Divergence & Curl 散度和旋度

DivergenceDivergence is an operation on a vector field that tells us how the field behaves toward or away from a point. Locally, the divergence of a vector field $\vec{F}$ in $\mathbb{R}^2$ or $\mathbb{R}^3$ at a particular point $P$ is a measure of the “outflowing-ness” of the vector field at $P$.If $\vec{F}$ represents the velocity of a fluid, then the divergence of $\vec{F}$ at $P$ measures the net rate of change with respect to time of the amount of fluid flowing away from $P$ (the tendency of the fluid to flow “out of” P). In particular, if the amount of fluid flowing into $P$ is the same as the amount flowing out, then the divergence at $P$ is zero. Definition: divergence in $\mathbb{R}^3$If $\vec{F} = \langle P,Q,R \rangle$ is a vector field in $\mathbb{R}^3$ and $P_x, , Q_y,$ and $R_z$ all exist, then the divergence of $\vec{F}$ is defined by $$\begin{aligned}\operatorname{div} F &=P_{x}+Q_{y}+R_{z} \&=\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\end{aligned}$$ Note the divergence of a vector field is not a vector field, but a scalar function. In terms of the gradient operator $$\vec{\nabla}=\left\langle\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right\rangle$$ divergence can be written symbolically as the dot product $$\operatorname{div} \vec{F}=\vec{\nabla} \cdot \vec{F}.$$ Note this is merely helpful notation, because the dot product of a vector of operators and a vector of functions is not meaningfully defined given our current definition of dot product.

Continue Reading →

Antenna 天线

Antennas are widely used in the field of telecommunications and we know many applications for them. Antennas receive an electromagnetic wave and convert it to an electric signal, or receive an electric signal and radiate it as an electromagnetic wave. Yagi Uda antenna In the past, dipole antennas were used for TV reception. Satellite dish antenna in detail Nowadays we have moved to dish TV antennas. These consist of two main components, a parabolic shaped reflector and a low noise block down converter. Microstrip antenna or Patch antenna The cellphone in your hand uses a completely different type of antenna, called a patch antenna. These types of antennas are inexpensive and fabricated easily onto a printed circuit board. A patch antenna consists of a metallic patch or strip placed on a ground plane with a piece of dielectric material in-between. Here, the metallic patch acts as a radiating element. The length of the metal patch should be half of the wavelength for proper transmission and reception.

