Calculus is a branch of mathematics, which studies the differential and integral of functions and related concepts and applications in higher mathematics. It is the basic subject of mathematics. The content mainly includes limit, differential calculus, integral calculus and its application. Differential calculus, including the calculation of derivatives, is a set of theories about the rate of change. It makes the function, velocity, acceleration and curve slope can be discussed with a set of universal symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume.
As a science, advanced mathematics has its inherent characteristics, namely, high abstraction, strict logic and wide application. Abstract and calculation are the most basic and remarkable characteristics of mathematics-high abstraction and unity, which can profoundly reveal its essential laws and make it more widely used. Strict logic means that in the induction and arrangement of mathematical theory, whether it is concept and expression, or judgment and reasoning, we must use the rules of logic and follow the laws of thinking. Therefore, mathematics is also a way of thinking, and the process of learning mathematics is the process of thinking training. The progress of human society is inseparable from the wide application of mathematics. Especially in modern times, the appearance and popularization of electronic computers have broadened the application field of mathematics. Modern mathematics is becoming a powerful driving force for the development of science and technology, and it has also penetrated into the field of social sciences extensively and deeply. Therefore, it is very important for us to learn advanced mathematics well.
The basic theorem of calculus points out that finding indefinite integral and derivative function are reciprocal operations [the integral value can be obtained by substituting the upper and lower bounds into indefinite integral, while differentiation is the product of derivative value and increment of independent variable], which is why the two theories are unified into calculus. We can discuss calculus from either of them, but in teaching, differential calculus is usually introduced first.
Calculus and integral calculus
Calculus is a general term for differential calculus and integral calculus. It is a mathematical idea, in which' infinite subdivision' is differential and' infinite summation' is integral. /kloc-In the second half of the 7th century, Newton and Leibniz completed the preparatory work involving many mathematicians and independently established calculus. Their starting point of establishing calculus is intuitive infinitesimal, but their theoretical foundation is not solid. Because the concept of "infinity" can't be calculated by the existing algebraic formula, it was not until Cauchy and Wilstras established the limit theory and Cantor established the strict real number theory in the19th century that the subject was considered rigorous.
limit
The first step in learning calculus is to understand the necessity of introducing "limit", because algebra is a familiar concept, but algebra can't handle the concept of "infinity". Therefore, in order to deal with the quantity representing infinity by algebra, the concept of "limit" is carefully constructed. In the definition of "limit", we can know that this concept bypasses the trouble of dividing a number by 0 and introduces an arbitrary small number of processes. That is to say, the divisor is not zero, so it makes sense. At the same time, the amount of this process can be arbitrarily small. As long as it is less than any small amount in the δ interval, we will say that its limit is this number-you can think it is opportunistic, but its practicality proves that this definition is relatively perfect and gives the possibility of correct inference. This concept is successful.
Connecting with practical application
Calculus is developed with practical application, and is more and more widely used in various branches of natural science, social science and applied science such as astronomy, mechanics, chemistry, biology, engineering and economics. In particular, the invention of computers is more conducive to the continuous development of these applications. Everything in the objective world, from particles to the universe, is always moving and changing. Therefore, after introducing the concept of variables into mathematics, it is possible to describe the movement phenomenon in mathematics. Due to the emergence and application of the concept of function and the needs of the development of science and technology, a new branch of mathematics has emerged after analytic geometry, which is calculus. Calculus plays a very important role in the development of mathematics. It can be said that it is the greatest creation in all mathematics after Euclidean geometry.