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Circular Integral or Contour Integral or Closed Integral

Circular integral is different from Integral over closed Contour. A C losed contour is topologically a circle but the curvature can be anything. It is expressed as  ∮ where as circular integral is expressed as Integral from 0 to 2pi. Circular Integral is used mostly in spherical and cylindrical co ordinates. It is also used in periodic functions. The Closed Contour Integral is found usefulness in electromagnetic theory to express the relation between electric current and magnetic field. As an Induction of current through a spiral wire creates a magnetic field and vice versa. In summary, while the symbol ∮ itself does not directly denote an integral over a circular region in the plane (like a disk), it represents a closed line integral around a closed curve or path in space. This concept is closely related to understanding circulation, flux, and work done by vector fields in contexts that involve circular or closed geometries.
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Sine, Cosine and Tangent - Simply heights and lengths.

Sines, Cosines and Tangents are simply different, complicated words for people who learns maths. Any math student will addition to be very useful and subtraction to be equally useful and at time multiplication is needed. while most may wonder why do we need to study trigonometry. Many people who learn pythagoras theorem will appreciate to calculate the unknown one side length of the triangle when 2 of the other sides are known with the formula c^2 = a^2 +b^2. Trigonometry: Tri+gono+metry  = Three+Angle+Measure From my point of view 1st 2D primitive and the only primitive 2D shape is traingle with least no. of sides. All other 2D shapes have more than 3 sides. we cannot form 2D shape with 2 lines alone, we require 3 lines. Based on sides triangles can be equilateral, isosceles and scalene. Based on angles triangles can be right angled, acute angled and obtuse angled. Equiangular triangle - equilateral by default whose angle is 60 degrees. Isoceles triangle - right angled by default....
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Russian Peasant method - Multiplication with Doubling and Halving why it works?

Russian Peasant method Given 2 Number, 1st Number need to be doubled and 2nd number need to be halved. Until the halved number becomes zero. Whenever reminder occurs during halving take the first doubling number and note it down separately. Add all the noted numbers. It is the required multiplication result. How is it useful? It is useful to make use of bitwise operator in algorithms instead of ALUs. Right shift halves and left shift doubles and reminder check can be done with bitwise AND operator. Very useful in Assembly language programming to do multiplication. Why does it work? We can see multiplication as increasing one entity with another entity. When we multiply one number by 2 and we are dividing another number by 2. Thereby we are balancing and equating the operations on 2 numbers. When the reminder arrives, 1*doubling number is kept separately and 1 is reduced from halving number. 1 times doubling number is kept separately to keep the balance or equating stuff. Anyway when we...
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