### Welcome to Talha's Physics Academy

To Help Teachers and Students.

Talha's Physics Academy is an exploration environment for concepts in physics which employs free Physics Books and other linking strategies to facilitate smooth navigationThe entire environment is interconnected with thousands of links, reminiscent of a neural network.

New content for Talha's Physics Academy will be posted as it is developed,It is my intent to keep this material continuously available except for brief maintenance times.

All the Branches of Physics are covered.

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### What is the difference between magneto and a.c generator?what is meant by frequency of alterneting current?

A magneto is an electrical generator that uses permanent magnets to produce periodic pulses of alternating current.

Hand-cranked magneto generators were used to provide ringing current in early telephone systems.
Magnetos adapted to produce pulses of high voltage are used in the ignition systems of some gasoline-powered internal combustion engines to provide power to the spark plugs. Use of magnetos for ignition is limited mainly to engines where there is no low-voltage electrical system available, such as lawnmowers and chainsaws. Magnetos have traditionally been used in aviation piston engines even though a low-voltage electrical system is usually available, to keep the ignition system independent of the battery and charging system and to keep the engine running in the event of alternator or battery failure. For redundancy purposes, virtually all piston engine aircraft are fitted with two magneto systems, each supplying power to one of two spark plugs in each cylinder.

Magnetos were rarely used for power generation, although they were for a few specialised uses.

A Generator is a device which converts mechanical energy into electrical energy.

A.C Generator works on the principle of electromagnetic induction (motional emf). In generator an induced emf is produced by rotating a coil in a magnetic field. The flux linking the coil changes continuously hence a continuous fluctuating emf is obtained.

A.C Generator consists of the following parts.
Powerful field magnet with concave poles.
Armature:It is a rectangular coil of large number of turns of wire wound on laminated soft-iron core of high permeability and low hysteresis loss.
Slip rings:The ends of the coil are joined to two separate copper rings fixed on the axle (S1 & S2).
Carbon brushes:Two carbon brushes remain pressed against each of the rings which form the terminals of the external circuit.

The frequency of alternating current depends on the speed of the alternator/generator at the power production station.

In India, its usually 220 Volts at 50 Hz.
In USA, it is 440 Volts at 60 Hz.
So the variation depends on the speed of the alternator at the time of generation and how the phase changes on the wave.

Hence, its possible to change the frequency of AC.

## The electrical power generated in a power station situated in a remote place is transmitted to different regions for domestic and industrial uses. Here we discuss the electric power grid, power generation, transmission, and distribution.

• ### Power Grid

What is a power grid?
The power grid forms a bridge between electrical suppliers and consumers through interconnected networks.
The electrical power grid consists of three main parts:
1. Generating plant for electric power.
2. Transmission of the electric power.
3. Distribution of the electric power.
• ### Electrical Power Generation in Power Stations

Normally, electric power generating plants are located near the source of power generation like dams, coal mines, etc. They may be in remote locations, and the generated power has to be supplied to the consumer in the city.
Sources for Power Generation:
The following types of resources are available for generating electrical energy for distribution:
Steam Power Plants
Looking For Used Processing Plants? Find Them Here! 30 Years Experience
• Conventional Methods
1.Thermal:
Thermal energy or Nuclear Energy used for producing steam for turbines which will drive the alternators (rotating AC generators).
2. Hydro-Electric:
Potential of water stored at higher altitudes is utilized as it is passes through water turbines which drive the alternators.
• Non- Conventional Methods:
1. Wind Power:
High velocities of wind are utilized in driving wind turbines coupled to alternators.

