Application of Mathematics

Generally, we consider mathematics as a different subject from science. But in fact, mathematics is an intimate part of basic science and daily life activities.

Mathematics can be used to find the rate of disintegration of radioactive isotopes and to find its half life.

It is used for periodic function and periodic motion such as:

• (i) simple harmonic motion,
• (ii)progressive sound waves and
• (iii) A.C. current.

Using logarithmic function it can be used to find the concentration of P.H. value of acidic, basic and neutral solution in chemistry.

The concept of increasing and decreasing function in mathematics can be used for the diagrammatic representation of Boyle's Law, Charles Law and Ohm's Law in chemistry.

Using the anti-derivative of mathematics we can find the gravitational potential and electrostatic potential as well.

The concept of anti-derivative can also be used in laws of thermos-dynamics and so on.

Here we will discuss how the periodic function of mathematics is used in Simple Harmonic Motion.

What is periodic function?

In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. Thus a function y=f(x) is said to be a periodic function of period P if P is the least positive real number such that f(x+P)=f(x).The most important example of periodic function are the trigonometric functions, whose values repeat over certain interval.

For example: sin(2π+x)=sinx, cos(2π+x)=cosx ,so sine and cosine are periodic functions of period 2π.

Also, tan(π+x)=tanx , so it is a periodic function of period π and so on.

Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity.

Example:

A graph of the sine function showing two complete periods

For example, the sine function is periodic with period 2π, since for all values of x, this function repeats on intervals of length 2π. (see the graph above).

Periodic functions frequently arise in various fields of science and engineering, business and in some daily life activities such as:

• i. Brain waves and heart beating are periodic.
• ii. Household voltage and electric current are periodic.
• iii. The cash flows in seasonal business are periodic.
• iv. The seasons and weather are periodic.
• v. The motion of wrist watch and pendulum clock are periodic.
• vi. The phases of the moon are periodic.

What is Simple Harmonic Motion?

In mechanics and physics, simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement.

Equation of simple Harmonic Motion

The mathematical representation of simple harmonic motion can be expressed by the relation y=A sint ; where y=displacement from mean position ,

A= Amplitude , = angular velocity with = 2π/T; T = time period & t = time.

This simple harmonic motion can be represented by the motion of simple pendulum.

Fig: sinusoidal Wave representation of simple Pendulum.

By the equation of Simple Harmonic Motion, i.e. y=A sint,

when t=0, then y=0

t=T/4, then y=A

t=T/2, then y=0

t=3T/4, then y=-A

t=T, then y=0

So from above, its clear that position of the bob of the pendulum repeats in every T/2 interval. Hence the motion of simple pendulum is periodic.

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The Importance of Mathematics

The study of mathematics is interesting and enjoyable. People like its challenge, its clarity, and the fact that we know when we are right. The solution of a problem has an excitement and a satisfaction. We will find all these aspects in mathematics .We should also be aware of the wide importance of Mathematics, and the way in which it is advancing at a spectacular rate. Mathematics is about pattern and structure; it is about logical analysis, deduction, calculation within these patterns and structures. When patterns are found, often in widely different areas of science and technology, the mathematics of these patterns can be used to explain and control natural happenings and situations. Mathematics has a pervasive influence on our everyday lives, and contributes to the development of our mind.

The everyday use of arithmetic and the display of information by means of graphs are an everyday commonplace. These are the elementary aspects of mathematics. The use of mathematics in our daily life is very much essential besides this advanced mathematics is widely used, some importance of mathematics often used in an unseen and unadvertised way are:

• The mathematics of error-correcting codes is applied to CD players and to computers.
• The stunning pictures of far away planets sent by Voyager II could not have had their crispness and quality without such mathematics.
• Voyager's journey to the planets could not have been calculated without the mathematics of differential equations.
• Whenever it is said that advances are made with supercomputers, there has to be a mathematical theory which instructs the computer what is to be done, so allowing it to apply its capacity for speed and accuracy.
• The development of computers was initiated by mathematicians and logicians, who continue to make important contributions to the theory of computer science.
• The next generation of software requires the latest method of mathematics that is called category theory, a theory of mathematical structures which has given new perspectives on the foundations of mathematics and on logic.
• The physical sciences (chemistry, physics, oceanography, astronomy) require mathematics for the development of their theories.
• In ecology, mathematics is used when studying the laws of population change.
• Statistics provides the theory and methodology for the analysis of wide varieties of data.
• Statistics is also essential in medicine, for analyzing data on the causes of illness and on the utility of new drugs.
• Travel by aeroplane would not be possible without the mathematics of airflow and of control systems.
• Body scanners are the expression of subtle mathematics, discovered in the 19th century, which makes it possible to construct an image of the inside of an object from information on a number of single X-ray views of it.
• Thus mathematics is often involved in matters of life and death.

