Calculus- part 2: Vector Calculus| Gradient, Divergence, Curl

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Calculus- part 1: Introduction

Calculus is the mathematical study of continuous change. Originally called infinitesimal calculus, it has two major branches, differential calculus and integral calculus.

Differential Calculus

The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point is the slope of the tangent to a curve. The derivative of a function f(x) with respect to x is 

The above expression is also called as fundamental theorem of differentiation.

A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant. The del symbol is used to denote the partial derivative. The chain rule is used to find the derivative of the multivariable function.

Integral Calculus

The concept of the integral was first started to calculate the area under a curve. The area under a curve can be thought of as a sum of rectangles of infinitesimal width. The integral of a function f(x) with respect to a variable x on an interval [a, b] is written as

If limits are specified, the integral is called a definite integral. When the limits are omitted, the integral is called an indefinite integral. There is no fundamental theorem of integration unlike differentiation.

Now let's look at the fundamental theorem of calculus which relates these two branches of calculus i.e. differential calculus and integral calculus. It states that the integral of f over an interval is equal to the antiderivative of f (a function whose derivative would be f). This result was kind of surprising and nonintuitive that somehow the problem of finding areas under a curve could be solved with differentials. 

Proof : For a given function f, define the function F(x) as

Differentiating on both sides






Integral from x to x+Δx is just the area under f in that interval. For a very small value of Δx, we can assume f(x) remains constant. Thus this area is equal to f(x)Δx.




Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread roots in every branch of physics and engineering.

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Calculating atomic mass, A practical approach

Dalton published his first table of relative atomic weights containing six elements (hydrogen, oxygen, nitrogen, carbon, sulfur and phosphorus), relative to the weight of an hydrogen atom. Since these were only relative weights, they do not have a unit of weight attached to them.

The basic approach for measuring mass of an hydrogen atom is discussed below. The amount of electricity passed through the electrolyte is directly proportional to the mass of any substance deposited or liberated at an electrode, according to Faraday’s law of electrolysis.

        m = Z q ;     m - mass,  q - charge,  Z - constant of proportionality

        

The constant 'Z' can be experimentally determined by taking ratio of hydrogen liberated at electrode and the charge transferred q = I × t . It is approximately equal to 96500 gm/C. 

So the amount of charge required for depositing one hydrogen atom is same as the charge of an H+ atom that is q = 1.6 × 10^-19. Thus the mass of one hydrogen atom liberated at an electrode should be simply Z times q.  

        Atomic mass =  96500  × 1.6 ×10^-19     = 1.6 ×10^-27 kg

        

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What is Centrifugal Force ?

Centrifugal is not a real force. It's just the equivalent of the force needed to be applied to overcome inertia as per Newton's second law. When a body is moving in a circular path with constant speed, its velocity is constant in magnitude but not in direction. Thus the rate of change of velocity is not zero. As the body keeps changing direction and has non-zero acceleration, there must be some force acting on it.

Let's assume a body of mass m moving on a circular path of radius r. The body moves by a small angle dθ in time dt. Distance traveled by the body during this time is given by the product of the radius of the circle and the angle subtended at the center that is rdθ. Therefore dt can be written as

The angle between two lines is the same as the angle between their perpendiculars. Therefore v1 and v2 are also having angle dθ between them. Taking the x-axis along v1 and the y-axis perpendicular to v1 and radially outwards

Change in velocity is difference of final and initial

Centrifugal force is oriented along the negative y-axis as per the assumed coordinate system which is the direction towards the center. So whenever the body is revolving in a circular motion with constant speed, a centripetal force towards the center is required to provide for the inertial force. Centrifugal is not one of the four fundamental forces or derived from them unlike centripetal force rather it is the result of newton's second law.

As a body in circular motion has acceleration towards the center, the person inside the body experiences a pseudo force in opposite direction i.e. radially outward. It is just like a person inside a lift moving upwards experiences a downward pull. This phenomenon leads to the common misconception that centrifugal force is something special force acting radially outward.

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Reality vs Fiction

Fiction comes with infinite possibilities. It is not difficult to be lost in these imaginary worlds. Some corner of the mind secretly wants to live such fairy tales. Be the detectives solving cases. Be overpowered demon-lord crushing every enemy coming their way or a mad scientist traveling through time. Fiction can be anything you want. It is different from the trivial way of life. There is excitement and adventure at each step. Medieval era of swords and weapons, horses, magic spells, kingdoms, mythical creatures, or cyberpunk atmosphere of flying cars, mecha suits, and space travel. All seems to incite every cell in the body with a sense of new.

In the movie Matrix, Morpheus presents Neo with two choices. If he takes the blue pill, he stays in the simulated world of the Matrix but if he takes the red pill, he awakes in the real world to find the harsh truth about machines. There are multiple reasons why reality sucks. 

• Everything is bound by physical laws. You cannot create anything out of nothing or use magic. 
• Reality is unpredictable. It does not have specific goals or an inbuilt story. You don’t always get to be the main character or be the center of attention. 
• Life is a slow process. Consistent efforts for years or even a lifetime are required to make a significant difference. Day-to-day life isn’t mysterious and most often is repetitive and boring. 
• Life is harsh. Failures are an essential part of life. You have to fight for survival and at times endure pain and misery. Some things cannot be fixed no matter what you do.

Sometimes truth is more fascinating than fiction. Despite so many cons, the reality is especially different. It is just like how truth should be. Balance of good and bad. Look at the stars in the night sky. Being alive in this world itself is no less than any fiction. Humanity has come so far. Inventions like vehicles, airplanes, the telephone, the computer, satellites, and spaceships, etc. are on equal terms with sorcery. Although expecting exact fiction from life would not be practical, many possibilities cannot be ignored. 

It does not do to dwell on dreams and forget to live. If you are a fan of fantasy then you may like to split time between work or job and then as a reward reading a novel or watching a movie or playing video games but running away from real-world problems is only going to make them worse. Just keep doing as much as possible. I think what world we live in is not going to change me because - It is our choices that show who we are, far more than our abilities. 

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