Compute following :Tangent , curvature k, torsion tau, normal n, binormal

To compute the tangent vector \(\mathbf{T}(t)\), curvature \(k\), torsion \(\tau\), normal vector \(\mathbf{N}(t)\), and binormal vector \(\mathbf{B}(t)\) for a space curve, we typically need the parametric equation of the curve \(\mathbf{r}(t)\), i.e., \(\mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle\), where \(t\) is the parameter.

Here is how each quantity is computed in general for a smooth curve \(\mathbf{r}(t)\):

1. **Tangent Vector \(\mathbf{T}(t)\)**:
   \[
   \mathbf{T}(t) = \frac{\mathbf{r}'(t)}{|\mathbf{r}'(t)|}
   \]
   The tangent vector is the normalized first derivative of \(\mathbf{r}(t)\).

2. **Curvature \(k(t)\)**:
   \[
   k(t) = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|}
   \]
   The curvature measures how quickly the direction of the tangent vector changes. It's computed using the derivative of the tangent vector and the magnitude of the first derivative of \(\mathbf{r}(t)\).

3. **Normal Vector \(\mathbf{N}(t)\)**:
   \[
   \mathbf{N}(t) = \frac{\mathbf{T}'(t)}{|\mathbf{T}'(t)|}
   \]
   The normal vector is the normalized derivative of the tangent vector.

4. **Binormal Vector \(\mathbf{B}(t)\)**:
   \[
   \mathbf{B}(t) = \mathbf{T}(t) \times \mathbf{N}(t)
   \]
   The binormal vector is the cross product of the tangent and normal vectors.

5. **Torsion \(\tau(t)\)**:
   \[
   \tau(t) = -\frac{\mathbf{B}'(t) \cdot \mathbf{N}(t)}{|\mathbf{r}'(t)|}
   \]
   The torsion measures how much the curve twists out of the plane formed by the tangent and normal vectors.

### Steps:
- Start with the parametric form of the curve \(\mathbf{r}(t)\).
- Compute the derivatives \(\mathbf{r}'(t)\), \(\mathbf{r}''(t)\), etc.
- Find the unit tangent, normal, and binormal vectors.
- Calculate the curvature and torsion from the derivatives.

Do you have a specific curve or parameterization in mind for these calculations? Or would you like a worked-out example?

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### Follow-up Questions:
1. Would you like an example calculation for a specific curve?
2. Do you want more details on how to compute each vector step by step?
3. Are you interested in applications of curvature and torsion in physics?
4. Would you like to know how curvature and torsion relate to 3D geometry?
5. Are you curious about Frenet-Serret formulas in this context?

**Tip**: Always simplify intermediate steps when computing derivatives for these quantities, as expressions can become complex quickly.

Compute the following: Tangent, curvature k, torsion tau, normal n, binormal

Learn how to compute the tangent vector, curvature, torsion, normal, and binormal for a space curve using Frenet-Serret formulas. This solution covers the key formulas and concepts in vector calculus and differential geometry, ideal for college-level students.

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