Intereting Posts

Counting primes by counting numbers of the form $6k \pm 1$ which are not prime
To find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$.
Circle Rolling on Ellipse
Proving that $f(x,y) = \frac{x^2y}{x^2+y^2}$ with $ f(0,0)=0$ is continuous
What's a non-standard model of Tarskian Euclidean geometry?
From a deterministic discrete process to a Markov chain: conditions?
Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?
Expected number of people to not get shot?
How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?
When is an accumulation point not the limit of some sequence in a topological space?
Exponential of the differential operator
Integral $ \int_0^1 \frac{\ln \ln (1/x)}{1+x^{2p}} dx$…Definite Integral
calculate arbitrary points from a plane equation
Who realized $\int \frac 1x dx =\ln(x)+c$?
Proving $\mathrm e <3$

I’ve seen the math, but… It just doesn’t make sense to me. How is the slope going to point perpendicular to the vector that is clearly a straight line not going in that direction?

- Find the limit $\lim \limits_{n\to \infty }\cos \left(\pi\sqrt{n^{2}-n} \right)$
- The first $4$ primes $p$ for which $15347$ has a square root mod $p$ are $2, 17, 23,$ and $29$
- Fractional Trigonometric Integrands
- A derivation of the Euler-Maclaurin formula?
- Generalized PNT in limit as numbers get large
- $\lim_{x\to0^{+}} x \ln x$ without l'Hopital's rule
- How to compute $\int_0^{\pi/2}\frac{\sin^3 t}{\sin^3 t+\cos^3 t}dt$?
- Evaluate $\int_0^\infty\frac{1-e^{-x}(1+x )}{x(e^{x}-1)(e^{x}+e^{-x})}dx$
- Is there a bounded function $f$ with $f'$ unbounded and $f''$ bounded?
- Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$

Take some point $P_1 = C(s_1)$ on a curve $C$. Let $T_1$ be the unit tangent at this point. Now move a tiny distance along the curve, to get to the point $P_2 = C(s_2)$, and let $T_2$ be the unit tangent at this point. Now draw the vector $V = T_2 – T_1$. If $P_1$ and $P_2$ are very close together, $T_1$ and $T_2$ will be almost equal, and $V$ will be roughly perpendicular to both of them (because the triangle with sides $T_1$ and $T_2$ will be isosceles and very skinny). Here’s a picture.

But, by the definition of derivative,

$$

\frac{dT}{ds} = \lim_{s_2 \to s1} \frac{T_2 – T_1}{s_2 – s_1}

$$

and this limit will also be roughly perpendicular to $T_1$.

The dot product of the unit tangent vector with itself is of course equal to 1.

$$ \mathbf{T} \cdot \mathbf{T} = \|\mathbf{T}\|^2 = 1^2 = 1. $$

Take the derivative of both sides, and remembering the product rule,

$$ \frac{d}{ds} ( \mathbf{T} \cdot \mathbf{T} ) = \frac{d}{ds}(1) $$

$$ \mathbf{T} \cdot \frac{d \mathbf{T}}{ds} + \frac{d \mathbf{T}}{ds} \cdot \mathbf{T} = 0 $$

$$ 2 \mathbf{T} \cdot \frac{d \mathbf{T}}{ds} = 0 $$

or

$$ \mathbf{T} \cdot \frac{d \mathbf{T}}{ds} = 0. $$

The essence of a derivative is the approximation of functions by linear equations: $\mathbf{T} (s + \delta s) \approx \mathbf{T} (s) + \delta s \mathbf{T}’ (s) $.

Performing this vector addition tip-to-tale, the three vectors $\mathbf{T} (s + \delta s),\space \mathbf{T} (s) \space \text{and} \space \delta s \mathbf{T}’ (s)$ form the sides of an isosceles triangle with two legs of length 1 and an arbitrarily small base with length proportional to $\delta s$. As $\delta s$ goes to zero, the angle between the two legs goes to zero, and the base angles go to right angles.

The unit tangent vector gives the instantaneous velocity. But unless you go in a straight line forever, you will turn. Suppose you turn left. The unit tangent vector still points forward at any given moment, but it is turning left — its derivative is leftward. The unit normal points left, to indicate the direction that the tangent is changing.

First of all, the unit normal vector $N$ is in general *not* the derivative of the unit tangent vector $T$. What *is* true is that the derivative of $T$ with respect to the arc length along the curve $s$ is a vector $T'(s)$ normal to $T(s)$ in the sense that $T'(s) \cdot T(s) = 0$, and if $T'(s) \ne 0$, so that $\Vert T'(s) \Vert \ne 0$, we can *define* the *unit normal* $N(s)$ by the equation

