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Condensed versions of the full Schaum's Outlines called "Easy Outlines" started to appear in the late s, aimed primarily at high-school students, especially those taking AP courses. These typically feature the same explanatory material as their full-size counterparts, sometimes edited to omit advanced topics, but contain greatly reduced sets of worked examples and usually lack any supplementary exercises. As a result, they are less suited to self-study for those learning a subject for the first time, unless they are used alongside a standard textbook or other resource.

Schaum's Outlines are part of the educational supplements niche of book publishing. They are a staple [ citation needed ] in the educational sections of retail bookstores, where books on subjects such as chemistry and calculus may be found. Many titles on advanced topics are also available, such as complex variables and topology , but these may be harder to find in retail stores. The "Demystified" series is introductory in nature, for middle and high school students, favoring more in-depth coverage of introductory material at the expense of fewer topics. The "Easy Way" series is a middle ground: more rigorous and detailed than the "Demystified" books, but not as rigorous and terse as the Schaum's series.

Schaum's originally occupied the niche of college supplements, and the titles tend to be more advanced and rigorous. The outline format makes explanations more terse than any other supplement. Schaum's has a much wider range of titles than any other series, including even some graduate-level titles.

From Wikipedia, the free encyclopedia. McGraw-Hill Professional. Archived from the original on 6 May Solutions to these can be of great value to the scientist and engineer. In a mathematical formulation of a physical problem, a mathematical model is chosen which often approximates the actual situation. For example, in treating rotation of the earth around the sun, we may consider that the sun and earth are points. If a mathematical model and corresponding mathematical formulation leads to fairly good agreement with that predicted by observation or experiment, then the model is good.

Otherwise a new model may have to be chosen. The following table lists some special applications of an elementary nature which arise in science and engineering. The quantity mv is often called the momentum. If m is constant, then this equation becomes where a is the acceleration. Various systems of units are available. The current I measured in amperes is the instantaneous time rate of change of charge Q on the condenser measured in coulombs, i.

For single loop circuits law a implies that the current is the same throughout the loop. A curve cutting each member of a one parameter family of curves at right angles is called an orthogonal trajectory of the family. A horizontal beam situated on the x axis of an xy coordinate system and supported in various ways, bends under the influence of vertical loads. The deflection curve of the beam, often called the elastic curve and shown dashed in Fig. Various problems of science and engineering pertaining to temperature, heat flow, chemistry, radioactivity, etc.

In such case various methods are available for obtaining an approximate or numerical solution. In the following we list several such methods. Step by step or Euler method. In this method we replace the differential equation of 8 by the approximation 9 so that. We choose h sufficiently small so as to obtain good approximations.

A modified procedure of this method can also be used. See Problem 2. Taylor series method. By successive differentiation of the differential equation in 8 we can find y' x0 , y" x0 , y'" x0 , Then the solution is given by the Taylor series Picard's method. The limit of this sequence, if it exists, is the required solution. However, by carrying out the procedure a few times, good approximations can often be obtained. Runge-Kutta method. These methods can also be adapted for higher order differential equations by writing them as several first order equations.

Classify each of the following differential equations by stating the order, the dependent and independent variables, and whether the equation is ordinary or partial. Check whether each differential equation has the indicated solution. Which solutions are general solutions? Since the number of arbitrary constants one equals the order of the differential equation one , it is a general solution.

However, it is not a general solution since the number of arbitrary constants one is not equal to the order of the equation two. Determine the particular solution of the differential equation of Problem 2. From Problem 2. We have. Thus the given relation is a solution to the boundary-value problem. Referring to Fig. The parabola is the envelope of the family of straight lines.

Schaums Outline of Advanced Mathematics for Engineers & Scientists

Through each point of the xy plane there passes one and only one member of the family. Note that if we integrate the differential equation twice, we have so that the solution represents a two parameter family of curves. The difficulty can be avoided on noting that actually x. Since 8 - Sy is negative, the result can then be written as. W where F y is the constant of integration which may depend on y. Then from 2 ,. T tion is exact.

