Summary
In the last semester and this semester we learned differential calculus, understand the limits with the idea to solve the problem.
In combing various points of time, we can increase the geometric dimensions of the relationship (I do not know the exact increase in the dimension of expression is not accurate, but I feel like this, so I wrote the following) or use and quality the idea of integration to sort out, but can not be used due to the triple integral dimension of elevated thoughts to help me understand the problem, so I decided to sort out common points with the idea of seeking quality, integration of the curve arc length of the area surface points.
Then combed by calculation for the integral curve scalar coordinates and coordinates of surface integrals.
Vector
Here we start thinking about the simplest and quality problems, and step by step to find quality problems become complicated, in order to sort out the knowledge we have learned.
In the beginning, we have a rope in a straight line, the mass per unit length of this string is ρ (ρ is a constant), his length L, then obviously, the quality of this rope is ρx, here we do not need use thinking in limit, but if we use here thinking limits, we can put a rope into an infinite number of copies that countless Δx, each Δx of quality ρΔx, then the quality of the entire string is
= ρx
now we have a problem becomes a little more complicated, the linear mass distribution of the rope is no longer uniform and the density of the rope and the rope at each have a certain relationship, i.e., at each of the cord density of f (x), then we can not use the simple multiplication, but if we use the idea of limits, the rope into infinitesimal part Δx, and every quality can be seen as f (x) Δx, then the total mass of the rope for all small portions add up, that is,
So Leibniz put this formula is defined as the (and math teacher and we emphasize that this symbol, the British do not want to use something gay Leibniz and Newton chose uncle, causing them to fall behind a lot of math), At this time, our integration has emerged.
For example, we want to calculate a rope x∈ [0, π] relationship quality, density and length of the rope the rope is: ρ = sin (x), we can be expressed as the mass of the rope:
Image showed the following figure: the arrow points to the area of the region can be seen as the quality of the rope (because I did not learn how to paint shadows, so we had to use arrows instead)
(O (╥﹏╥) o) ...
Here we do not discuss the computational method of integration, we consider only the integral function, and then we will further complicate the issue: If this line is not a straight line it? We also calculated the quality of this rope just thinking of using it? Obviously it can, where we introduced the integration of the curve arc length, AKA, first class curve integral, since this is the curve, so our formula becomes
That is: the mass of the rope (x, y) is f (x, y), then subjected to a first type integral curve, where no clear image quality can help understand solved here, so the drawing is not performed we can also be understood that this embodiment is integral to straighten a curve, so that it becomes a straight line, then the equation f (x, y) may correspond to f (x), the formulas can become ds dx, which is my understanding of ds = method.
Then we consider finished in a straight line and a curve for the quality process, we began to think about seeking quality method area, we still continue just practice, first think about how to find a simple quality plane here because we in the calculation of plane quality , so the use of a double integral:
Of course, here we can also graphically represented it, the requirements for the quality to be understood that the volume is a perspective bottom area D, for example, we require the quality of a surface, the surface area is -3≤x≤3, -3≤ y≤3, density and x, y is the relationship
z = x2 + y2
so we can mass as follows:
The image showed the volume of the object following:
So if this becomes a curved surface of it? So we had this image of the right quality. We should be introduced to the area of surface integrals, AKA, first class surface integral:
Since no surface expression of this type of volume is easy to use way, then we will not be associated with the geometry.
Then we continue to follow the original idea of thinking, when it becomes a three-dimensional graphics, and quality we should use the triple integral, but many go on this:
In the previous process, we use the object of seeking a way to sort out the quality Common integral, the integral of arc length of the curve, the integral of the surface area of the way in the process of triple integral and double integral is referenced. As long as our relationship is given density, and may be found suitable method shows the shape of the object request, then we can use various methods to obtain more than the mass of the object.
