18. Calculation of triple integral in a case of arbitrary area.

Triple integral taken in arbitrary area can easy be reduced to considered case of parallelepiped area. For proving it’s necessary to consider function f* that coincide with function f in given area and equals to zero in other points of parallelepiped area contained given area. We investigate two partial cases. Let body (V) is bounded by planes x=a and x=b, and each section by plane parallel to this planes represents itself some figure with projection (P_x) onto plane yz. Then (7*)

Another case – let’s body (V) will be cylindrical column bounded by surfaces z₁(x,y) and z₂(x,y) at top and bottom. And these surfaces projects onto plane xy into some figure (D). (V) is bounded by cylindrical surface with elements parallel to z. Then formula (10) takes place. If (D) represents itself curvilinear trapezium bounded by two curves y₁(x) and y₂(x) and by two lines x=a and x=b, then body (V) satisfies to both considered cases and changing double integral by repeated integral in formula (7*) or (10) we have got (11).

19. Change of variables in triple integrals.

Let’s we have space with rectangle system of coordinates xyz and another space with system of coordinates . Suppose that these areas are connected with equations. Then, analogically with plane case, it can be proved that formula(11) takes place. Here jacobian. There are two systems of curvilinear coordinates used for triple integral.

1. Cylindrical coordinates. Its represent itself combination of polar system with ordinary applicate . Here. Jacobian will be

. It’s considerable to use cylindrical coordinates when the integrand or surfaces bounded area (V) contain expression of the kind , i.e. sum of squares of two variables.

2. Spatial polar coordinates (spatial spherical coordinates) This system is connected with Cartesian’s coordinates by formulas , wherehave the same sense as in polar system andmeans angle positive direction of axis z and position vector of point. Here. Let’s calculate jacobian

. This system usually is used when expression of the kind are contained in integrand.

20. Harmonic analysis and periodic values.

We often have to do with periodic values in science and engineering, i.e. such values that repeated in the same kind through definite time interval T called period. Different values connected with considered periodic process return to its last values after a lapse of time T, therefore its represents itself periodic functions of time T, describable by equality .

The simplest of periodic functions (excluding constant) is sinusoidal value: , whereis frequency, connected with period T by ratio(1). More complicated functions may be composed from such simplest ones. It is clear that their components must have different frequencies because adding of sinusoidal values with the same frequencies result to old function with the same frequency. On the contrary, if we add some values of the kind(2) which have (excluding constant) frequenciesdivisible by the least of itand periodsthen we have got periodic function with period T but significantly different from values of the kind (2).Now it’s naturally to ask reverse question – may we represent given periodic function with period T as sum of finite or infinite set of sinusoidal values of the kind (2)? As we shall see below, there is positive answer for wide class of functions but only if we use just infinite set of values of the kind (2). For functions of this class expanding in “trigonometric” series(3)

takes place, where are constant having particular meanings for any function and frequency described by formula (1). Geometrically it means that periodic function graph come in by superposition of sinusoids. But if we consider every sinusoidal value in mechanical sense, as representing harmonic oscillation then we may say that functionis complicated oscillation expanding in separate harmonic oscillations. Because that separate sinusoidal values in expanding (3) are called harmonics of function(first harmonic, second harmonic and so on). And the process of periodic function expanding in harmonics is called harmonic analysis. If we takeas independent variable then we got function of x:, it is periodic too, but with standard period. Then expanding (3) will take form

(4)

Using formula for sines’ sum and taking we have got final form of trigonometric expanding

(5).