91Ó°ÊÓ

In vector calculus, the velocity vector is an essential element in understanding the motion of a particle along a path. It represents the rate of change of the position vector with respect to time. For any vector function \( \mathbf{r}(t) \), its velocity vector \( \mathbf{v}(t) \) is given by the derivative \( \frac{d\mathbf{r}(t)}{dt} \).
This derivative must be computed component-wise, meaning calculating the derivative of each component of the position vector separately:

• The \( x \)-component derivative might look like \( \frac{d}{dt} (e^t \cos t) \).
• The \( y \)-component derivative as \( \frac{d}{dt} (e^t \sin t) \).
• The \( z \)-component derivative simply is \( \frac{d}{dt} (e^t) \).

Plugging in these derivatives yields a vector that describes instantaneously how fast, and in what direction, the particle is moving. Evaluating this vector at a specific time, like \( t = \ln 2 \) in our example, provides the precise velocity at that moment.

Much like the velocity vector, the acceleration vector reflects another layer of motion detail, which is how the velocity itself changes over time. Mathematically, the acceleration vector \( \mathbf{a}(t) \) is the derivative of the velocity vector: \( \mathbf{a}(t) = \frac{d\mathbf{v}(t)}{dt} \).

• Compute the \( x \)-component by differentiating the result from \( \frac{d}{dt}(e^t \cos t - e^t \sin t) \).
• For the \( y \)-component, use \( \frac{d}{dt}(e^t \sin t + e^t \cos t) \).
• The \( z \)-component will just be \( \frac{d}{dt}(e^t) \).

The acceleration vector provides a deeper insight into the forces acting on the particle since it's essentially describing how the particle's speed and direction are changing. Evaluating \( \mathbf{a}(t) \) at a specific instant like \( t = \ln 2 \) gives a snapshot of these changes in motion for that precise moment.

Curvature and torsion provide a profound description of a curve's geometry in space. The curvature \( \kappa \) measures how fast a curve deviates from being a straight line, while torsion \( \tau \) describes the rate of twist of the curve out of the plane of curvature.

• The formula for curvature is \( \kappa = \frac{||\mathbf{v} \times \mathbf{a}||}{||\mathbf{v}||^3} \). This formula uses the cross product of the velocity and acceleration vectors, showing how these vectors interact to create curvature.
• The formula for torsion is \( \tau = \frac{(\mathbf{v} \times \mathbf{a}) \cdot \frac{d\mathbf{B}/dt}{||\mathbf{v} \times \mathbf{a}||}}{||\mathbf{B}||} \), which involves the binormal vector and its derivative.

Curvature gives insight into the local circularity of the path, while torsion reveals how the path spirals in 3D space. Calculating these at a point like \( t = \ln 2 \) provides concrete numbers to assess how these concepts manifest at that moment.

The unit tangent vector \( \mathbf{T}(t) \) and the unit normal vector \( \mathbf{N}(t) \) are crucial for understanding the orientation and curved nature of a particle’s path.

• The unit tangent vector \( \mathbf{T}(t) = \frac{\mathbf{v}(t)}{||\mathbf{v}(t)||} \) points in the direction of the trajectory and helps in identifying its orientation. It's a normalized form of the velocity vector.
• The unit normal vector is perpendicular to the tangent vector and lies in the plane of the curvature, computed as \( \mathbf{N} = \frac{d\mathbf{T}/dt}{||d\mathbf{T}/dt||} \).

The tangent vector is useful for determining direction at a given point, while the normal vector is vital for understanding how the path bends. Each tells a part of the story of motion, offering clarity apart from just looking at speed or acceleration. Evaluating these vectors at \( t = \ln 2 \) helps visualize the curve's orientation and curvature precisely at that time.