Continue Reading →

Electromagnetic Radiation 电磁辐射

In physics, electromagnetic radiation (EM radiation or EMR) refers to the waves (or their quanta, photons) of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays. In the modern world, we humans are completely surrounded by electromagnetic radiation. Have you ever thought of the physics behind these traveling electromagnetic waves? The great scientist, Heinrich Hertz, was the first man to transmit and detect electromagnetic waves. In his famous experiment, a high voltage current was applied to the two ends of two metal wires, which generated a spark in the gap between them. This spark resulted in the radiation of electromagnetic waves. Those electromagnetic waves traveled through the air and created a spark in a metal coil located over a meter away. If you had placed an LED in that gap, the bulb would have glowed. This was a clear case of electromagnetic wave propagation and detection. However, before Hertz, the brilliant mathematician, James clerk Maxwell, had already laid out the foundations for electromagnetic radiation by formulating for mathematical equations. However, these equations and the Hertz experiment raised a question, how do electromagnetic fields detach themselves from wires and propagate through a space? More specifically, what we need is a traveling electromagnetic wave and not a fluctuating one. Let’s explore this logically. Consider an electric charge, which is moving at a constant speed. The electric field around it is shown. Now imagine for a fraction of a second it accelerates after that, it continues its uniform motion at a higher speed. What we need to understand is the effect of this acceleration on the electric field. The interesting thing is that the information does not travel at an infinite speed, instead, it travels at the speed of light. Similarly, the information about the sudden variation of velocity of the charge does not get conveyed to the whole electric field region. The field near it knows about it, but the field far away still has no idea that the charge has accelerated and it is still in the old state. Let’s separate out these regions with the help of two circles. Since the electric field cannot break the field between these distances must transition. This transition field is known as a kink. The kink moves or radiates outwards at the speed of light. To show the kink animation in a clear way, let’s move the camera along with the charge. We can say here that the acceleration of the charge has caused an electromagnetic disturbance or electromagnetic radiation. Based on this understanding, we will be able to understand the most important experiment in the field of antenna technology, the oscillating electric dipole. The interesting fact about this simple oscillating dipole is that it produces electromagnetic radiation in a perfectly sinusoidal manner. Let’s see how it is achieved. Before getting into the electromagnetics, let’s understand how velocity and acceleration vary in this simple case. It is clear that at both ends the velocities should be zero and in the middle the velocity should be at the maximum. This means that this is a case of continuous acceleration and deceleration. The electric field pattern is drawn here when the chargers are far apart, and when the velocity is zero. In order to have a better understanding, let’s examine one of the electric field lines. Let’s observe the electric field line at t by eight. You can see that the electric field line is deformed. The reason for this deformation is simple. This time period is the region with the highest acceleration. As we saw earlier, accelerating or decelerating charges cause kinks in the electric field. In short, the old electric field does not get adjusted to the new field very well. This deformation is continuous since there is continuous acceleration in the charge. When two charges meet at the central point, the deformed line also meets there. After that, it detaches and radiates. This radiation travels at the speed of light. If you applied an electric field intensity variation with respect to length, you can see that the radiation we have produced is perfectly sinusoidal in nature. Please note that this varying electric field will automatically generate a varying magnetic field perpendicular to it. Now let’s have a look at how this applies to an antenna. A time varying voltage is applied to the metal wire is shown due to the effect of the voltage the electrons will be displaced from right to left and create positive and negative charges. With a continuous variation of voltage, the positive and negative charges will shuttle back and forth in the wire. The simple arrangement is known as a dipole antenna. The dipole antenna produces the same radiation as we saw in the previous section. In this case, the antenna works as a transmitter. The frequency of the transmitted signal will be the same as the frequency of the applied voltage signal. The same antenna can act as a receiver if the operation of the antenna is reversed. When propagating electromagnetic waves strike the antenna, the oscillating fields of waves create positive and negative charges at the ends of the antenna. The varying charge accumulation means a varying voltage signal is produced at the center of the antenna. This voltage signal is the output when the antenna works as a receiver. We can note here that for perfect transmission or reception, the length of the antenna should be half of the wavelength. This is the first antenna design criteria for proper reception or transmission. The second most important design criteria is a term called impedance matching. Perfect impedance matching will make sure that the waves are radiated in the most efficient way. When an alternating current passes through a circuit, it faces opposition from the combined effects of resistance, inductance and capacitance. This combined effect is known as impedance. According to the maximum power transfer theorem, to transfer the maximum amount of power the load impedance should match with the source impedance. For further understanding, let’s take an example of a circuit containing an alternator as a source and a motor bulb, et cetera, as a load. In this setup to achieve maximum power transfer from alternator to the load, the impedance of the load must match with the impedance of the alternator. A similar impedance balance is required in the case of an antenna system. Since an antenna works on high frequency signals, the impedance of the transmission lines also becomes important. Hence to achieve maximum power, the impedance of an antenna should match to the impedance of the source and transmission line as well. If the impedances do not match, some portion of the power would be reflected back to the source instead of radiating outwards from the antenna. A free space has an impedance value of 377 ohms. In a parabolic antenna, a wave guide is used as a transmission line, which has a different impedance value from the free space. That’s why a feedhorn is also included in a parabolic antenna. This way, the impedance of the wave guide is matched with the impedance of the free space so that the EM waves can be received properly.

Continue Reading →

Colored Pencils 彩铅

There’s a lot to like about drawing with colored pencils. They’re utterly convenient—a handful of colored pencils and a pad of paper are all you really need to start creating. Prep time and cleanup are practically non-issues, the materials are light and portable, and you don’t need messy or toxic solvents. At the same time, colored pencil drawing lends itself to highly refined and exquisite works of art that rival those created with any other medium. Colored pencils are relatively inexpensive, and the palette is extensive. The color is pure, clean and bright. The medium is permanent, and colored pencil drawings do not require elaborate care or storage. Along with hard and soft colored pencils, watercolor pencils and oil-based colored pencils offer more options for beginning artists. Aside from their convenience and versatility, much of the appeal of colored pencils is the control they offer. You can do loose work, tight work or anything in between. You can use colored pencil to tint a drawing with light strokes that let the color of the paper show through, or you can use colored pencil to create a solid deposit of many layers of color. Because colored pencil is primarily a dry medium, there’s no drying time to worry about. You can walk away from the work and come back and pick up right where you left off. You can start and stop at any time. Colored pencil offers the pleasures and rewards of both drawing and painting. Whatever other medium you enjoy, you’ll find colored pencil a worthwhile addition to your repertoire. Colored pencil offers the pleasures and rewards of both drawing and painting. Whatever other medium you enjoy, you’ll find colored pencil a worthwhile addition to your repertoire.