These applications have often developed from the study of general ideas for their own sake: numbers, symmetry, area and volume, rate of change, shape, dimension, randomness, patterns of mathematics and many others. Mathematics makes an especial contribution to the study of these ideas and makes a right decision. The study of mathematics can satisfy a wide range of interests and abilities. It develops the imagination. It trains in clear and logical thought. It is a challenge, with varieties of difficult ideas and unsolved problems, because it deals with the questions arising from complicated structures. Yet it also has a continuing drive to simplification, to finding the right concepts and methods to make difficult things easy, to explaining why a situation must be as it is. In so doing, it develops a range of language and insights, which may then be applied to make a crucial contribution to our understanding and appreciation of the world, and our ability to find and make our way in it.

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Some Nepali Sweets

 सेल

सामग्रीः

• मसिनो पिसेको चामलको पीठोः १ किलो
• खस्रो चामलको पीठो ः ५०० ग्राम
• मैदाः ५०० ग्राम
• चिनीः ५०० ग्राम
• अलैँची पिसेकोः २० ग्राम
• घिउः आवश्यकता अनुसार

विधिः
एउटा गहिरो भाँडामा चामलको पीठो, मैदा, चिनी तथा अलैँचीको धूलो मिसाउने र त्यसमा पानी हालेर मुुछ्ने । त्यसलाई आधा घण्टा जति त्यत्तिकै छाड्ने । सेल पकाउने तावाँमा सेल डुुब्ने गरी घिउ तताउने । घिउ तातेपछि एक डाडुु जति मुछेको पीठोमा हालेर चलाउने । सोलीको सहायताले घिउमा सेल आकारमा पकाउने । दुवै साइडमा खैरो भइन्जेल पल्टाई पल्टाई गरेर पकाउने ।

लाखमरी

सामग्रीः

• ताइचिन चामलको पीठोः १ किलो
• घिउः ७५० ग्राम
• मासः ५०० ग्राम
• चास्नीः डुबाउन मिल्ने गरी

विधिः
चामललाई दिनभरि भिजाएर रातभरि ओभाउन दिने र भोलिपल्ट पिस्ने । मास पनि रातभरि भिजाएर बोव्रmा निकाली बिहान पिस्ने । पिसेको मासमा अलिअलि पानी हारेर फिट्ने । घिउ, चामलको पीठो तथा मास मिसाएर राम्रोसँग फिट्ने । नरम भएपछि स–साना प्वाल भएको कपडामा बाँध्ने । त्यसपछि ताईमा हालेर पकाउने र सेप बनेपछि कराहीमा घिउ तताएर खैरो कुने गरी पकाउने । खैरो भएपछि झिकेर प्लेटमा हाल्ने र सेलाएपछि चिनीको बाक्लो झोललाई ब्रसले पालिस गरेजस्तै गरी रोटीमा लगाउने । यसरी तयार भएको लाखमरी तिहारमा बढी प्रयोग हुन्छ । यसलाई लामो समय सम्म राखेर खान पनि सकिन्छ ।

ल्वाङ्गफूल

सामग्री

• मैदाः १ किलो
• घिउः ७५० ग्राम
• चास्नीः आवश्यकता अनुसार
• नरिवलः १२५ ग्राम
• केशरः आधा चम्चा
• पानीः १ लि. जति

विधि
घिउलाई तताएर मनतातो बनाई १ लिटर पानी हाल्ने । उक्त घिउ र पानीमा मैदा हालेर मुछ्ने । पीठो तयार भएपछि गोलो बनाएर तीनकुने आकारमा बेल्ने । बेल्दा अलिअलि पानी हाल्दै फोल्ड गर्दै थिच्दै जाने । त्यसपछि यसलाई घिउमा पकाउने । ५–६ मिनेट पकाएपछि निकालेर सेलाउन दिने । सेलाएपछि चास्नीमा डुबाउने । चास्नीबाट निकालेपछि माथीबाट नरिवल र केशर हालिदिने ।

निम्किन

सामग्रीः

• मैदाः ४ कप
• नुनः स्वाद अनुसार
• घिउः १ कप मुछ्नको लागि
• जिराको दानाः १ कप
• कालो तिलः १ चम्चा
• घिउः १ केजी
• खान मिल्ने सुन्तला रंगः थोरै
• पानीः आवश्यकता अनुसार

