Question

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

Short answer

Question: Show the equivalence between the two formulas for torsion: \(\tau = -\frac{d\mathbf{B}}{ds} \cdot \mathbf{N}\) and \(\tau=-\frac{1}{|\mathbf{v}|}\cdot\frac{d\mathbf{B}}{dt}\cdot\mathbf{N}\). Answer: By applying the chain rule and using the magnitude of the velocity vector formula, we showed that the two formulas for torsion are equivalent: \(\tau = -\frac{d\mathbf{B}}{ds} \cdot \mathbf{N}\) and \(\tau = -\frac{1}{|\mathbf{v}|} \cdot \frac{d\mathbf{B}}{dt} \cdot \mathbf{N}\).

Step-by-step solution

Recall formulas

In this step, let's recall the formulas we will be using: 1. Torsion: \(\tau = -\frac{d\mathbf{B}}{ds} \cdot \mathbf{N}\) 2. Magnitude of velocity vector: \(|\mathbf{v}| = \frac{ds}{dt}\) 3. Chain rule: \(\frac{d(f\circ{g})}{dx} = \frac{df}{dg}\cdot\frac{dg}{dx} \)

Apply the chain rule to find \(\frac{d\mathbf{B}}{dt}\)

Using the chain rule to differentiate \(\mathbf{B}\) with respect to \(t\), we get: \(\frac{d\mathbf{B}}{dt} = \frac{d\mathbf{B}}{ds} \cdot \frac{ds}{dt}\)

Substitute the magnitude of velocity vector formula

Now, let's substitute the magnitude of the velocity vector formula (\(|\mathbf{v}| = \frac{ds}{dt}\)) into the expression from Step 2: \(\frac{d\mathbf{B}}{dt} = \frac{d\mathbf{B}}{ds} \cdot |\mathbf{v}|\)

Solve for \(\frac{d\mathbf{B}}{ds}\)

Next, let's solve the expression from Step 3 for \(\frac{d\mathbf{B}}{ds}\): \(\frac{d\mathbf{B}}{ds} = \frac{1}{|\mathbf{v}|} \cdot \frac{d\mathbf{B}}{dt}\)

Substitute the expression for \(\frac{d\mathbf{B}}{ds}\) into the torsion formula

Finally, let's substitute the expression for \(\frac{d\mathbf{B}}{ds}\) (from Step 4) into the torsion formula: \(\tau = -(\frac{1}{|\mathbf{v}|} \cdot \frac{d\mathbf{B}}{dt}) \cdot \mathbf{N}\) Thus, we have shown that the given torsion formula, \(\tau=-\frac{d\mathbf{B}}{ds}\cdot\mathbf{N}\), is equivalent to \(\tau=-\frac{1}{|\mathbf{v}|}\cdot\frac{d\mathbf{B}}{dt}\cdot\mathbf{N}\) as required. The latter formula is easier to use since it does not require the computation of arc length.

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the rate at which a function is changing at any given point. If you think of a curve, differentiation helps you determine the slope of the curve at a specific point. This slope is essentially how steep or flat the line is. In this exercise, differentiation is key to understanding the torsion formula and how we can convert it from one form to another.
When you differentiate a vector, like the binormal vector \(\mathbf{B}\) in our exercise, you get another vector that tells you how \(\mathbf{B}\) changes over time or space. In our solution, this means finding how \(\mathbf{B}\) changes with respect to the arc length \(s\), and later with respect to the time \(t\). Tool like the chain rule comes in handy as you navigate these differentiated expressions.
Understanding the importance of differentiation not only helps here but also in various fields like physics, engineering, and economics where change is a crucial concept.

Chain Rule

The chain rule is a method in calculus used to compute the derivative of a composition of functions. Think of it as a "rule of thumb" for working with nested functions. When you have a function composed within another, the chain rule lets you differentiate step by step.
In our exercise, the chain rule is employed to bridge the gap between differentiation with respect to different variables—specifically, we go from \(\frac{d\mathbf{B}}{ds}\) to \(\frac{d\mathbf{B}}{dt}\). The key is to recognize that \(ds\) and \(dt\) are linked through the velocity, altering how \(\mathbf{B}\) and in essence, the torsion itself behaves over time. The rule is expressed as:

• \(\frac{d(f\circ{g})}{dx} = \frac{df}{dg} \cdot \frac{dg}{dx}\)

where each part corresponds to a piece of the differentiated function. By applying this rule, we simplify complicated processes and translate them into manageable calculations.

Velocity Vector

A velocity vector represents the speed and direction of an object in motion. It's a crucial part of the torsion formula transformation because it links spatial changes to temporal changes.
The magnitude of the velocity vector \(|\mathbf{v}|\) signifies the speed of the moving object. In our context, it assists in converting derivatives from spatial \(s\) to temporal \(t\) variations. Remember that the relationship \(|\mathbf{v}| = \frac{ds}{dt}\) helps us manipulate the torsional formula from one that relies on the arc length to one that uses time, simplifying our calculations.
Using velocity vectors is widespread in physics when assessing movement, mechanics, and dynamics. Here in our problem, it's employed to establish an equivalency, making the application of torsion easier and more intuitive.