Continue Reading →

Inception 盗梦空间

A thief who steals corporate secrets through the use of dream-sharing technology is given the inverse task of planting an idea into the mind of a C.E.O.


Dom Cobb is a skilled thief, the absolute best in the dangerous art of extraction, stealing valuable secrets from deep within the subconscious during the dream state, when the mind is at its most vulnerable. Cobb’s rare ability has made him a coveted player in this treacherous new world of corporate espionage, but it has also made him an international fugitive and cost him everything he has ever loved. Now Cobb is being offered a chance at redemption. One last job could give him his life back but only if he can accomplish the impossible, inception. Instead of the perfect heist, Cobb and his team of specialists have to pull off the reverse: their task is not to steal an idea, but to plant one. If they succeed, it could be the perfect crime. But no amount of careful planning or expertise can prepare the team for the dangerous enemy that seems to predict their every move. An enemy that only Cobb could have seen coming.

多姆·柯布 是一位经验老道的窃贼,他在这一行中算得上是最厉害的,因为他能够潜入人们精神最为脆弱的梦境中,窃取潜意识中有价值的秘密。柯布这一罕见的技能使他成为危险的企业间谍活动中最令人垂涎的对象,但这也让他成为了一名国际逃犯,失去自己的所爱。如今柯布有了一个赎罪的机会,只要完成最后一项任务他的生活就会回复本来面目。与以往不同的是,柯布和他的团队这一次的任务不是窃取思想,而是植入思想。如果他们成功,这就是一次完美犯罪。但是即使提前做好了细致专业的安排,也无法预料到危险的敌人对他们的行动早已了如指掌,而只有柯布能够预料到敌人的行踪。

Nunchaku 双截棍

The Handcrafted Nunchaku or nunchucks (Japanese: ヌンチャク nunchaku, often “nunchuks“, “chainsticks“, “chuka sticks” or “karate sticks” in English) is a traditional Okinawan martial arts weapon consisting of two sticks connected at one end by a short chain or rope. The two sections of the weapon are commonly made out of wood, while the link is a cord or a metal chain. The nunchaku is most widely used in martial arts such as Okinawan kobudō and karate, and is used as a training weapon, since it allows the development of quicker hand movements and improves posture. Modern-day nunchaku can be made from metal, wood, plastic or fiberglass. Toy and replica versions made of polystyrene foam or plastic are also available. Possession of this weapon is illegal in some countries, except for use in professional martial art schools.

双截棍或双节棍(日语:ヌンチャク 英文中通常为“nunchuks”,“chainsticks”,“chuka sticks” 或 “karate sticks”)是一种传统的中国武术武器,短链或绳索连接有两根棍子。 武器的两部分通常由木头制成,而链节则是绳索或金属链。



Pencil Sketch 素描

A pencil is an implement for writing or drawing, constructed of a narrow, solid pigment core in a protective casing that prevents the core from being broken or marking the user’s hand. Pencils create marks by physical abrasion, leaving a trail of solid core material that adheres to a sheet of paper or other surface. They are distinct from pens, which dispense liquid or gel ink onto the marked surface.

Line art

A line drawing is the most direct means of expression. This type of drawing without shading or lightness, is usually the first to be attempted by an artist. It may be somewhat limited in effect, yet it conveys dimension, movement, structure and mood; it can also suggest texture to some extent.


Line gives character but shading gives depth and value-it is like adding an extra dimension to the sketch.

Oil Painting 油画

Oil painting is the process of painting with pigments with a medium of drying oil as the binder. Commonly used drying oils include linseed oil, poppy seed oil, walnut oil, and safflower oil. The choice of oil imparts a range of properties to the oil paint, such as the amount of yellowing or drying time. Certain differences, depending on the oil, are also visible in the sheen of the paints. An artist might use several different oils in the same painting depending on specific pigments and effects desired. The paints themselves also develop a particular consistency depending on the medium. The oil may be boiled with a resin, such as pine resin or frankincense, to create a varnish prized for its body and gloss.

Fast Fourier Transform 快速傅立叶变换

The fast fourier transform or FFT is without exaggeration one of the most important algorithms created in the last century. So much of the modern technology that we have today such as wireless communication, GPS and in fact anything related to the vast field of signal processing relies on the insights of the FFT.

