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August 26, 2010

My Six Favourite Formulas – #2

Filed under: mathematics — ckrao @ 2:21 pm

In this post I would like to say more about the generalised Stokes Theorem:

\displaystyle \int_M d\omega = \int_{\partial M}\omega

Here:

  • M is an oriented smooth n-manifold (a space whose local neighbourhoods look like n-dimensional Euclidean space)
  • \omega is an (n-1)-form with compact support on M (further explanation will follow)
  • \partial M is the boundary of M with induced orientation (sign/orientation matter!)
  • d\omega is the exterior derivative of \omega (further explanation will follow)

This gem of a formula shows a duality between the boundary operator and the exterior derivative, where the two are linked by the integration operation. It relates the algebraic structure of differential forms (integrands) to the topology of manifolds. This duality is understood better by studying homology (of chains) and (de Rham) cohomology.

Apart from this duality, another reason this formula appeals is that through the language of differential forms it generalises many integration formulas:

1. The fundamental theorem of calculus (FTOC):

\displaystyle \int_a^b f(x)\ dx = F(b)-F(a) where F' = f

2. The gradient theorem (FTOC for line integrals):

\displaystyle \int_p^q \mathbf{\nabla} \Phi . d\mathbf{r} = \Phi(q) - \Phi(p)

3. Green’s theorem in the plane:

\displaystyle \oint_C M dx + N dy = \int \int_R \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}\right)\ dx dy

4. Cauchy’s integral theorem (can be proved from Green’s theorem and Cauchy-Riemann conditions if we assume continuous partial derivatives):

\displaystyle \oint_\gamma f(z)\ dz = 0

5. The curl theorem (Stokes’s theorem), a more general form of the previous two equations:

\displaystyle \int_S \mathbf{\nabla}\times\mathbf{F}. d\mathbf{S} = \oint_{\partial S} \mathbf{F}.d\mathbf{r}

6. The divergence theorem:

\displaystyle \int_V \mathbf{\nabla}.\mathbf{F}\ dV = \int_S \mathbf{F}.d\mathbf{S}

One only needs to recall the general form and 1-6 all drop out as special cases.

About Differential Forms

Élie Cartan was the genius who recognised that differential forms were the right objects to study when generalising the fundamental theorem of calculus. These can be viewed simply as things that are integrated and combine the ideas of smooth functions and multilinear forms (i.e. many-variable real-valued functions that are linear in each variable).

A 0-form is simply a smooth function on a manifold M. For k > 0 a differential k-form assigns to each point p of M an alternating multilinear map on k tangent vectors in the tangent space at p. Think of the tangent space at p as the vector space of directions in which one could pass through p locally. A map is alternating if it evaluates to 0 when two of its variables are the same. The wedge product of k vectors is such an example and it turns out by linearity that a k-form can be written as a linear combination of wedge products of k 1-forms. Alternating forms generalise the calculation of the signed volume of a parallelepiped spanned by k vectors.

For example the 1-form dx associates to any point p a map which assigns to any tangent vector at p its x-coordinate. A 1-form multiplied by a smooth function is also a 1-form.

Higher k-forms can be formed by taking wedge product of two forms. The 2-form dx\wedge dy (also written dxdy) evaluated at a point p maps a pair of tangent vectors at p to the signed area of the projection onto the x-y plane of the parallelogram spanned by the tangent vectors.

In general a k-form may be written as

\displaystyle \alpha = \sum_I f_I\ dx_I

where the multi-index I represents a k-tuple of increasing integers from 1 to n. The notation dx_I is short for

\displaystyle dx_{i_1} \wedge dx_{i_2} \wedge \ldots \wedge dx_{i_k}.

Here are some examples of differential forms on the manifold \mathbb{R}^3. Note that one cannot add a k-form and an l-form if k and l are not equal.

0-form: x - \cos y + xyz

1-form: xdx - yzdy + xdz

2-form: (z-xy)dxdy + z dydz + dzdx

3-form: (xy + 3)dx dy dz

Exterior derivative

The exterior derivative unifies the grad/div/curl operations of vector calculus and generalises the notion of a differential of a function. In fact the exterior derivative of a 0-form (smooth function) f is defined as its differential:

\displaystyle df = \sum_{i=1}^n \frac{\partial f}{\partial x_i}\ dx_i

This is seen as a linear combination of 1-forms dx_i and so df is a 1-form. With a general differential k-form \alpha = \sum_I f_I\ dx_I as given above, its exterior derivative d\alpha is a (k+1)-form that may be defined as

\displaystyle d\alpha = \sum_I df_I \wedge dx_I

The differentiation operator d obeys the product rule and commutativity of partial derivatives implies that d^2 = 0. This is dual to the result that the boundary of a boundary is 0. The wedge product operation is alternating and therefore skew-symmetric, meaning \alpha \wedge \beta = -\beta \wedge \alpha whenever \alpha and \beta are 1-forms.