• ### Transmission of Electric Power

Once the power is generated from the alternator, it is send to the typical substation in the power plant where they step up the voltage by using the step-up transformers for transmission purposes.
As the voltage is stepped up, it reduces the transmission losses. It is then sent to the power grid from where it is then transmitted to different cities. All the power generated in different places by different methods is stepped up and sent to a common place called the grid.
Why do we need to step up the voltage for transmission?
For long distance transmission, power lines are made of conducting material like aluminum. There is always some power loss associated with these lines.
If I is the current through the wire and R is the resistance, a considerable amount of electric power (I2R) is dissipated as heat. Hence, the power at the receiving end will be much less than the actual power generated.
However by transmitting the electrical energy at higher voltage, power loss can be controlled as is evident from the following cases:
Case 1:
Power Transmitted At Lower Voltage:
A Power of 11,000 Watts is transmitted at 220 Volt.
Formula for Power (P) = V × I
V-Voltage
I-Current
Therefore, Current (I) =P / V
Current (I) =11,000 / 220.
Current (I) =50 Ampere.
If R is the Resistance of the line wire,
Then the power Loss:
Formula, Power Loss = I2 × R
Power Loss = 502 × R watts = (25000 × R) Watts.
Case 2:
Power Transmitted At Higher Voltage:
If 11,000 Watts power is transmitted at 22,000 Volt
Current (I) =P/V = 11,000/22,000 =0.5 Ampere.
Power loss = I2 × R = (.5)2× R = (0.25 × R ) watts.
From the case 1 and 2, we found that when the power is transmitted at higher voltage, the power losses are less. Hence it is evident that if power is transmitted at a higher voltage, the loss of energy in the form of heat can be considerably reduced.
For transmitting electric power at 11,000W at 220 V the current capacity of line wire has to be 50 A and if transmission is done at 22,000V it is only 0.5 A.
Thus for carrying larger current (50A), thick wires have to be used. This increases the cost of transmission to support these thick wires, and stronger poles have to be erected, which further adds on to the cost. On the other hand, if transmission is done at high voltages, the wires required are of lower current carrying capacity, so thicker wires can be replaced by thin wires, thus reducing the cost of transmission considerably.
An example would be if 400MW power is produced at 15,000V in a power station, it could be stepped up to 230,000V before transmission. The power is then transmitted through the transmission lines or network to the power grid.

• ### Distribution of Electric Power

The electric power grid connects different parts of the country. And the grid distributes the power to the different parts of the country or state through the transmission lines or network connecting different cities.
Outside the city, the transmitted power from the grid is stepped down in voltage to 110,000 Volt from 230,000 Volt by a step-down auto transformer. Again the power is stepped down to 11,000 Volt from 110,000 Volt by a Step Down power transformer located in the streets and it is distributed to the industrial uses.
Finally before distribution to the domestic user, the power is stepped down to 230 V or 440 V depending upon the needs of the domestic user.

### Wave Nature of Particle - the de Broglie Hypothesis( College Level)

In experiments like photoelectric effect and Compton effect, radiation behaves like particles. de Broglie, a french physicist asked whether in some situations, the reverse could be true, i.e., would objects which are generally regarded as particles (e.g. electrons) behave like waves ? In 1924 de Broglie postulated that we can associate a wave with every material object. In analogy with photons, he proposed that the wavelength associated with such a matter wave is related to the particle momentum through the relationship

where is the Planck's constant

Wavelike behaviour of a macroscopic object is difficult to detect as the wavelength is very small.
However, wave nature of particles may be detected in diffraction experiments where the dimensions of the obstacles are comparable with the wavelength of matter wave incident on the obstacle.

Example :Electron Diffraction from a Crystal

Consider a beam of electron with a speed m/s corresponding to a wavelength nm. Such a wave may be diffracted by gratings with separation of similar order as that of the wavelength. Crystals provide such natural gratings.
Davisson - Germer Experiment :
Experimental confirmation of de Broglie hypothesis was provided in 1926 by Davisson and Germer, who studied diffraction of a beam of electrons from the surface of a nickel crystal.

A beam of electrons from a heated filament, accelerated through a potential difference is made to strike the surface of a crystal of Ni. Electrons are scattered in all directions and may be detected by an array of detectors located at various angles of scattering. It is found that the intensity of scattered beam is maximum at some particular angles of incidence, in the same manner as the case when a beam of x-rays strikes the crystal
In Davisson - Germer experiment, the electron beam was accelerated through a potential difference of volts. The kinetic energy of the electron is thus 54 eV. The de Broglie wavelength associated with an electron accelerated through a potential difference may be expressed as

Notes:

The dual nature not only is exhibited by radiation but is also associated with matter. In some experiments matter shows wave character.
de Broglie hypothesis poastulates a wavelength of with a particle having a momentum .
Experimental confirmation of wave nature of matter comes from experiments such as Davisson Germer experiments
on electron diffraction from crystals. It is seen that the intensity of scattered beam is maximum at those points where one would expect Laue spots in x-ray diffraction assuming the electronsare waves with de Broglie wavelength.
Bohr model can be understood by postulating that stable orbits in atoms are those which are standing waves of
electrons.
One can perform double slit experiment with electrons, similar to the way Young's double slit experiment is
performed with light waves. The intensity pattern obtained on a screen is very similar in both cases.
According to the principle of complementarity one cannot obtain information on both the wave nature and particle
nature of matter or radiation in the same experiment.
Heisenberg uncertainty principle states that one cannot precisely measure both position and momentum of a
particle in the same experiment.