But it’s also one of the most beautiful albums you’ll ever see. The depth and sheer number of brilliant ideas that went into it is just astounding it’s easy to miss the beauty aspect of the FFT since it’s often introduced in fairly complex settings that require a lot of prerequisite knowledge such as the discrete fourier transform time domain to frequency domain conversions and much more.

The Fast Fourier Transform (FFT) is an efficient $O(N \log N)$ algorithm for calculating DFTs. The FFT exploits symmetries in the $W$ matrix to take a “divide and conquer” approach. We will first discuss deriving the actual FFT algorithm, some of its implications for the DFT, and a speed comparison to drive home the importance of this powerful algorithm.

To derive the FFT, we assume that the signal’s duration is a power of two: $N=2^l$ . Consider what happens to the even-numbered and odd-numbered elements of the sequence in the DFT calculation.

$$\begin{align}S(k) &=s(0)+s(2) e^{(-j) \frac{2 \pi 2 k}{N}}+\ldots+s(N-2) e^{(-j) \frac{2 \pi(N-2) k}{N}}+s(1) e^{(-j) \frac{2 \pi k}{N}}+s(3) e^{(-j) \frac{2 \pi \times (2+1) k}{N}}+\ldots+s(N-1) e^{(-j) \frac{2 \pi(N-2+1) k}{N}} \nonumber \ &=s(0)+s(2)e^{(-j)\frac{2\pi k}{\frac{N}{2}}} + \ldots + s(N-2)e^{(-j) \frac{2 \pi\left(\frac{N}{2}-1\right) k}{\frac{N}{2}}} +\left( s(1)+s(3) e^{(-j) \frac{2 \pi k}{\frac{N}{2}}}+\dots+s(N-1) e^{(-j) \frac{2 \pi\left(\frac{N}{2}-1\right) k}{\frac{N}{2}}}\right) e^{\frac{-(j 2 \pi k)}{N}} \end{align}$$

Each term in square brackets has the form of a $\frac{N}{2}$ -length DFT. The first one is a DFT of the even-numbered elements, and the second of the odd-numbered elements. The first DFT is combined with the second multiplied by the complex exponential $e^{\frac{-(j 2\pi k)}{N}}$. The half-length transforms are each evaluated at frequency indices $k \in{0, \ldots, N-1}$. Normally, the number of frequency indices in a DFT calculation range between zero and the transform length minus one. The computational advantage of the FFT comes from recognizing the periodic nature of the discrete Fourier transform. The FFT simply reuses the computations made in the half-length transforms and combines them through additions and the multiplication by $e^{\frac{-(j 2 \pi k)}{N}}$, which is not periodic over $\frac{N}{2}$, to rewrite the length-N DFT. The formula above illustrates this decomposition. As it stands, we now compute two length-$\frac{N}{2}$ transforms (complexity $2 O\left(\frac{N^{2}}{4}\right)$), multiply one of them by the complex exponential (complexity $O(N)$), and add the results (complexity $O(N)$). At this point, the total complexity is still dominated by the half-length DFT calculations, but the proportionality coefficient has been reduced.

Now for the fun. Because $N=2^l$, each of the half-length transforms can be reduced to two quarter-length transforms, each of these to two eighth-length ones, etc. This decomposition continues until we are left with length-2 transforms. This transform is quite simple, involving only additions. Thus, the first stage of the FFT has $\frac{N}{2}$length-2 transforms (see the bottom part of fomula). Pairs of these transforms are combined by adding one to the other multiplied by a complex exponential. Each pair requires 4 additions and 4 multiplications, giving a total number of computations equaling $8\frac{N}{4}=\frac{N}{2}$. This number of computations does not change from stage to stage. Because the number of stages, the number of times the length can be divided by two, equals $\log_2 N$, the complexity of the FFT is $O(N \log N)$.

The initial decomposition of a length-8 DFT into the terms using even- and odd-indexed inputs marks the first phase of developing the FFT algorithm. When these half-length transforms are successively decomposed, we are left with the diagram shown in the bottom panel that depicts the length-8 FFT computation.


DFT 是指 Discrete Fourier Transform,离散傅里叶变换。

FFT 是指 快速傅里叶变换,它是 DFT 的一种简化计算。

Baduk 围棋入门 4.0

胡翼飞 的围棋入门视频。