Integration

Differential forms are integrated by mapping the region of integration on the manifold to open sets in \mathbb{R}^n (via a parametrisation), then performing the integration there as a multi-dimensional integral on chains (linear combinations of simplices). Intuitively, to find \int_M \alpha:

(i) divide M into infinitesimal parallelepipeds (which reduce to segments or parallelograms in lower dimensions)

(ii) for each vertex of the subdivision evaluate \alpha at the k tangent vectors spanning the parallelepiped to obtain an infinitesimal scalar

(iii) sum the infinitesimal scalars over all infinitesimal parallelepipeds in the entire region of integration to obtain a scalar. Take the limit as the largest infinitesimal parallelepiped tends to 0 in volume.

Example: Green’s Theorem

We will show some of the manipulations of differential forms required in establishing Green’s theorem (3., above) from the generalised Stokes theorem. Here \omega is the 1-form Mdx + Ndy in a subset of the plane and

\begin{array}{lcl}d\omega & = & d(Mdx + Ndy)\\ & = & d(Mdx) + d(Ndy)\\& = & dM \wedge dx + dN \wedge dy\\ & = & \left(\frac{\partial M}{\partial x}dx + \frac{\partial M}{\partial y}dy \right) \wedge dx + \left(\frac{\partial N}{\partial x}dx + \frac{\partial N}{\partial y}dy \right) \wedge dy\\ & = & \frac{\partial M}{\partial x}dx\wedge dx + \frac{\partial M}{\partial y}dy \wedge dx + \frac{\partial N}{\partial x}dx\wedge dy + \frac{\partial N}{\partial y}dy\wedge dy\\& = & \frac{\partial M}{\partial y}dy \wedge dx + \frac{\partial N}{\partial x}dx\wedge dy\\ & = & \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx \wedge dy\\ & = & \left(\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dx dy \end{array}

Substituting this into the generalised Stokes theorem gives Green’s theorem.

Apart from Stokes’s formula, the language of differential forms is useful for making the change of variable formula for integration straightforward. More about that in a future post perhaps!

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August 21, 2010

The Russian Heat Wave of 2010

Filed under: climate and weather,geography — ckrao @ 7:37 am

Now that the Russian heat wave appears to be over finally, it’s time to capture some of the phenomenal temperature records set. Firstly let’s focus on Moscow, the largest city to have suffered such unusual conditions.

I have plotted its maximum and minimum temperatures in Moscow over the period of the heat wave between June 20 and August 19 (2 months exactly, or 61 days). Not once did it fall below its average temperature during that period. The heat was at its worst in the three week period between July 22 and August 11, when the average maximum temperature was 35.5°C over the 21 days! The historical average maximum for June-August is 21-23°C and the average minimum 12-14°C.

  • In July 2010 the average temperature (including max and min) was 26.1°C, when the previous record for the warmest ever month in Moscow was 23.3°C! A typical July in Moscow has an average temperature of just 17°C.
  • A total of 22 date records were set for the maximum temperature. In the brutal 3 week period mentioned above, the date record was set an amazing 16 times!
  • The maximum temperature exceeded 30°C every day between July 14 and August 15… 32 straight days!
  • Prior to this year the maximum temperature ever recorded was 36.8°C way back in 1920. This temperature was exceeded no fewer than 6 times this summer (within 12 days), with the new record of 38.2°C set on July 29.
  • A temperature of at least 30°C was set 43 times this summer. To put this number in perspective, this temperature was not even reached once in all of 2009!

Turning to outside Moscow, this image from the NASA Earth Observatory shows fairly well how unusually hot it has been over a large part of western Europe during the brunt of the heat wave.

Heatwave in Russia (click to see image in new window)

Yashkul (Jaskul) near the Caspian Sea possibly recorded the highest maximum temperatures in Russia (in °C), but not as far above its average as Moscow. On the 11th of July it recorded 44.0°C, the highest ever temperature recorded in Russia. Furthermore it had the following hot streak to begin August (when its historical mean maximum is around 32°C):
42.3, 39.8, 40.4, 40.7, 40.4, 41.2, 42.2, 40.0, 40.3, 43.5, 42.5, 41.0, 39.4, 39.8, 39.9

Over the four week period between 21 July and 17 August its daily maximum was always at least 35°C with a mean maximum of 39.3°C!

Sources and further information

August 14, 2010

Best consistency streaks in sport

Filed under: sport — ckrao @ 5:43 am

The streaks below reflect consistency rather than necessarily winning. The next best of its kind is also shown.

