BARD/GPT - comparison 1/3
What is the difference between Newtonian and Non-Newtonian calculi?
BARD -
Newtonian calculus is the calculus that is based on the work of Isaac Newton in the 17th century. It is based on the idea of derivatives and integrals, which are used to calculate the rates of change and areas under curves.
Non-Newtonian calculus is a family of calculi that are not based on the work of Newton. They are based on other ideas, such as the idea of differential forms or the idea of geometric calculus.
The main difference between Newtonian and Non-Newtonian calculus is that Newtonian calculus is linear, while Non-Newtonian calculus is nonlinear. This means that in Newtonian calculus, the derivative of a function is a linear operator, while in Non-Newtonian calculus, the derivative of a function may be a nonlinear operator.
Non-Newtonian calculus is often used in areas where Newtonian calculus is not sufficient, such as in the study of turbulence or in the study of materials that are not solid or fluid.
ChatGPT -
Newtonian calculus refers to the standard calculus developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. It deals with the study of rates of change and accumulation of quantities. Newtonian calculus is based on the assumption that functions are continuous, differentiable, and have a constant rate of change.
Non-Newtonian calculi, on the other hand, are a family of alternative calculi that relax one or more of the assumptions of Newtonian calculus. Non-Newtonian calculi have been developed to deal with situations where the assumptions of Newtonian calculus are not satisfied, such as in the case of discontinuous or non-differentiable functions.
There are several types of non-Newtonian calculi, including:
Fractional calculus: This deals with derivatives and integrals of non-integer order, which can be used to describe systems with long-range interactions or memory effects.
Infinitesimal calculus: This deals with infinitesimal quantities that are smaller than any real number, which can be used to describe phenomena that occur on a very small scale.
Ultradifferentiable calculus: This deals with functions that are infinitely differentiable but not necessarily analytic, which can be used to describe functions with singularities or rapid growth.
Quantum calculus: This deals with operators and functions that behave like quantum mechanical operators, which can be used to describe systems in quantum mechanics.
In summary, while Newtonian calculus assumes continuity, differentiability, and a constant rate of change, non-Newtonian calculi relax one or more of these assumptions to deal with more complex systems.
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