### Uses of DC Load line

The below figure shows the output characteristic curves for the transistor in CE mode. The DC load line is drawn on the output characteristic curves.

Load line To draw load line, we have to find saturation current and the cutoff voltage. After plotting these values on the vertical and the horizontal axes, a line is drawn joining these two points, which represents DC load line. It represents all possible combinations of the collector current Ic and the collector voltage Vc (or Vce) for the given load resistorRc.

Saturation point The point at which the load line intersects the characteristic curve near the collector current axis is referred to as the saturation point. At this point of time, the current through the transistor is maximum and the voltage across collector is minimum for a given value of load. Therefore saturation current for the fixed bias circuit, Ic (sat) =Vcc/Rc

Cutoff point The point where the load line intersects the cutoff region of the collector curves is referred as the cutoff point (i.e. end of load line). At this point, collector current is approximately zero and emitter is grounded for fixed bias circuit.Therefore, Vce (cut) = Vc = Vcc

### Basic Unit Conversions

Ans:In the field of science, the metric system is used in performing measurements. The metric system is actually easier to use than the English system, as you will see shortly. The metric system uses prefixes to indicate the magnitude of a measured quantity. The prefix itself gives the conversion factor. You should memorize some of the common prefixes, as you will be using them on a regular basis. Common prefixes are shown below:

 Prefix Symbol Power Prefix Symbol Power mega- M 106 centi- c 10-2 kilo- k 103 milli- m 10-3 hecto- h 102 micro- 10-6 deca- D 101 nano- n 10-9 deci- d 10-1 pico- p 10-12

Metric - Metric Conversions
Suppose you wanted to convert the mass of a 250 mg aspirin tablet to grams. Start with what you know and let the conversion factor units decide how to set up the problem. If a unit to be converted is in the numerator, that unit must be in the denominator of the conversion factor in order for it to cancel.

Notice how the units cancel to give grams. I've shown the conversion factor numerator as 1 x 10-3 because on most calculators, it must be entered in this fashion, not as just 10-3. If you don't know how to use the scientific notation on your calculator, try to find out as soon as possible. Look in your calculator's manual, or ask someone who knows. Also, notice how the unit, mg is assigned the value of 1, and the prefix, milli-, is applied to the gram unit. In other words, 1 mg literally means 1 x 10-3 g.
Next, let's try a more involved conversion. Suppose you wanted to convert 250 mg to kg. You may or may not know a direct, one-step conversion. In fact, the better method (foolproof) to do the conversion would be to go to the base unit first, and then to the final unit you want. In other words, convert the milligrams to grams and then go to kilograms:

English - Metric Conversions
These conversions are accomplished in the same way as metric - metric conversions. The only difference is the conversion factor used. It would be a good idea to memorize a few conversion factors involving converting mass, volume, length and temperature. Here are a few useful conversion factors:
length: 2.54 cm = 1 inch (exact)

mass: 454 g = 1 lb

volume: 0.946 L = 1 qt

temperature: oC = (oF - 32)/1.8

All of the above conversions are to three significant figures, except length, which is an exact number. As before, let the units help you set up the conversion.
Suppose you wanted to convert mass of my 23 lb cat to kilograms. One can quickly see that this conversion is not achieved in one step. The pound units will be converted to grams, and then from grams to kilograms. Let the units help you set up the problem:

Let's try a conversion which looks "intimidating", but actually uses the same basic concepts we have already examined. Suppose you wish to convert pressure of 14 lb/in2 to g/cm2. When setting up the conversion, worry about one unit at a time, for example, convert the pound units to gram units, first:

Next, convert in2 to cm2. Set up the conversion without the exponent first, using the conversion factor, 1 in = 2.54 cm. Since we need in2 and cm2, raise everything to the second power:

Notice how the units cancel to the units sought. Always check your units because they indicate whether or not the problem has been set up correctly.

