Applications of Vector Analysis and Fourier Series and Its Transforms




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Chance Harenza

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Applications of Vector Analysis and Fourier Series and Its Transforms

Chance Harenza

Imagine a world where oil spills cannot be contained, falling overboard from an ocean conveyance is an automatic death sentence by drowning, buildings can be routinely expected to simply fall over from high winds, ships routinely sink from having entered whirlpools and electricity that cannot be reliably gathered, to name a few of our dependencies, and it becomes painfully obvious that our day to day existence is tremendously improved by Vector Analysis along with Fourier Series and Transforms.

Vectors and Fourier series, chapters 6 and 7 of Mathematical Methods in the Physical Sciences, have innumerable applications some of which will be presented here. Some of the realms of our world that use and scientifically benefit from our knowledge of vectors are engineering, meteorology, and of course our wonderful world of physics and research. Fourier series and transforms are valuable although possibly a little less obvious. Fourier series is often used in electrical engineering and spectrum estimation and analysis research but also in studying behavior of heat in space.

Vectors are to a large extent the heart and soul of much in the field of physics. Vectors are a scientific or mathematical expression of magnitude and direction. These are used to explicitly express movement of mass in space, three dimensional space or two dimensional space, and force in directions. A vector can be in terms of V(2D)=|F|cosØi+|F|sinØj= ai+bj=[x, y] two dimensions or three V(3D) = ai+bj+ck= [x, y, z]. They can be in unit direction form i (position along the x axis) j (position along the y) k (position along the k), the k is dropped in two dimensions, with a, b, c= the distance on the corresponding axis or they can be in coordinate form in brackets. The brackets are to let the reader know the expression is of vectors instead of routine coordinates of a single point.

Vectors apply to anything or situation that one requires an analysis of movement and force. One aspect of vector analysis is the curl. “ The curl of a vector field, denoted or (the notation used in this work), is defined as the vector field having magnitude equal to the maximum "circulation" at each point and to be oriented perpendicularly to this plane of circulation for each point. More precisely, the magnitude of is the limiting value of circulation per unit area”. The curl of a vector field is the amount of "rotation" or angular momentum of a substance in a given area of space (Weisstein, “curl”). Another vector tool is divergence which gives the "density" is decreasing in a section of space (Weisstein, “divergence”). They apply to anything that has a density and flow such as charge density in an electric field, air density and the atmosphere, and body of water with its water density and current flow. The gradient is the direction of a function in which the directional derivative has the largest value and the negative gradient is the direction of minimum value (Weisstein, “gradient”). It is the three dimensional version of finding max and min.

Anything with a flow movement and substance can be studied with these means. Therefore things as dissimilar and diverse as Meteorology, Oceanography, Astrophysics, Geology, and Engineering, all share the use of vectors analysis techniques in common.

Meteorology is the study of the weather even though the word sounds like it is the study of meteors. The study of weather is really the applied study of air and water vapor movements. The wind is air moving from a high pressure area to a low pressure area like people leaving a crowded room to go to an empty one. This movement is what we define as wind. The wind and wind currents are analyzed by representing them with vectors. Lots of little vectors accumulate to become a strong wind. With sufficient strong winds or vectors there can be a storm or a front. A storm is many strong winds pushing on each other and if they hit at the right angle they make a circle of wind or whirlwind. If the combined wind cycle is sustained and stable we have a hurricane or tornado (Saucier 245-373).

Oceanography, specifically the study of ocean currents, while similar to meteorology, uses vectors to instead analyze the movements of water. These oceanic movements are very important to nautical enterprises and ships. Ocean currents, like water, can be stormy especially deep under the surface. As meteorology can predict air storms, oceanography can predict troubled waters warning ships to avoid them.



The undersea current’s direction and flow are needed to be known references should something inadvertently fall overboard or leak from a boat. People usually being the “things” that fall overboard can then appropriately be rescued. Also pollutants, such as an oil spill, can be stopped before they are able to spread and dissolve into the sea killing the sea-life and the humans that fish them.

Vectors in Astrophysics are an absolute necessity. An object in space has a path of motion and its path along with present location can be determined by using vectors. “Every particle in the universe attracts every other particle with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This force acts along the line joining the two particles.” That inverse force proportion is known as the law of universal gravitation. The resulting movement is accurately described in vectors. The vector form of Newton’s Law of universal gravitation is:

F12= - G (m1*m2) r G=6.67e-11 Nm²/kg² and r is the unit vector

(R21)²

Collisions and orbits in space also require vectors to accurately analyze and predict outcomes. In space, collisions are elastic and momentum is conserved. The direction of the objects is determined by the angle of collision which is the angle between the vectors of the objects. (Giancoli 135-37,206-217) Launching something into space or controlling and monitoring the orbit of a satellite require computing and analyzing vectors. The object has a discernable direction and a defined magnitude so it is now described as a vector. Its movement can be predicted and corrected with techniques affecting pitch and yaw so that it appropriately arrives at orbiting distance.