Only the first leads to favorable results, i. Multiplying by this factor, the equation becomes. As in Problem 2. Similar reasoning analogous to that of Problem 2. By eliminating p from 2 we can obtain the general solution in the form.. However, it cannot be obtained from the general solution 3 by any choice of c. Neglecting air resistance, a the maximum height reached and b the total time taken to return to the starting point.

Choose the upward direction as positive. Thus the maximum height reached is cm. Choose the positive direction downward. Solve Problem 2. Net force. Then 2 which is the velocity at any time. A boat of mass m is traveling with velocity v0. At t — 0 the power is shut off. Assuming water resistance proportional to vn where n is a constant and v is the instantaneous velocity, find v as a function of distance traveled. A uniform chain of length a is placed on a horizontal frictionless table so that a length b of the chain dangles over the side.

How long will it take for the chain to slide off the table? Suppose that at time t a length x of the chain is dangling over the side [Fig. Assume that the density mass per unit length of the chain is a. Then Fig. An inductor of L henries and a condenser of C farads are connected in series [Fig.

Potential drop across Potential drop across Then Since. Since the slope of each member of the orthogonal family must be the negative reciprocal of this slope, we see that the slope of the orthogonal family is. To solve 1 differentiate both sides with respect to x, so that. The curve 3 which is a hypocycloid [Fig. A cylindrical tank has 40 gallons gal of a salt solution containing 2 Ib dissolved salt per gallon, i. Rate of change of amount of salt. A right circular cone [Fig. In what time will the water empty through an orifice O of cross-sectional area a at the vertex?

We have Change in volume of water z. Radium decays at a rate proportional to the instantaneous amount present at any time. The half life is the time T when the amount present is half the original amount, i. Chemical A dissolves in solution at a rate proportional to the instantaneous amount of undissolved chemical and to the difference in concentration between the actual solution Co and saturated solution Cs. A porous inert solid containing 10 Ib of A is agitated with gallons water and after an hour 4 Ib of A is dissolved. If a saturated solution contains. Let x Ib of A be undissolved after t hours.

Newton's law of cooling states that the time rate of change in temperature of an object varies as the difference in temperature between object and surroundings. A beam of length L is simply supported at both ends [Fig. Let x be the distance from the left end A of the beam. To find the bending moment M at x, consider forces to the left of x. A cantilever beam [Fig.

Find a the deflection and 6 maximum deflection of the beam assuming its weight to be negligible. To obtain the entries in the second line use h —. Better accuracy can be obtained by using smaller values of h or proceeding as in Problem 2. Show how to improve the accuracy of the method of Problem 2.

The method which we shall use is essentially the same as that of Problem 2. The first line in the table below is the same as the first line in the table of Problem 2. Similarly the entries corresponding to y1 and y[ in line two of the table below are the same as those in line two of the table of Problem 2. We now refer only to the table given below.

Since this agrees with 4 , the process ends. Continuing in this manner we finally obtain y. This agrees closely with the true value 1. We have on integrating the differential equation using the boundary condition, V x. Better approximations can be obtained by using two or more applications of the method with smaller values of h. The equation can be solved exactly and the solution is y.

Work Problem 2. Classify each of the following differential equations by stating the order, the dependent and independent variables and whether the equation is ordinary or partial. Find the differential equation for a the family of straight lines which intersect at the point 2,1 and b the family of circles tangent to the x axis and having unit radius. Each of the following differential equations has an integrating factor depending on only one variable.

Find the integrating factor and solve the equation. Solve each of the following, determining any singular solutions. An object moves along the x axis, acted upon by a constant force. A 64 Ib object falls from rest. An electric circuit contains an 8 ohm resistor in series with an inductor of. An electric circuit contains a 20 ohm resistor in series with a capacitor of. At t — 0 there is no charge on the capacitor. A curve passing through 1, 2 has the property that the length of the perpendicular drawn from the origin to the normal at any point of the curve is always equal numerically to the ordinate of the point.

Find its equation. The tangent to any point of a certain curve forms with the coordinate axes a triangle having constant area A. Find the equation of the curve. A tank contains gallons of water. A salt solution containing 2 Ib of salt per gallon flows in at the rate of 3 gallons per minute and the well-stirred mixture flows out at the same rate, a How much salt is in the tank at any time?

