrightarrow{q_j}}{ |\overrightarrow{v_k}||\overrightarrow{q_j}|})^2##
I discovered this fact by coincidence but it turns out that it may have a nice link to the quantum mechanics.
For example, if the cosine of the angle...
Thank you, I think this should take me to my proposed solution under the assumption that the rate of the salt leaving the bowel is proportionated to the total salt in the bowel. (The proportion constant should be negative sign because the leaving salt reduces the total amount of salt in the...
Sorry that I was not clear in the description of the problem. I like the analogy of salt and water, so I will consider it here.
Let`s consider a tank has water and salt. The hole near the bottom leaks water and salt. To keep the volume of the fluid constant, we add from the top a volume of water...
Homework Statement
Suppose there is solute ##s## in a bowel containing fluid. There is a tiny hole near the bottom which leaks a small fixed volume of solute ##\lambda## per unit time ##dt##. In addition, there is a small added solute to the fluid in a constant rate ##\alpha## so as the volume...
Homework Statement
Calculating the area of equilateral triangle using calculus.
Homework Equations
The Attempt at a Solution
The area of the triangle is the area of the circle minus 3 times the area of the sector shown in (light blue). So, the target is to calculate the pink area first...
From some help, I found a way out.
First the velocity should be represented by the total derivatives not partial derivatives.
##\frac{dx}{dt}=\frac{dx}{dx`}\frac{dx`}{dt`}\frac{d t`}{dt}##
Now ##\frac{dx}{dx`}## and ##\frac{d t`}{dt}## are expressed in term of partial derivatives...
Homework Statement
How to obtain the famous formula of velocity transformation using a chain rule.
I know that there is a straightforward way by dividing ##dx## as a function of ##dx`## and ##dt`## on ##dt## which is also a function of them. But I would rather try using the chain rule.
Homework...
Homework Statement
Given that matrix, A can be decomposed using SVD (Singular Value Decomposition) into ##A=USV^T##`, why does always the sum of the square of cosines between v` vectors and any other column vector q representation of arbitrarily column vector Q vector sum up to 1?
Homework...
In this post, I adopted the convention of column vector instead of row because it is more conventional for me.
Yes, I am looking for a closed analytic expression for ##x_n##. This is important especially if n is larger.
Homework Statement
Given an initial distribution state vector that represents the probability of the system to be in one of its states. Also given a Markov transition matrix. How to calculate the state vector of the system after n-transition?
Homework Equations
Assuming the initial state...
d is still fixed and represents half length of the needle.
I followed the same reasoning of the classical Buffon needle. In classical version, the needle crosses the vertical line when the projection of the needle on the horizontal axis is not larger than $$2dcos\theta$$ . In other words, the...
So if we taking the area under the curve, p(theta), it should represent the desired integration, right!.
Remember Buffon needle problem, a needle with a length 2d and the distance between the vertical lines is L crosses those lines with a probability = $$ 2d/ \pi L$$, One of smart solution to...
I am not familiar with elliptic integrals so I don not know what z and k in this solution are related to my problem.
For example, $$ \int_0^{\pi/2} r + d\cos\theta - \sqrt{d^2\cos^2\theta + r^2 - d^2 } \ d\theta = {\pi/2} r + d - f(d, r)$$. So how to represent f(d,r)? If it is not elementary...
This is nice step that: ##d^2\cos^2\theta + r^2 - d^2 = r^2 - d^2\sin^2\theta## , I overlooked it at all.
Nothing wrong with $$ \int_0^{\pi/2} r + d\cos\theta - \sqrt{d^2\cos^2\theta + r^2 - d^2 } \ d\theta$$ because I know how to evaluate the the first two terms but I stuck at the third one...
I am interested to find the length shown in red in the attached figure. I want this length as a function of d (shown in blue) and the angle θ. Then I will integrate this length to dθ from 0 to π/2.
Firstly, I used the law of the triangle to determine the length s which when subtracted from the...