### Projectile Motion (College Level)

Projectile is a body thrown with an initial velocity in the vertical plane and then it moves in two dimensions under the action of gravity alone without being propelled by any engine or fuel.Its motion is called projectile motion.The path of a projectile is called its trajectory.
Examples:
1. A packet released from an airplane in flight.
2. A golf ball in flight.
3. A bullet fired from a rifle.
4. A jet of water from a hole near the bottom of a water tank.
Projectile motion is a case of two-dimensional motion .Any case of two dimensional motion can be resolved into two cases of one dimensional motion -one along the x-axis and the other along the y-axis.The two cases can be studied separately as two cases of one dimensional motion.The results from two cases can be combined using vector algebra to see the net result
What is important to remember is that the motion along the horinzontal direction does not affect the motion along the vertical direcion and vice versa.Horizontal motion and vertical motion are totally independent of each other .
A body can be projected in two ways :
1. Horizontal projection-When the body is given an initial velocity in the horizontal direction only.
2. Angular projection-When the body is thrown with an initial velocity at an angle to the horizontal direction.
We will study the two cases separately.We will neglect the effect of air resistance.We will take x-axis along the horizontal direction and y-axis along the vertical direction.

A body is thrown with an initial velocity u along the horizontal direction.We will study the motion along x and y axis separately.We will take the starting point to be at the origin.
 Along x-axis Along y-axis 1. Component of initial velocity along x-axis. ux=u 1. Component of initial velocity along y-axis. uy=0 2. Acceleration along x-axis ax=0(Because no force is acting along the horizontal direction) 2.Acceleration along y-axisay=g=9.8m/s2It is directed downwards. 3. Component of velocity along the x-axis at any instant t. vx=ux + axt =u + 0 vx=u This means that the horizontal component of velocity does not change throughout the projectile motion. 3. Component of velocity along the y-axis at any instant t. vy=uy + ayt =0 + gt vy=gt 4. The displacement along x-axis at any instant t x=uxt + (1/2) axt2 x=uxt + 0 x=u t 4. The displacement along y-axis at any instant t y= uyt + (1/2) ayt2 y= 0 + (1/2) ayt2 y=1/2gt2
Equation of a trajectory(path of a projectile)
We know at any instant x = ut
t=x/u
Also, y= (1/2)gt2
Subsituting for t we get
y= (1/2)g(x/u)2
y= (1/2)(g/u2)x2
y= kx2 where k= g/(2u2 )
This is the equation of a parabola which is symmetric about the y-axis.Thus,the path of projectile,projected horizontally from a height above the ground is a parabola.

We know ,at any instant t
vx= u
v
y= gt
v= (v
x2+ vy2)1/2 = [u2 + (gt)2]1/2
Direction of v with the horizontal at any instant :
(angle) = tan-1 (vy/vx)= tan-1 (gt/u)

Time of flight (T):It is the total time for which the projectile is in flight ( from O to B in the diagram above)
To find T we will find the time for vertical fall
From
y= uyt + (1/2) gt2When , y= h , t=T
h= 0 + (1/2) gt2
T= (2h/g)1/2

Range (R) :It is the horizontal distance covered during the time of flight T.
From x= ut When t=T , x=R
R=uT
R=u(2h/g)1/2