$N(s) = \kappa^{-1}T'(s), \tag{1}$

where

$\kappa = \Vert T'(s) \Vert \tag{2}$

is the *magnitude* of $T'(s)$. (1) is then equivalent to

$N(s) = \kappa T'(s), \tag{3}$

the way (1) is usually written; it is one of the Frenet-Serret formulas which are the standard differential geometric means of addressing the properties of curves in $\Bbb R^3$. It is very important, in utilizing these formulas, to take derivatives with respect to $s$, the arc length of the curve $\gamma(s)$ under consideration. Otherwise, though we will still have $T'(s) \cdot T(s) = 0$, since $T(s) \cdot T(s)$ is constant, implying $T'(s) \cdot T(s) = \frac{1}{2}(T(s) \cdot T(s))’ = 0$, $\kappa$ will not necessarily reflect the true *rate of bending* of the curve $\gamma(s)$. This is perhaps most easily seen *via* the observation that, if $r$ is another parametrization of $\gamma$, then along $\gamma$, $\frac{d}{dr} = \frac{ds}{dr} \frac{d}{ds}$. This means, for example, that

$\frac{dT}{dr} = \frac{ds}{dr} \frac{dT}{ds} = \frac{ds}{dr} \kappa N; \tag{4}$

if we accept $\Vert \frac{dT}{dr} \Vert$ as the definition of curvature, it’s value can vary extensively with the choice of $r$, and its geometrical meaning is obscured. Taking the curve parameter as arc length $s$ standardizes the definition of $\kappa$ by isolating it from the somewhat arbitrary choices of possible parametrizations of $\gamma$.

In the above, the unit vector $N$ gives the *direction* of bending of the curve $\gamma$, and the curvature $\kappa$ gives its *magnitude*. Intuition can be developed for this, in terms of elementary physics, by imagining one driving in a car at a fairly high but constant rate of speed. If the vehicle is traveling at say, $100 \,\text{km/hr}$, we can “unitize” this rate by defining a new distance measure, the *umpla*: $1 \, \text{umpla} = 100 \, \text{km}$; thus “unit speed” in umpla-units is

$100 \, \text{km/hr}$; quite rapid, actually. We imagine the car as turning while maintaining this speed; then the car is undergoing an acceleration in a direction perpendicular to its motion; the magnitude of this acceleration is $\kappa$ and its direction is $N$. Now imagine, if you will, for any $s_0$ a circle of radius $\kappa^{-1}(s_0)$ centered at $\gamma(s_0) + \kappa^{-1}(s_0)N(s_0)$, where we hold $s_0$ fixed. This circle touches the curve $\gamma(s)$ at $\gamma(s_0)$. A particle traveling at unit speed on such a circle has velocity vector $T(s_0)$ at $\gamma(s_0)$ and will experience a centripetal acceleration, in the direction $N(s_0)$, given by the classic formula $\frac{v^2}{r}$ with $v = 1$ and $r = \kappa^{-1}(s_0)$, exactly the same as the acceleration $\kappa N$ of $\gamma$ at $s = s_0$; the greater $\kappa(s_0)$ is, smaller $r$ will be, and *vice versa*; tight turns produce large such accelerations; you can almost feel yourself thrown against the door on a tight right.

An example from physics: gravity pulls Earth towards the sun, and the earth travels in a (vaguely) circular orbit. So, the attractive force acting on the earth is always a radial vector (pointing towards the center of the orbit), yet the velocity vector that Earth travels with is always tangent to the circle traced out by the orbit.

Force is related to acceleration, and acceleration is the derivative of velocity.

If you have a curve in $\mathbb{R}^3$, you can translate each unit tangent vector to this curve so that its foot point starts at the origin in $\mathbb{R}^3$.

When you view all of these unit tangent vectors with foot points now starting at the origin, the collection of head points produces a curve in $S^2$.

Now, let’s say your curve in $\mathbb{R}^3$ is $c(t)$ and that the unit tangent field to your curve is $T(t)$, that is, $$T(t) = \frac{\dot c(t)}{|\dot c(t)|}.$$

Now, the vector $T'(t)$ is found by computing the velocity vector to the curve $T(t)\in S^2$. Since the velocity vector to any curve that lives in $S^2$ must be tangent to $S^2$, we must have $T(t)$ and $T'(t)$ are perpendicular.

- What IS conditional convergence?
- Largange polynomial second order derivative
- Multi variable integral : $\int_0^1 \int_\sqrt{y}^1 \sqrt{x^3+1} \, dx \, dy$
- Integrating $e^{a\cos(\phi_1-\phi_2)+b\cos(\phi_1-\phi_3)+c\cos(\phi_2-\phi_3)}$ over $^3$
- Show that there exists a vector $v$ such that $Av\neq 0$ but $A^2v=0$
- How many permutations of a multiset have a run of length k?
- Boy and girl paradox is driving me crazy
- dual of $H^1_0$: $H^{-1}$ or $H_0^1$?
- Fourier series for $\sin x$ is zero?
- Are bimodules over a commutative ring always modules?
- Tensor product in dual-space
- Possible ways to walk to school
- The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$
- Differentiation, using d or delta
- Why do we need the partial derivative $\frac {\partial F}{\partial t}$ in the definition of an envelope?