Unfortunately no. Looking into this page more, I think what I'm looking for is normal curvature. I found a webpage with a formula:
http://wordpress.discretization.de/geometryprocessingandapplicationsws19/a-quick-and-dirty-introduction-to-the-curvature-of-surfaces/
I'm not sure how to use it...
I know curvature (k) of a 2 dimensional graph y(x) is equal to y''/(1+(y')^2)^(3/2), were y' is the first derivative of y with respect to x, and y'' is the second derivative of y with respect to x.
Is there a formula for the curvature at a point on a 3 dimensional graph z(x,y)? The curvature...
Thanks for the response. I was looking into general relativity, and was learnong about geodesics. After viewing some videos on youtube showing geodesics on three dimensional graphs I became curious. I don't necessarily want to focus on relativity, but rather, 3d surfaces. If I can figure out...
Great! Could someone help walk me through finding the next point and new directional vector using the geodesic equations you recommended? I'm unfamiliar with determining the metric and Christoffel symbols for this kind of thing. Could you point me in the right direction of how to set it up? :)
Suppose there is a three dimensional graph (such as z=x^2+y^2).
Suppose there is a point on the surface of the 3 dimensional graph, for example at (x,y,z)=(1,1,2).
Suppose the point is moving along the surface (along a geodesic) according to a unit vector, such as <0,1,0>.
Is there a...
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 isn't 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...