We will now consider the case when the object is projected with an initial velocity u at an angle to the horizontal direction.
We assume that there is no air resistance .Also since the body first goes up and then comes down after reaching the highest point , we will use the Cartesian convention for signs of different physical quantities.The acceleration due to gravity 'g' will be negative as it acts downwards.
We will separate the motion into horizontal motion (motion along x-axis) and vertical motion (motion along y-axis) .We will study x-motion and y-motion separately.
 X axis Y axis 1. Component of initial velocity along x-axis. ux=u cosÎ¦ 1. Component of initial velocity along y-axis. uy=u sinÎ¦ 2. Acceleration along x-axis ax=0(Because no force is acting along the horizontal direction) 2.Acceleration along y-axisay= -g= -9.8m/s2(g is negative as it is acting in the downward direction) 3. Component of velocity along the x-axis at any instant t. vx=ux + axt =ucosÎ¦ + 0= ucosÎ¦ vx=ucosÎ¦ This means that the horizontal component of velocity does not change throughout the projectile motion. 3. Component of velocity along the y-axis at any instant t. vy=uy + aytvy=usinÎ¦ - gt 4. The displacement along x-axis at any instant t x=uxt + (1/2) axt2 x=ucosÎ¦.t 4. The displacement along y-axis at any instant t y= uyt + (1/2) ayt2 y= usinÎ¦.t - (1/2)gt2
Equation of Trajectory (Path of projectile)
At any instant t
x= ucosÎ¦.t
t= x/(ucosÎ¦)
Also , y= usinÎ¦.t - (1/2)gt2
Substituting for t
y= usinÎ¦.x/(ucosÎ¦) - (1/2)g[x/(ucosÎ¦)]
2
y= x.tanÎ¦ - [(1/2)g.sec2.x2 ]/u2
This equation is of the form y= ax + bx2 where 'a' and 'b are constants.This is the equation of a parabola.Thus,the path of a projectile is a parabola .
Net velocity of the body at any instant of time t
vx=ucosÎ¦
v
y=usinÎ¦ - gt
v= (v
x2+ vy2)1/2
Î¦= tan-1(vy/vx) Where Î¦ is the angle that the resultant velocity(v) makes with the horizontal at any instant .
Angular Projectile motion is symmetrical about the highest point.The object will reach the highest point in time T/2 .At the highest point,the vertical component of velocity vy becomes equal to zero .
vy=usinÎ¦ - gtAt t=T/2 , vy= 0
0= usinÎ¦ - gT/2
T= (2usinÎ¦)/g
Equation for vertical distance (y component)y= uyt - (1/2)gt2
At t=T/2 , y=H
H= usinÎ¦.T/2 - (1/2)g(T/2)
2
substituting T
H= usinÎ¦.usinÎ¦/g - (1/2)g(usinÎ¦/g)
2= (u2sin2)/g - (u2sin2)/2g
H= (u2sin2)/2g
Range is the total horizontal distance covered during the time of flight.
From equation for horizontal motion, x=uxt
When
t=T , x=R
R= u
xT = ucos.2usinÎ¦/g
= u
22sinÎ¦cosÎ¦/g = u2sin2Î¦/g
using 2sinÎ¦cosÎ¦= sin2Î¦
R= (u2sin2Î¦)/g

### Introduction to Space Physics - M. Kivelson, C. Russell

Description: Designed as a text for upper-level undergraduate and first-year graduate students, this volume attempts to establish the curriculum for the modern student entering the fledgling field of space physics. The book provides a broad, yet selective, treatment of the subject, covering virtually all aspects of space plasmas in the solar system. There are sections on the sun and solar wind, the magnetized and unmagnetized planets, and the fundamental process of space plasmas including shocks, plasma waves, ULF waves, wave-particle interactions, and auroral processes. In addition to emphasizing analysis, the authors also place importance on underlying phenomenology with extensive attention to observations.

### Physics for Scientists and Engineers 6th ed. (College Text) - Serway and Jewett

Description:

This best-selling, calculus-based text is recognized for its carefully crafted, logical presentation of the basic concepts and principles of physics. PHYSICS FOR SCIENTISTS AND ENGINEERS, Sixth Edition, maintains the Serway traditions of concise writing for the students, carefully thought-out problem sets and worked examples, and evolving educational pedagogy. This edition introduces a new co-author, Dr. John Jewett, at Cal Poly Pomona, known best for his teaching awards and his role in the recently published PRINCIPLES OF PHYSICS, Third Edition, also written with Ray Serway. Providing students with the tools they need to succeed in introductory physics, the Sixth Edition of this authoritative text features unparalleled media integration and a newly enhanced supplemental package for instructors and students!