Geology is the study of Earth and its materials. The most famous aspect and one of the most concept-shattering are the movements of the continental plates. Our lands are the parts or segments of giant continents that are not covered in water. These continents float on the sludgy thick liquidly melted rock that makes up the earth’s mantle. Currents push and pull on the floating continental plates. The movement of the plates is described by vectors. The continents have a magnitude and a direction of movement. An example of which is:

“Consider the ridge-ridge-ridge triple junction that brings the Indian, Australian, African, and Antarctic plates into contact (figure 10.63A).” it can be represented in the form of figure B. (Davis, Reynolds 610)



The movements of the continents can be tracked and, at least, retrospective or historically plotted. “Coney was able to show that the relative velocity between the western Northern America and the oceanic plates to the west was anomalously high (namely, 14 cm/yr) during the interval 80 Ma-40…his calculations revealed that the 14cm/yr relative velocity vector was oriented almost perfectly perpendicular to the boundary.” Coney concluded it was the convergence had actually created the Rocky Mountains (613). The Earth is spinning and its core is also spinning (Hawaiian). This causes a coriolis effect in the liquid outer core. Coriolis affects the behavior of vectors giving the particles or objects appearing to arc away from a median line of the spinning larger object the subject objects are moving on (Giancoli 292-91). There are currents similar to those in the air and water

The exacting world of engineering would have a much higher failure rate if it weren’t for the use vectors. Vectors are critical to safety and functionality. Within the field of engineering the plans and specification sheets must be sound and specific. In order for them to be sound and adequately expressed engineers by necessity build models and calculate the forces their structure must be expected to endure well. Those calculations are done with the applications that use the understanding of vectors. The wind responses and fire damages in emergency theoretical scenarios must end with the building still standing or the occupants having had enough time to escape, preferably both(Harris).

Electrical engineers, to be effective, have to be aware of and have an understanding of electric fields, magnetic fields, and, of course, current. All of which, depending on the application, require a knowledge of vectors to be used. Current, for starters, is a moving charged particle. So it is represented by a vector of the flow of electrons (which is negative to positive) which is opposite to the flow of current (I) (being positive to negative) (Giancoli 636-37). An electric field, or electromotive force (E), surrounds a charged particle and E=Force/charge= (kqQ/r²)/q k=1/(4πєo) and particle in E will accelerate (554,564). If charged partials line up such as within wire with a current they produce a magnetic field (B) B=(µoI)/(2πd) d=distance from the wire; µo=permeability of free space=4πE-7. The movement of charged partials can be described in vector form so then currents are shown as an arrow (710-11). One vector tool is the Laplacian. The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics. It is defined by Weisstein at the mathworld website as







The Fourier Series, on the other hand, are used on periodic or intermittent behaving things such as harmonic waves, electromagnetic waves, and heat conduction. The latter being the inspiration for the Fourier Series. Jean-Baptiste Joseph Fourier was figuring out how to analyze the movement of heat through materials when he came up with what has become known as the Fourier Series. His methods were published in 1822 in Théorie analytique de la chaleiur. (Vretblad 9).Although some say Euler published in the 1700’s long before Fourier began his work (Price 1). To say something is periodic means it is repeating and action or behavior according to a pattern to infinity so how do you add all those periods up? Fourier series is extremely useful and very commonly used to break up any arbitrary periodic function into sections of simple terms that can be plugged in, solved, and put back together to reach the solution to the original problem or an approximation to it to whatever accuracy is required (Weisstein). A series is kind of like a power series but with sines and cosines, the Fourier Series. The wave function doesn’t necessarily have to be a simple sine or cosine just as long as it is periodic in a repeating format. Fourier Series is also helpful to determine the average value from the cumulative aspect of the periodic repetition. The average value of sine squared of nx over a period equals the average value of cosine squared of nx over a period (Boas 349). They both equal one half.

