Lecture. Curvilinear integrals
With this lecture, we conclude the section on functions of several variables and conclude it with a new discovery in integral theory. It turns out that integrals may be curved, and they are related to the concept of a function of two variables, and of course they are related to certain integrals. To begin with, we will analyze curvilinear integrals of the first kind. On the plane, a continuous curve of finite length is given, which is rectified, and at each point of this curve a function of two variables is defined, the value of z is calculated by the function f (x,y). What will we do? We divide this curve by points into n parts: M0, M1 and so on, Mn is point B.
We denote deltai is the length of the i-th part. Form the integral sum by multiplying the value of any selected point Mi* in the i-th arc by the length of the i-th arc, which this selected point belongs to. Summing up these values and get sigma. We call this value sigma as the integral sum of the function f on curve AB. The greatest of the lengths of partial arcs will be called partitioning step. There is a limit process if there is a finite limit of the integral sum with partitioning step tending to zero, and this limit is independent of the method of division of curve in the partial arcs or from selecting process of points Mi *, then the function f is called integrable by curve AB, and the value limit is called the curvilinear integral of the first kind of function f along curve AB.
Denote this integral as follows (see the video). We can specify curve AB, sometimes we denote this curve simply by L and the factor is dl. The simplest example: the integral of the differential of curve length of dl. Look integrand here is a unity, all its values at any point (xi, yi) are equal to 1. And what do we get? The sum of all partial lengths of arcs.it is clear that the result is 1, the length of curve L, when the partioning step tends to zero, the limit is equal to l, thus, the differential of curve length in calculating the integral of curve gives us 1.
This is an important fact that we often use. What properties should we know and use when calculating a curvilinear integral of the first kind? The curvilinear integral does not depend on the direction of the integration path. We can write AB, we can write BA, curvilinear integral will not change. The property of linearity, which we considered both for definite, indefinite, and also for double integrals. First, uniformity, a constant factor can be taken as the sign of the integral. Second, additivity, the curvilinear integral of the sum of functions is equal to the sum of the curvilinear integrals of the summands. The second property.
We can split curve L into two parts, such L1 and L2, which have a common end connecting these two arcs, then the integral of curve L is equal to the sum of the integrals of curves L1 and L2. Properties related to inequalities. If the function is non-negative, then the curvilinear integral of curve L is also non-negative, if it is less than or equal to 0, then the integral is also associated with 0 by the same sign of inequality. When we considered applications of a definite integral, there were formulas for calculating the length of curve and the differential of curve is calculated using the following formulas (see the video).
It all depends on how curve is defined, in which coordinates - cartesian, polar, or parametric. It is these formulas that are used in the calculation of the curvilinear integral of the first kind. As we remember, there is a multiplier dl in the notation of the curvilinear integral. Example (see the video). Calculate the curvilinear integral of curve L of the functions y2. L is an arc of a circle, of radius a, centered at the origin, which is located in the first quarter. And so the circle is set parametrically (see the video). When calculating dl, we simply look at the task.
Look, there stands dl, we calculate dl, as it is set parametrically, then we apply the formula and get a. We turn to calculation. Do not forget that calculation is carried out on curve, and the function is defined on curve, then y is a sin (x), we substitute, take out of the brackets a2 and get the integral of the square of the sine, use the formula reducing the power and further it is simple to calculate, you can do it independently and check the answers. We will consider the curvilinear integral of the second kind. Again, a continuous curve is given, it is rectifiable, and let us define at any point of curve the function of two variables, P (x, y). Split curve on a finite number of partial arcs M0 = A points, M1 and so on Mn = B.
On each partial arc again choose a point, let's say that the value of xi is i, but now the situation is the following look at the axis Ox. The projections of the partial arc ends have the coordinates xi-1 and xi, form the difference form the delta x1, form the integral sum as a sum of products of function P values of selected points on the delta xi. Partitioning step is the greatest of the values xi delta. If there is a finite limit of the integral sum, partitioning step tends to zero, depends neither on the partition curve in partial ones nor on selecting points on these partial arcs, this limit is called the curvilinear integral of the second kind of function P on variable x along curve AB.
Remember, we also considered only delta xi and the notation in this case is also related to variable x, which is indicated by the multiplier dx. A completely similar situation occurs when we have the value of a function, in this case let's consider another function Q (x,y) defined again on arc AB, the value of the function of the selected point Mi* we multiply by delta yi, where yi is the projection of the selected point of division Mi on the axis Oy. And so the integral sum will have a slightly different form. Passing to the limit when the partition step tends to zero, we get a new curvilinear integral in exactly the same way, and we say that this is already an integral over the variable y.
And so, a curvilinear integral of the second kind is associated with variables. And the general situation leads to an integral, which is called a curvilinear integral of general or combined form. Its recording on the left side of the equation looks like this (see the video), and it is defined as the sum of two curvilinear integrals of the second kind, each of which is respectively on variable x, on variable y on curve AB. We move on to properties. Note that the curvilinear integral of the second kind, unlike the first ones, depend on the direction on curve, so if we changed curve, it was from A to B, and it became from B to A, then the sign of the integral will change to the opposite.
Therefore, when calculating curvilinear integrals of the second kind, we indicate the direction on curve. The sign of the integral depends on this. If the beginning and end of curve coincide with points A and B, then curve is closed. Let a closed curve be a limit of a certain area (see video). The traversal direction is called positive if, when moving along the contour along this curve, the area bounded by this curve is on the left. In the picture shown, the specified direction is positive and in this case the curvilinear integral is denoted as follows (see in the video) and is called a curvilinear integral over a closed contour. By analogy with integrals of the first kind, we can split curve AB with a point, and then the curvilinear integral is split into the sum of two integrals.
The integral on curve AB is equal to the sum of the same integrals of curve AC and CB. The calculation of all curvilinear integrals is reduced to a certain integral. Let's see an example of this. A curvilinear integral of the general form, where L is the arc of a circle of a unit radius centered at the origin, located in the first quarter (see the video). And let the direction of this curve correspond to the direction of increasing parameter, t equal to zero corresponds to a point on the Ox axis and t equal to π /2 corresponds to a point on the Oy axis of this curve.
What are we doing? We practically do what is written in the integral itself. Don't forget that we are in curvilinear integrals on a curve. And on the curve, x and y are calculated according to the specified formulas, so everywhere instead of x we substitute acos(t), everywhere instead of y, we substitute asin(t). Look, it remains to calculate the differentials of the function of one variable, we still need to change the limits of integration from 0 to π /2, and we get a definite integral of one variable t on the segment from 0 to π / 2. We see the integrand here will be zero, and the integral is zero too.