Question

Show that the formula defining the torsion, $\tau =-\frac{d\mathbf{B}}{ds}\cdot \mathbf{N},$ is equivalent to $\tau =-\frac{1}{|\mathbf{v}|}\frac{d\mathbf{B}}{dt}\cdot \mathbf{N}.$ The second formula is generally easier to use.

Short answer

Question: Show the equivalence of the two formulas for torsion, τ: 1. τ = - $\frac{d\mathbf{B}}{ds}\cdot \mathbf{N}$ 2. τ = - $\frac{1}{|\mathbf{v}|}\frac{d\mathbf{B}}{dt}\cdot \mathbf{N}$ Answer: The equivalence of the two formulas for torsion can be demonstrated by relating the derivatives with respect to arc length (s) and time (t) using the definition of velocity. This relationship can be expressed as $\frac{d}{ds}=\frac{d}{dt}\cdot \frac{1}{|\mathbf{v}|}$. Substituting this into the first formula for torsion, τ, we arrive at the second formula, thus showing their equivalence.

Step-by-step solution

Derivative with respect to arc length and time

To find the relationship between the derivative with respect to arc length (s) and to time (t), we start by noting the definition of velocity: $\mathbf{v}=\frac{d\mathbf{r}}{dt}$, where $\mathbf{r}$ represents the position vector of an object moving in a curve. Now, if we differentiate with respect to arc length (s), we can write: $$\frac{d\mathbf{r}}{ds}=\frac{d\mathbf{r}}{dt}\frac{dt}{ds}=\mathbf{v}\frac{dt}{ds}$$.

Relate the time and arc length derivatives

To find the relationship between $\frac{d}{ds}$ and $\frac{d}{dt}$, we need to find the value of $\frac{dt}{ds}$. From the equation in step 1, we have: $$\frac{d\mathbf{r}}{ds}=\mathbf{v}\frac{dt}{ds}$$. So, $$\frac{dt}{ds}=\frac{1}{|\mathbf{v}|}.$$ Now, we can express the derivatives with respect to arc length (s) and time (t) as: $$\frac{d}{ds}=\frac{d}{dt}\frac{dt}{ds}=\frac{d}{dt}\cdot \frac{1}{|\mathbf{v}|}.$$

Substitute the derivatives in the torsion formula

Now, we can substitute the derivatives into the first formula for torsion, τ: $$\tau =-\frac{d\mathbf{B}}{ds}\cdot \mathbf{N}.$$ Using the relationship between the derivatives found in Step 2, we have: $$\tau =-(\frac{d}{dt}\cdot \frac{1}{|\mathbf{v}|})\mathbf{B}\cdot \mathbf{N}.$$ Simplifying, we get: $$\tau =-\frac{1}{|\mathbf{v}|}\frac{d\mathbf{B}}{dt}\cdot \mathbf{N}.$$ This is the second formula for torsion, demonstrating their equivalence.

Key concepts

Arc Length Derivative

Understanding the concept of arc length derivative is crucial when studying curves in calculus. Picture a smooth curve, like a bend in a road. As you drive along, you're essentially tracing the arc length. In mathematical terms, the arc length can be defined as the distance along the curve from one point to another. Now, if we consider a particle moving along this path, the arc length 's' varies as time 't' progresses.

The arc length derivative $\frac{d}{ds}$ is essentially the rate of change of some quantity with respect to this distance 's'. It plays a significant role in understanding various aspects of curve motion, such as curvature or torsion of the curve. The derivative of a position vector $\mathbf{r}$ with respect to 's' is crucial because it gives us insight into the particle's velocity $\mathbf{v}$ along the curve without the direct influence of time.

From a solution standpoint, instead of measuring the rate of change as time varies, the arc length derivative measures it as the distance along the curve varies - giving a sense of 'pure' geometric progress, un-tainted by the potentially non-uniform passage of time.

Position Vectors in Calculus

Position vectors are foundational in the study of motion within calculus. Think of them as arrows pointing from a fixed origin to a particle's location in space. Every point on a curve can be associated with a position vector $\mathbf{r}$. When a particle moves along a curve, its position vector changes over time.

In calculus, these vectors are not just static entities; they can be manipulated to show changes in motion, rate of spin, and other dynamics of the curve in space. By taking derivatives of position vectors with respect to time, $\frac{d\mathbf{r}}{dt}$, we get the velocity vector which shows how fast and in what direction the particle is moving at any given instance. Similarly, differentiating the position vector with respect to arc length, as seen in the exercise, links geometric changes in the curve directly to the properties of motion.

These position vectors are key players when we deal with more complex curves, where the motion may not just be along a flat plane but in three-dimensional space, making them indispensable tools for understanding the subtleties of motion on complex paths.

Time Derivative Calculus

In the realm of calculus, time derivative plays a pivotal role as it measures how a quantity changes over time. For motion along a curve, this could mean understanding how fast a car moves around a track, or how quickly a planet orbits the sun. In our problem, the time derivative is a central concept in transitioning from one formula for torsion to another.

The time derivative of a vector, such as the binormal vector $\mathbf{B}$ in our exercise, $\frac{d\mathbf{B}}{dt}$, tells us how this vector changes as time progresses. This measure is powerful, especially when compared to the static frame of a simple position vector. It gives us the rate at which the orientation of the curve is changing, thus being a clear indicator of the motion's dynamics.

When analyzing curves in three dimensions, the time derivative connects the changing position vectors with physical quantities like velocity and acceleration, making it integral to our understanding of motion and change along these curves.