विधि
एउटा भाँडा मैदा राखेर नुन, तिल, जिरा, रंग, घिउ र पानी हाले मज्जाले मुछ्ने । त्यसपछि त्यसलाई स–साना गोला डल्ला बनाउने । त्यसलाई पातलो र बाटुलो हुने गरी बेल्ने र आधा पट्याउने । फेरि तीन चुच्चे तथा तीन पत्रे हुने गरी पट्याउने र त्यसपछि पान आकारमा बेल्ने । यसलाई तताएको घिउमा दुवै तर्फ हुने गरेर राम्रो सँग पकाएर निकाल्ने ।

मसला लड्डु

सामग्री

• खुवाः १ किलो
• घिउः ४ पाउ
• मसलाका लागि नरिवलको धूलो, गुँड, काजु तथा चिनीको धूलो स्वाद अनुसार

विधि
खुवालाई घिउमा हालेर घोट्ने । घिउ र खुवा मज्जाले मिसाएपछि केही समय त्यत्तिकै राखेर चिसो हुन दिने । त्यसपछि भाँडामा निकालेर किस्तीमा राख्ने । त्यसमा नरिवल, बदाम, गुँद, काजुआदि मसलाको धूलो हाल्ने । सबै मसला हालेर केहीबेर मोलेपछि चिनीको धूलो हालेर डल्लो पार्ने । यसरी मसलाको लड्डु घरमै पनि तयार पार्न सकिन्छ ।

कलाकन्द

सामग्री

• दूधः ८ लिटर
• चिनीः ५ सय ग्राम
• कागतीको रसः ४ थोपा

विधि
एउटा गहिरो कराहीमा दूध उमाल्ने । त्यसमा केही थोपा कागतीको रस मिसाउने । त्यसपछि आवश्यकता अनुसार चिनी हाल्ने । सानो आँचमा टाइट नभइन्जेल पकाइरहने । जब कलाकन्द टाइट हुँदै जान्छ तब त्यसलाई फराकिलो किस्तीमा निकालेर ५–६ घण्टासम्म जम्न दिने । यसलाई आफ्नो इच्छानुसार टुक्रा पारेर दाजुभाई तथा दिदीबहिनीलाई सगुनस्वरुप खुवाउन सकिन्छ ।

मुंगको बर्फी

सामग्री

• मुंगको दालः १ केजी
• घिउः १ केजी
• चिनीः १ केजी
• पानीः आधा लिटर
• अलैँची पाउडरः थोरै
• बरकः १ टुक्रा
• पिस्ता–बदामः थोरै

विधि
मुंगको दाललाई सफा गरेर ३ घण्टा जति भिजाउने । मिक्सरले खस्रो हुने गरी पिस्ने । पिसेको दाललाई घिउमा पकाउने । खैरो रंगको भएपछि निकाल्ने र चिनीको पाक गाढा बनाई डुबाएर निकाल्ने । अघि बनाएको सबै चीज मिसाउने, अलैँची पाउडर पछि राख्ने । किस्तीमा खन्याएर १२ घण्टापछि माथिबाट बरक पेपर राख्ने । काटेको पिस्ता, बदाम छर्कने र बर्फी आकारमा काट्ने ।

Poon Hill: The Popular Trekking Destination in Nepal

Overview:

Poon Hill is one of the popular trekking destination in the massif of Annapurnas. This easy but mostly up and down route follows the main trail to Annapurna Sanctuary and traverses through scenic rich fields, lush forest of rhododendron and beautiful villages mostly inhabited by Gurungs and Magars. Himalayan Peaks of Annapurna such as Annapurna South (7,219m), Annapurna II (7,937m), Annapurna III (7,555m), Annapurna IV (7,525m) and Mt. Fistail (Machhapuchchhre) (6,997m) becomes visible from the very first day of the Ghorepani and Pun Hill Trek.

Some Glimpses of Mountains

Sunrise Early in the morning

Golden Fishtail (Mt. Machhapuchchhre)

Fishtail (Mt. Machhapuchchhre)

Mt. Dhaulagiri Range

Mt. Annapurna Range

These majestic mountains appear different every day. From Ghorepani and Poon Hill, you can get an even better and close up views of two ranges, Annapurna and Dhaulagiri. An hour's hike up from Ghorepani to Poon Hill early in the morning bestows you with magnificent sunrise views over more than 10 peaks of Annapurna and Dhaulagiri massif.

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