### What are the uses of capacitor?

Ans:One of the most ubiquitous passive components used is the capacitor, found in nearly every electronic device ever made. Capacitors have a number of essential applications in circuit design, providing flexible filter options, noise reduction, power storage and sensing capabilities for designers.
Filter Applications

Combined with resistors, capacitors are often used as the main element of frequency selective filters. The available filter designsand topologies are numerous and can be tailored for frequency and performance by selecting the proper component values and quality. Some of the types of filter designs include:
High Pass Filter (HPF)
Low Pass Filter (LPF)
Band Pass Filter (BPF)
Band Stop Filter (BSF)
Notch Filter
All Pass Filter
Equalization Filter

Decoupling/By-Pass Capacitors - Capacitors play a critical role in the stable operation of digital electronics by protecting sensitive microchips from noise on the power signal which can cause anomalous behaviors. Capacitors used in this application are called decoupling capacitors and should be placed as close as possible to each microchip to be most effective, as all circuit traces act as antennas and will pick up noise from the surrounding environment.Decoupling and by-pass capacitors are also used in any area of a circuit to reduce the overall impact of electrical noise.

Coupling or DC Blocking Capacitors - Since capacitors have the ability to pass AC signalswhile blocking DC, they can be used to separate the AC and DC components of a signal. The value of the capacitor does not need to be precise or accurate for coupling, but it should be a high value as the reactance of the capacitor drives the performance in coupling applications.

Snubber Capacitors - In circuits where a high inductance load is driven, such as a motor or transformer, large transient power spikes can occur as the energy stored in the inductive load is suddenly discharged which can damage components and contacts. Applying a capacitor can limit, or snub, the voltage spike across the circuit, making operation safer and the circuit more reliable. In lower power circuits, using a snubbing technique can be used to prevent spikes from creating undesirable radio frequency interference (RFI) which can cause anomalous behavior in circuits and cause difficulty in gaining product certification and approval.

Pulsed Power Capacitors

At their most basic, capacitors are effectively tiny batteries and offer unique energy storage capabilities beyond those of chemical reaction batteries. When lots of power is required in a short period of time, large capacitors and banks of capacitors are a superior option for many applications. Capacitor banks are used to store energy for applications such as pulsed lasers, radars, particle accelerators, and railguns. A common application of the pulsed power capacitor is in the flash on a disposable camera which is charged up then rapidly discharged through the flash, providing a large pulse of current.

Resonant or Tuned Circuit Applications

While resistors, capacitors and inductors can be used to make filters, certain combinations can also result in resonance amplifying the input signal. These circuits are used to amplify signals at the resonant frequency, create high voltage from low voltage inputs, as oscillators, and as tuned filters. In resonant circuits, care must be taken to select components that can survive the voltages that the components see across them or they will quickly fail.

Capacitive Sensing Application

Capacitive sensing has recently become a common feature in advanced consumer electronics devices, although capacitive sensors have been used for decades in a variety of applications for position, humidity, fluid level, manufacturing quality control and acceleration. Capacitive sensing works by detecting a change in the capacitance of the local environment through a change in the dielectric, a change in the distance between the plates of the capacitor, or a change in the area of a capacitor.

### Explain whether atomic number can increase during nuclear decay .Support your answer with example.

Ans.When a beta particle is emitted out of any nucleus,then it's mass number does not undergo any change but it's charge number increase by one.Negative Beta is an electron and its emission from the nucleus becomes an in comprehensive enigma, as there is no electron in the nucleus >how ever , the emission of electrom from the nucleus can be thought of as neutron emitting an electron and becoming a proton, although the modern explanation is not that simple.
This means that beta particle is formed at the time of emission.That is why at the time of emission of a beta particle the charge number of nucleus increases by one.

For Example:

### Schaum's Outline of Vector Analysis Murray R. Spiegel

Description:

Confusing Textbooks? Missed Lectures? Not Enough Time? . . Fortunately for you, there's Schaum's. More than 40 million students have trusted Schaum's Outlines to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. . . This Schaum's Outline gives you. . Practice problems with full explanations that reinforce knowledge. Coverage of the most up-to-date developments in your course field. In-depth review of practices and applications. . Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time-and get your best test scores!. . Schaum's Outlines-Problem Solved..

# Directional Derivative

The directional derivative  is the rate at which the function  changes at a point in the direction . It is a vector form of the usual derivative, and can be defined as
 (1) (2)
where  is called "nabla" or "del" and  denotes a unit vector.
The directional derivative is also often written in the notation
 (3) (4)
where  denotes a unit vector in any given direction and  denotes a partial derivative.
Let  be a unit vector in Cartesian coordinates, so
 (5)
then

 (6)