Finding Curvature and Torsion: Derivatives and Unit Vectors Explained

In summary, the conversation discusses the process of calculating the unit vector for dN/ds, taking a derivative of N, and simplifying the resulting equation. The conversation ends with the individual figuring out the next steps in the process.
  • #1
dimpledur
194
0

Homework Statement



http://www.mathhelpforum.com/math-help/attachments/f6/22423d1317129472-curvature-torsion-untitled.png



The Attempt at a Solution



What I did was I calculated the unit vector for dN/ds={.21i+0.91j-0.42k}/ sqrt(.21^2+.91^2+0.42^2)=.205i+0.889j-.4102k


then, I took a derivative of N

since N=BxT,

dN/ds=(dB/ds)xT+Bx(dT/ds)
then taking T and doing dot product on both sides:
T⋅(dN/ds)=T⋅(dB/ds)xT+Bx(dT/ds) which simplifies to:
T⋅(dN/ds)=dB/ds
T⋅(dN/ds)=-τN

From here I have no clue what I should be doing..

Any suggestions will be greatly appreciated
 
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  • #2
nvm figured it out
 

Related to Finding Curvature and Torsion: Derivatives and Unit Vectors Explained

1. What is the difference between curvature and torsion?

Curvature is a measure of how much a curve deviates from being a straight line. It is calculated by evaluating the rate of change of the tangent vector along the curve. Torsion, on the other hand, measures the amount of twisting or bending in a curve. It is calculated by evaluating the rate of change of the normal vector along the curve.

2. How are curvature and torsion related to each other?

Curvature and torsion are related through the Frenet-Serret equations, which describe the geometric properties of a curve in three-dimensional space. These equations show that the curvature and torsion of a curve are dependent on each other, and cannot be changed independently.

3. What are some real-life applications of curvature and torsion?

Curvature and torsion have numerous applications in fields such as engineering, physics, and biology. In engineering, they are used to design and analyze curved structures, such as bridges and pipelines. In physics, they play a crucial role in describing the motion of particles in space. In biology, they are used to study the shape and movement of biological structures, such as DNA molecules.

4. How do you calculate curvature and torsion for a given curve?

The mathematical formulas for calculating curvature and torsion depend on the specific representation of the curve. In general, curvature can be calculated by finding the second derivative of the curve's parametric equations, while torsion can be calculated by finding the third derivative. However, for more complex curves, such as those found in three-dimensional space, more advanced mathematical techniques may be required.

5. Can negative curvature or torsion exist?

Yes, negative curvature and torsion can exist in certain situations. Negative curvature can be found in saddle-shaped surfaces, such as a Pringles chip, while negative torsion can be found in certain types of helical structures. However, in most practical applications, positive curvature and torsion are more commonly encountered.

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