Four Series transform with a parameter order {a} can be generalized into a fractional Fourier transform. These fractional Fouriers are so useful in optics that the study of optical systems is called Fourier optics. It is also used in optical information processing, signal analysis and signal processing (Kutay 1-2). "The diffraction of light can be viewed as a process of continual fractional Fourier transformation as light propagates, its distribution evolves through fractional transforms of increasing order" and there also exists a relationship between the amplitude distribution of light on two reference spherical surfaces that is a fractional Fourier transform(326). If filters are inserted at fractional Fourier transform planes provides the basis for various operations in signal processing to be performed (372). “Filter circuits” are based on the concept of filtering in fractional Fourier domains. “Within this framework one can flexibly and efficiently realize general shift-variant linear filters for a variety of applications” (387); applications such as signal recovery. Filtering is also used to eliminate distortions to images (413).

(Davidson)

The Fourier transform is often employed in geophysics to process and filter two dimensional data of seismic motion. It allows the waves to be separated based on the wave’s speed, frequency and wavelength so that only waves of a certain phase speed can get through designed filters called fan-filters (Gubbins chapter 5). It is also used to look for oil. The process for getting the image of oil under the earth and filtering out noise plus clarifying the images all are applications of the Fourier Series (Price). The Fourier series allows us to synthesize sounds. Synthesizers combine various pure tones (harmonics) “to create a richer sound through emphasizing certain harmonics by assigning larger Fourier coefficients (and therefore higher corresponding energies” (Stewart). Fourier series is one of the most important tools in solving differential equations, so it is frequently used in a differentiated form (Harrell).

Our world is intricately woven with physic application. The modern age is increasingly full of wave behaviors and vector motion. Our growing use of satellites, microwaves, radio waves, and electrical devices ensures the necessity for Fourier Series, Fourier transforms and vector analysis.

Works Cited

Boas, Mary L. Mathematical Methods in the Physical Sciences. New Jersey: John Wiley & Sons, 2005.

Davidson, Russ ,Spring. Basic Concepts in Digital Image Processing. Molecular Expressions. Jul 16, 2004 at 08:16 AM.November 13, 2005.

<http://www.microscopy.fsu.edu/primer/digitalimaging/imageprocessingintro.html >

Davis, Reynolds. Structural Geology of Rocks and Regions. New Jersey: Wiley, John & Sons, Incorporated. November 1995

Giancoli, Douglas C. Physics for Scientists & Engineers. New Jersey: Prentice Hall, 2000.

Gubbins, David. Time Series Analysis and Inverse Theory. United Kingdom: Cambridge U. Press, 2004.

Harrell II, Herod. Linear Methods of Applied Mathematics. 1997. November 13, 2005.

< http://www.mathphysics.com/pde/ch5wr.html >

Harris, Wood, Minson. “Towards an expert system for estimating wind loads on building attachments using detailed local velocity data.” Elsevier Science B.V. Journal of Wind Engineering and Industrial Aerodynamics. Volume 62, Issue 1 , August 1996, Pages 11-36. October 8, 2005 < http://continuum.uta.edu >.

Kutay, Ozaktas, Zalevsky. The Fractional Fourier Transform. Great Britain: Antony Rowe Ltd, 2001.

Price, John F. Fourier Techniques and Applications. New York: Plenum Press, 1985.

Saucier, Walter J. Principles of Meteorological Analysis. New York: Dover Publications, 2003.

Significate Wave Height and Direction valid for Oct-09-2005 06:00GMT. Current Marine Data. Oceanweather Inc. 2001.October 8, 2005 <www.oceanweather.com>

Stewart. “Fourier Series.” <www.stewartcalculus.com/data/CALCULUS%20Early%20Transcendentals/upfiles/FourierSeries5ET.pdf >

Vretblad, Anders. Fourier Analyses and Its Applications. New York: Springer-Verlag, 2003.

Weisstein. "Curl." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Curl.html>

"Divergence." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Divergence.html>

"Gradient." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Gradient.html>

"Laplacian." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/Laplacian.html>

"Fourier Series." From MathWorld--A Wolfram Web Resource. <http://mathworld.wolfram.com/FourierSeries.html>

Whalerb, Davisa. “The 1969 geomagnetic impulse and spin-up of the Earth's liquid core.” Elsevier Science B.V. Physics of The Earth and Planetary Interiors. 15 November 1997: Volume 103, Issues 3-4, Pages 181-194.October 8, 2005 < http://continuum.uta.edu >

Hawaiian Volcano Observatory. “Earth's spinning core provides magnetic protection and disaster movie material.“ October 17, 2003. November 13, 2005.

<http://hvo.wr.usgs.gov/volcanowatch/2003/03_10_09.html >

Davis, Reynolds. Structural Geology of Rocks and Regions. New Jersey: Wiley, John & Sons, Incorporated. November 1995

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