  • One of the most famous streaks in American sport is in Joe Dimaggio getting a base hit in 56 consecutive games in 1941. (The next best is 45.)
  • Cal Ripken, Jr. played an astonishing 2632 consecutive games of major league baseball between 1982 and 1998. (The next best is 2130 and the best in modern times is 1152.)
  • Brett Favre has the record for the most consecutive starts by a non-kicker – 307 including playoff games – dating back to 1992. (The next best is 282.)  This is more impressive when you consider he is a quarterback. After Peyton Manning with 210 the next best such streak by a starting quarterback is 128.
  • Quarterback Johnny Unitas had 47 consecutive games with a touchdown pass between 1956 and 1960. (The next best is 36.)
  • Tiger Woods made the cut in 142 straight professional golf tournaments (i.e. was roughly in the top half at the midway point) from 1998 to 2005. (The next best is 113.) Note that 31 of the 142 tournaments did not have a cut.
  • In test cricket Alan Border played his last 153 matches consecutively from 1979 to 1994. (The next best is 107.)
  • In Australian Rules football (AFL) Jim Stynes played in 244 consecutive games between 1987 and 1998. (The next best is 226.)
  • Roger Federer reached the semi finals or better in 23 consecutive tennis grand slam tournaments between 2004 and 2010.  (The next best is 10.)
  • Chris Evert did the same in 34 consecutive grand slams that she played in between 1971 and 1983. (The next best is 18.) She also reached the US Open semi finals or better 16 consecutive times.
  • Rafael Nadal reached the quarter final or better in 21 consecutive ATP World Tour Masters 1000 tournaments, from 2008 to 2010. (The next best I could find is 11.)
  • A. C. Green played 1192 consecutive games of NBA basketball from 1986 to 2001. (The next best is 906.)
  • In the 1983/4 season Wayne Gretzky played 51 consecutive (ice) hockey games with a point (assist or goal). (The next best is 46.)
  • Micheal Williams shot 97 straight free throws in the NBA in 1993. (The next best is 78.)
  • Orel Hershiser pitched 60 straight scoreless innings from 1988 to 1989. (The next best is 58.)

More here:

http://bleacherreport.com/articles/36934-untouchable-the-greatest-streaks-in-sports

August 4, 2010

My Six Favourite Formulas – #1

Filed under: mathematics — ckrao @ 1:59 pm

Going back to my earlier post listing my favourite formulas, it’s time to explore the beauty in them. First of all let’s look at Euler’s identity.

e^{i\pi} + 1 = 0

Why is this a favourite of mine and so many others? For me it is because it is highly unexpected on first look, and it brings together no fewer than nine of the most important mathematical concepts in a manner more succinct than one could imagine:

  1. the number e \approx 2.7183, which is the limit of the sequence a_n = (1 + 1/n)^n as n tends to infinity
  2. the number \pi \approx 3.1416, which is the ratio of a circle’s circumference to its diameter
  3. the imaginary unit i (defined by i^2 = -1)
  4. the multiplicative identity 1
  5. the additive identity 0
  6. the exponentiation operation a^b
  7. addition (+)
  8. multiplication (\times)
  9. equality (=)

This equality is a special case of Euler’s formula, but it can also be seen with physical ideas without the use of trigonometric functions. The exponential function y = e^z has the property that its rate of change is equal to itself (i.e. \frac{dy}{dz} = e^z), so for z = it,

\frac{d}{dt} e^{it} = i\frac{d}{dz}e^{z} = ie^{it}.

We visualise e^{it} as the position of a point particle in the two-dimensional complex plane, with position e^0 = 1 at time t = 0. The above equation describes the motion of the particle and says that its velocity is always i times its position.

Since multiplication by i corresponds to an anti-clockwise rotation by 90 degrees in the complex plane, this is saying that the instantaneous motion is in a direction perpendicular to its position relative to 0, which will neither increase nor decrease its distance from 0. (Here I visualise turning a handle that you might use when opening a car window, in the days before power windows! The incremental motion is always perpendicular to the length of the handle which is fixed in size.)

Hence as t increases from 0, e^{it} will trace out a circular arc, centred at 0, in an anticlockwise direction, at speed equal to the magnitude of its velocity, or

|dz/dt| = |i e^{it}| = |e^{it}| = |e^0| = 1.

Here the second last equality comes from the earlier statement that e^{it} does not change in magnitude with t (see footnote below for a more rigorous argument). Therefore z = e^{it} traces out a unit circle at unit speed. At t = \pi, the particle will have covered length \pi along the circular arc. By the definition/property of \pi, it will have covered half the circumference of the circle, and be diametrically opposite 1, at -1. Hence e^{i\pi} = -1 from which our magical formula arises.

Footnote: If e^{it} = x(t) + iy(t) where x and y are real functions of t, then taking derivatives or multiplying both sides by i gives ie^{it} = dx/dt + i dy/dt = ix(t) -y(t). Matching real and imaginary parts leads to dx/dt = -y, dy/dt = x, and so

\frac{d}{dt} |e^{it}|^2 = \frac{d}{dt} (x^2(t) + y^2(t)) = 2(x \frac{dx}{dt} + y \frac{dy}{dt}) = 2(-xy + yx) = 0.

Therefore when t is real, the magnitude of e^{it} is constant and equal to its value at t=0.

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