This is a common question:

1, the general math teacher's teaching method is:

(1), the two sides may be integrated first, and the solution of an integral constant is obtained;

(2) Then, according to the initial conditions, the integral constant is solved;

(3) Substituting the integral constant into the solution containing the integral constant to obtain the final result.

This method is often used to solve ordinary differential equations.

2, general physics teacher, astronomy, geology, meteorology, hydrology, engineering,,, the teacher's answer is:

(1), both sides are integrated at the same time, and the final result is obtained in one step.

Description:

There is no essential difference between the first and second solutions, that is, the solution of indefinite integral and definite integral;

Second, there is a corresponding problem between engineering and natural science, that is, the condition of definite solution.

Like the initial time t? , the corresponding speed is v? ; The last moment t corresponds to the speed v,

Both sides merge at the same time, v from v? Integral from t to v, t? The integral product of t.

Third, math teachers teach math in order to teach math. They don't think it's a waste of time to use indefinite integral.

It is a waste of time to solve practical problems with mathematics and use indefinite integral for scientific and engineering applications.

Fourthly, if you use indefinite integral when writing a paper, it is a waste of space, which is the writing method of chin beauty. Professional journals are

It is impossible for you to waste space by indefinite integral method. Just use definite integral in one step.

The most common example can be used in college physics, that is, general physics. The method of definite integral is always carried out.

In general physics and theoretical physics, if indefinite integral is used, it is a method that incompetent teachers will adopt.

Only by using definite integral can concrete physical meaning and physical process be embodied;

The indefinite integral can't reflect the physical process, let alone the exact physical meaning.

Integral has two real physical meanings, each of which has two meanings:

The first is the summation of state quantities, such as volume, mass, electricity, energy and so on;

The second is the accumulation of process quantities, such as work, enthalpy change, entropy change, potential difference and so on.

The second one is the sum of extensional quantities, such as mass, electricity, energy, moment of inertia, etc.

The second is the accumulation of intensity, such as electric field intensity, magnetic induction intensity, temperature, pressure and so on.

This final method of power accumulation is superposition in English and superposition principle in Chinese translation.

Note: Ordinary math teachers can't recognize these two differences in integral because:

1, few people really understand science and engineering, ordinary high school math teachers,

Almost completely incomprehensible, incomprehensible. Integral is specific in various fields of science and engineering.

Application, but also can't make an overall summary analysis of scientific application and engineering application. Fundamentally speaking,

They just lack interest.

Even a university math professor can use English, read, write and speak freely without a dictionary.

Few people can comment on mathematics and science in English. Most of them are not in the dictionary.

Blind, with a dictionary or dumb, everywhere. Middle school teachers, generally speaking, are math teachers.

Almost all of them are paralyzed by English, especially in middle schools below the county level. A school can have one math course on average.

It is impossible for teachers to use English freely.

Therefore, the above two categories, the average teacher, who teaches for a lifetime, are doomed not to be involved, because these are also involved.

To methodology, philosophy, logic,,,.

The more advanced journals, the more professional teachers and the more advanced courses, the more definite integral methods are adopted.

The reason is: focusing on major is the key. There is no time, and there is no need to waste time on gestures.

In particular, two-dimensional and three-dimensional problems must be solved by definite integral.

As long as you get into the habit, save time, have clear concepts, concise answers and strong professionalism!

The math teacher's method of indefinite integral can only be used as an introduction, and he will use definite integral later.

To show that you have the ability to solve practical problems. After double integral and triple integral,

Space surface integral, space curve integral,,, must use definite integral.

Conclusion:

1, indefinite integral, only a general conclusion can be drawn, and a specific integral constant must be obtained. This process

It is better to get the integral directly. In fact, the process of determining the integral constant is the idea of definite integral.

It is the method of definite integral, which is reflected in the lower limit of both sides when integrating, and uses the undetermined integral constant

Is the lower limit of the integral. In this regard, the landlord should carefully consider, beginners, it is difficult to understand at the moment.

2, the embodiment of the physical meaning has two aspects:

First: the equation before integration. If this equation is a mathematical identity, then this integral has only mathematics.

Meaning, not physical meaning, is actually just a mathematical game, or a mathematical skill.

It's just a coincidence from integral to integral, just the form of integral, just the improvement of skills.

It is essential only when the two sides are not identities, and this kind of integral must involve specific physical elements.

Principles of science and engineering. Sometimes, in order to simplify the integration, the two sides can be transformed identically and then integrated.

Only in this case, the integral of math teacher's identity is valuable.

So the physical meaning is embodied in the equation before integration, not in the equation.

Second, the determination of the constant after indefinite integral is the determination of the lower limit before definite integral, which is essentially a unified problem.

The determination of the upper and lower limits of definite integral itself is the second embodiment of physical meaning, that is, certain

What is the physical quantity corresponding to an initial moment and what is the physical quantity corresponding to a final moment? such

Correspondence can be time correspondence or boundary correspondence. This kind of problem adds up to

The definite solutions of ordinary differential equations and partial differential equations. Teaching of Ordinary Differential Equations and Partial Differential Equations

Professors, who basically graduated from the Department of Mathematics, have a common defect, that is, they can't master astronomy, geology,

Meteorology, hydrology, oceanography, machinery, electronics, electricity, theoretical physics, theoretical chemistry and other disciplines,

Determining the boundary conditions is their collective Achilles heel, and only a few rare exceptions can be determined.

In elementary mathematics, if you can solve equations, everything will be fine. But in advanced mathematics, solving differential equations,

In particular, partial differential equations can only be solved according to certain solving conditions. For beginners of advanced mathematics,

This is an incredible thing for elementary mathematics learners. Thanks to our teaching, we have cultivated a great.

A group of students who like to talk big, when they put forward new theories, they don't think calmly, but think systematically.

A kind of reflexive refutation, this irrational refutation emotion is not for individual students, it doesn't know the tomb.

How many students were sent to the future. However, our teachers have done many such demonstrations themselves.

Encourage. Therefore, in the fields of modern mathematics, modern science and modern engineering, internationally,

We are all third-rate and have no right to speak. All the theories are imported, and we have nothing.

Whether we know it or not, we will never get tired of it.

Therefore, the embodiment of physical meaning, the second is reflected in the conditions of definite solution. The embodiment of the definite solution condition is

Is the integral definite integral in one go.

Attachment: the English of the definite solution condition is the initial value problem, and the landlord can search it online.

The initial value problem seems to be the "initial value problem" in free translation, and there is nothing wrong with this translation.

Extension translation is the final solution from initial value problem to ordinary differential equation and partial differential equation.

Solution, that is, the problem of determining the final solution, so the significance of the initial value problem itself is also

Is a definite solution condition.