Lecture ¹11

# Lecture ¹11. The curvilinear integrals

Exercises for Lecture #11

The purpose of work:

The acquaintance with the curvilinear integrals and its properties.

Theoretical information and examples

Content

2. 9 Line Integrals

2.9.1 Line Integrals of the first type

2.9.2 Line integrals of the second type

2.9.3. The case of a total differential

2. 9 Line Integrals

2.9.1 Line Integrals of the first type. Let f(x, y) be a continuous function and y = j(x) (a £ x £ b) the equation of some smooth curve C.

Construct a system of points Mi(xi, yi)(i = 0, 1, 2, ··· , n), which subdivide the curve C into elementary arcs

and form the integral sum

The limit of this sum, when n ® ¥ and max Dsi ® 0, is called a line integral of the first type

(ds is the arc differential) and is evaluated from the formula

In the case of the parametric representation of the curve C: we have

One also considers line integrals of the first type of functions of three variables f(x, y, z) taken along a space curve. These integrals are evaluated in a corresponding manner. A line integral of the first type does not depend on the direction of the path of integration; if the integrand f is interpreted as a linear density of integration curve C, then this integral represents the mass of the curve C.

Example 1. Evaluate the line integral

where C is the contour of the triangle ABO with vertices A (1, 0), B(0, 1), C(0, 0) (Fig. 15).

Solution. Here, equation AB is y = l - x, equation OB is x = 0, equation OA is y = 0, whence

2.9.2 Line integrals of the second type. If P(x, y) and Q(x, y) are continuous functions and y = j (x) is a smooth curve C which runs from a to b as x varies, then the corresponding line integral of the second type is expressed as follows:

In the more general case when the curve C is represented parametrically, x=j(t), y=y(t), where t varies from a to b, we have

Similar formulae hold for a line integral of the second type taken over a space curve.

A line integral of the second type changes sign when the direction of the integration path is reversed. This integral may be interpreted in terms of Mechanics as the work of an appropriate variable force {P(x, y), Q(x, y)} along the curve of integration C.

Example 2. Evaluate the line integral

where C is the upper half of the ellipse x = a cos t, y = b sin t, traversed in the clockwise direction.

Solution. We have

2.9.3 The case of a total differential. If the integrand of a line integral of the second type is a total differential of some single-valued function U=U(x, y), i.e., P(x, y)dx+ Q(x, y)dy = dU(x, y), then this line integral does not depend on the path of integration and we have the Newton-Leibnitz formula

(1)

where (x1, y1) is the initial and (x2, y2) the final point of the path. In particular, if the contour of integration C is closed, then

(2)

If 1) the contour is contained entirely within some simply-connected region S and 2) the functions P(x, y) and Q(x,y), together with their partial derivatives of the first order, are continuous in S, then a necessary and sufficient condition for the existence of the function U is the identical satisfaction (in S) of the equality

(Total Differential, Integration).

If Conditions 1) and 2) are not met, the presence of Condition (3) does not guarantee the existence of a single-valued function U and (1) and (2) may prove to be wrong . We will present a method of finding a function U(x, y) from its total differential based on the employment of line integrals (which is yet another method of integration of a total differential). Let the contour of integration C be a broken line P1P2,M (Fig. 16), where P0(x0,y0) is a fixed and M(x, y) a variable point. Then, we have along P0P1 that y = y0 and dy = 0, and along P1M dx =0. We find

Similarly, integrating with respect to P0P2M, we have

Example 3. Find U for

Solution. Let x0 = 0, y0 = 0. Then,

or

where C = U(0, 0) is an arbitrary constant.