Curvature Proof Problem

In summary: Triple_product_relationshipsIn summary, the conversation discusses how to relate the curvature of a path parametrized by arc length to the equation k = ||c'(t) x c"(t)||/||c'(t)||3. It is suggested to use the vector identity (c'(t).c'(t))*c''(t)-c'(t)(c'(t).c''(t)) = c'(t)x(c''(t)xc'(t)) and to take the norm of c'(t) x (c"(t) x c'(t)) to simplify the problem. It is also mentioned that to find the curvature, dT/ds is needed, where s is arclength.
  • #1
MichaelT
25
0

Homework Statement


If c is given in terms of some other parameter t and c'(t) is never zero, show that
k = ||c'(t) x c"(t)||/||c'(t)||3

The first two parts of this problem involved a path parametrized by arc length, but this part says nothing about that, so I assume that this path is not parametrized by arc length.

Homework Equations


I have found that
T'(t) = c"(t) = [||c'(t)||2c"(t) - c'(t)(c'(t) dot c"(t))]/||c'(t)||3

I am having trouble figuring out how to relate the curvature to the equation
k = ||c'(t) x c"(t)||/||c'(t)||3

The Attempt at a Solution


I can express the equation as the components so
F(x,y,z) = [(z"y'-y"z')2+(x"z'-x'z")2(y"x'-x'y")2]1/2/(x'2+y'2+z'2)3/2

Where should I go from here, and is the above equation useful at all? Do I need to find
||c"(t)|| as well (and is that even possible)?
 
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  • #2
What you have the numerator there is (c'(t).c'(t))*c''(t)-c'(t)(c'(t).c''(t)). There is a vector identity that tells you that that is the same as c'(t)x(c''(t)xc'(t)). When you take the norm of that note that c'(t) and c''(t)xc'(t) are orthogonal. So the sin(theta) in the cross product is one. Is that enough of a hint?
 
  • #3
I am definitely getting there with this hint.

so I take the norm of c'(t) X (c"(t) X c'(t)) and get

||c'(t) X (c"(t) X c'(t))|| = ||c'(t)||* ||c"(t) X c'(t)||sin(theta) where theta = pi/2

so ||c'(t) X (c"(t) X c'(t))|| = ||c'(t)||* ||c"(t) X c'(t)||

Now I need to relate this to k = ||c"(t)|| correct? I have taken the norm of the numerator, but not of the denominator ||c'(t)||^3

Thanks for your help!
 
Last edited:
  • #4

1. What is the "Curvature Proof Problem"?

The "Curvature Proof Problem" is a mathematical problem that involves proving or disproving the existence of a particular type of curvature on a given surface. It is commonly studied in the field of differential geometry.

2. What are the main types of curvature that are studied in the "Curvature Proof Problem"?

The main types of curvature that are studied in the "Curvature Proof Problem" are Gaussian curvature, mean curvature, and principal curvatures. These concepts are used to describe the shape and properties of a surface.

3. How is the "Curvature Proof Problem" relevant in real-world applications?

The "Curvature Proof Problem" is relevant in various fields such as physics, engineering, and computer graphics. It is used to analyze and understand the curvature of surfaces in real-world objects, such as the Earth's surface, and to design and simulate curved structures.

4. What are some techniques used to solve the "Curvature Proof Problem"?

Some techniques used to solve the "Curvature Proof Problem" include differential equations, tensor analysis, and geometric methods. These techniques involve using mathematical tools to analyze and manipulate the equations and properties of surfaces.

5. What are some open questions or challenges in the study of the "Curvature Proof Problem"?

One open question in the study of the "Curvature Proof Problem" is the existence and uniqueness of solutions. Additionally, finding general techniques for solving these problems on different types of surfaces is a challenge that researchers continue to work on.

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