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Is calculus a cloud?
No. Calculus if a field of mathematics.
Is every field an integral domain?
Yes, every field is an integral domain.
Mathematical field calculus?
Calculus is mainly about limits, which in turn are used to calculate the slope of a line (known as the "derivative"; lots of applications for that), and to calculate the area under a curve (the "integral" - also lots of applications for that). For more details, read the Wikipedia article on "Calculus", or read an introductory book on calculus. As prerequisites, you should be well-acquainted with high-school algebra.
Why is calculus so useful in science and engineering?
Speaking as someone who is in the engineering field i can tell you that calculus serves no purpose at all for me. Everything is just done with a calculator or a computer so it's all just a bunch of hogwash they put you through for Engineering school that you will never use in the real world unless you are to become a program writer or work in physics.
Who are the famous mathematicians in the field of calculus?
Leibniz and Newton are usually considered the most famous in the field of calculus.
