Why put the differential here in the definite integral here? It is because differentiation is actually inseparable from integration, and finding the derivative is just a calculation method (the same is true for indefinite integral).
Differentiation is the calculation of straight algebra. The relationship with the derivative requires the geometric properties of its slope so that in a small enough range, the slope of the derivative and the height of the triangle formed by dx can be used to replace dy to obtain an approximation of a small range. Area-the whole uses the basic theorem of calculus (Newton Leibniz formula) to connect the definite integral and the original function of the function.
The definite integral is the sum limit, and the indefinite integral is to find all possible original functions (function family) different in that the definite integral is to find a certain limit value. So although they are all called points, they are not the same thing.
The definition of definite integral is derived from the limit. When investigating the limit of the sum of definite integrals, it is necessary to clearly divide the points that are close to infinity (tested in 2021), so the value of anomalous integral flaws can be obtained with the limit.
The double integral can be obtained by two definite integrals. The difference from finding the area is that the points on each surface are weighted. In our elementary mathematics, the weight is a constant 1, which varies with x and y.
Differential equations are similar to the existence of implicit function derivation (differentiation on both sides at the same time). Due to the large amount of calculation, the general solution is often used to find the undetermined coefficients. There are many transformations that are not understandable in initial mathematics (integration on both sides at the same time) ), and the processing skills of logarithms and arbitrary constants. In addition, dynamic substitution points to find undetermined coefficients should also be mastered.