I can't use Latex because I'm on mobile and there's no 'backslash' key.
Thanks for the advice!
I was able to prove that the
double integral of dx*dy/(1-(xy)^2)
written as the infinite series you mentioned does indeed become the
sum of 1/n^2 from n=1 to infinity
However I am having...
Yes, that double integral is related to my integral. If you evaluate the inner dx from 0 to 1 you get my integral.
I'm having trouble with the final logarithmic integral I mentioned.
Summary:: Using an integral and taylor series to prove the Basel Problem
The Basel problem is a famous math problem. It asked, 'What is the sum of 1/n^2 from n=1 to infinity?'. The solution is pi^2/6. Most proofs are somewhat convoluted. I'm attempting to solve it using calculus.
I notice...
So if i know that the gradient vector is <-a,-b,1>, and i want that vector to start at (xo,yo,zo) and move towards the plane, I can set up parametric equatioms to describe this movement through 3d space.
x=xo-a*t
y=yo-b*t
z=zo+(1)*t
Since I want to determine where this parametric point crosses...
Yes.
The normal vector will be
<-a,-b,1> right?
I imagine this unit vector is orthogonal to the plane and should pass through both points being considered, right?
Suppose there is a 3d plane z=a*x+b*y+c.
Suppose there is a point in space near, but not on the plane. (xo, yo, zo).
What is the coordinate (x1,y1,z1) on the plane that is nearest the original point?
My attempt uses minimization but the result is blowing up into large answer. I wonder if...
*Moving this thread from 'General Math Forum' to 'General Relativity Forum' in order to generate more discussion.*
Any object will move through spacetime along its geodesic. Since mass bends spacetime, an object initially at rest near the mass will move towards the mass along a geodesic. It...
Suppose I have a three dimensional unit Vector A and two other unit vectors B and C. If B is rotated a certain amount in three dimensions to get vector C, how do I find what the new Vector D would be if I rotated Vector A the same direction by same amount?
Just to add to why this method isnt working for me in three dimensions....
I could, for example, be moving in the x direction along the three dimensional surface. I could calculate the curvature in the zx plane and apply it to the vector. However, the graph could also be curving in the zy...
I would like to determine how a point (xo,yo,zo) moves along a geodesic on a three dimensional graph when it initially starts moving in a direction according to a unit vector <vxo,vyo,vzo>. So, if I start at that point, after a very small amount of time, what is its new coordinate (x1,y1,z1) and...
Thank you all. Just another seemingly chaotic math problem that ends up being ordered and solvable. How beautiful.
One more thing, there seems to be some people on this thread who don't care for Jordan Peterson. I know a lot of people who get a lot from what he says; why do some of you not...
Thank you all for your well thought out responses.
1. Apologies for not being clearer. The rule I attempted to articulate was : if a person is at zero money, they can't lose any more, and they may win money again as long as they win coin flips.
2. mfb , would you mind showing some more steps...
I wasn't sure which forum to post this topic to, so feel free to move it, mods. The topic seems to cover a lot of fields of study: economics, physics, statistics, calculus, etc. :)
I was listening to a YouTube video the other day, and the speaker presented situation where there are a thousand...
I'd like to understand the movement of a particle along the surface of a three dimensional graph. For example, if there is a flat two dimensional plane (z=2 for all x and y), and a unit vector describes its initial direction of movement (<sqrt(2)/2i+sqrt(2)/2j> for example), then the vector...
I've lately been interested in series and how they converge to interesting values. It's always interesting to see how they end up adding up to something involving pi or e or some other unexpected solution.
I learned about the Leibniz formula back in college : pi/4 = 1/1-1/3+1/5-1/7+1/9-...
Thanks for your responses. I agree that we should try to simplify the problem a bit more by changing how I we are setting it up. Instead of the sphere made of infinitely thin discs being placed a distance "d" away from the origin, let's put this series of objects centered at the origin and...
I am attempting to prove Newton's shell theorem. There are multiple solutions to this problem, but I am attempting a solution involving adding up the gravitational force of an infinite number of infinitely small disks that are placed together (the discs facing a point mass "m") to form a...
Thank you very much for your response. The complimentary error function of x/sqrt(t) does indeed satisfy the differential equation dT/dt=kd^2T/d^2x. How beautiful. This equation seems to suggest that heat diffusion throughout a rod or a substance is really slow, especially since the time...
I have recently been curious about heat diffusion. If there is space in one dimension with any kind of temperature dispersed throughout, then the heat equation states that the derivative of the temperature with respect to time at any point equals some constant (k) multiplied by the second...
If a beam of light is shot from a fast space ship, it travels a distance c*t1 according to their reference frame.
The same beam of light seen from an outside observer goes at an angle and travels a distance c*t2.
The distance the space ship travels is equal to v*t2.
Using the triangle made from...
Hello, I am trying to find a function x(t) that describes a particle falling into a gravitational well from a certain distance. So, for example, I am trying to figure out the differential equation: F = m*(dx^2/dt^2) = -GMm/x^2.
Or simply, dx^2/dt^2=-GM/x^2
or even more simply,
x''(t)=-k/x^2...
I am having difficulty solving Q*y - y^P = Q - 1 for y.
If P = 2 or 0.5 or 0, I can solve it quadratically. However, I'd like to find a general solution for y. Any tips?
Consider this, and tell me where my thinking goes wrong. I'm confused.
There is a person observing a spaceship launch from earth, and another person on the spaceship. There are space rocks and asteroids stationary with respect to the earth for the spaceship to observe as it flies.
The person on...