In Problem 2. A right circular cylinder of radius 8 ft and height 16 ft whose axis is vertical is filled with water. The rate at which bacteria multiply is proportional to the instantaneous number present. After 2 days, 10 grams of a radioactive chemical is present. Three days later 5 grams is present. How much of the chemical was present initially assuming the rate of disintegration is proportional to the instantaneous amount which is present?

Chemical A is transformed into chemical B at a rate proportional to the instantaneous amount of A which is untransformed. At P. Assume Newton's law of cooling. A beam of length L ft and negligible weight is simply supported at the ends and has a concentrated load W Ib at the center. Find a the deflection and 6 the maximum deflection.

Use the step-by-step or Euler method to solve numerically each of the following. U, ind. V order 2, dep. T, ind. If R x , the right side of J , is replaced by zero the resulting equation is called the complementary, reduced or homogeneous equation. Example 3. Example 4. Then we have the following important theorem, sometimes referred to as the superposition principle or theorem. The general solution of 4 is obtained by adding the complementary solution Yc x to a particular solution Yp x of 4 , i.

Because of this theorem it is clear that we shall have to consider separately the problems of finding general solutions of homogeneous equations and particular solutions of nonhomogeneous equations. A set of n functions yi x , yz x , Otherwise the set of functions is said to be linearly independent. This last theorem is important in connection with solutions of the homogeneous or reduced equation as seen in the following Theorem Particular simplifications occur when the equation has constant coefficients and we now turn to this case.

Two general procedures are available in this case, namely those which do not involve operator techniques and those which do. For each case methods exist for finding complementary and particular solutions and use is then made of the fundamental Theorem Three cases must be considered. Case 1. Roots all real and distinct. Some roots are complex. If ao, 01, Case 3. Some roots are repeated. Method of Undetermined Coefficients. In this method we assume a trial solution containing unknown constants indicated by a, b, c, The trial solution to be assumed in each case depends on the special form of R x and is shown in the following table.

The above method holds in case no term in the assumed trial solution appears in the complementary solution.

If any term of the assumed trial solution does appear in the complementary solution, we must multiply this trial solution by the smallest positive integral power of x which is large enough so that none of the terms which are then present appear in the complementary solution. Method of Variation of Parameters. Since, to determine these n functions, we must impose n restrictions on them and since one of these is that the differential equation be satisfied, it follows that the remaining n -1 may be taken at will.

From these Ki,Kz,.. The method is applicable whenever the complementary solution can be found, including cases where oo, This is not true if 00,0,1,.. The constants mi, To obtain particular solutions, the following operator methods will be found useful.

Method of Reduction of Order. By continuing in this manner, y can be obtained. This method yields the general solution if all arbitrary constants are kept, while if arbitrary constants are omitted it yields a particular solution. Method of Inverse Operators. In using these we often employ the theorem that. Table of Inverse Operator Techniques. This can also be evaluated by expanding the inverse operator into partial fractions and then using entry A.

In the following we list a few important methods. Cauchy or Euler Equation.


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Case where one solution is known. If n - 2, the equation can then be solved exactly. See Problem 3. Reduction to canonical form. See Problems 3. By solving for p and the other constants, a series solution can often be obtained. The series is called a Frobenius series and the method is often called the method of Frobenius. Solutions obtained should be checked by substitution into the original differential equations to insure that the proper number of arbitrary constants are present.

In general, however, the commutative law for multiplication does not hold for operators with nonconstant coefficients, as seen in Problem 3. In a similar way we can show that D2, D3, Prove that if yi,yz, Then using Problem 3. Prove Theorem , page Show that the functions cos 2x, sin2 x, cos2 x are linearly dependent. Suppose the contrary, i. Then there are n constants c l Prove that if the functions yi, Consider the system of equations. Since the Wronskian is zero we see that this system has a solution c,, Since ylt This problem and Problem 3. Use Problem 3. From 2 of Problem 3.

There are two methods which can be used to get the general solution. Method 1. This method, called the method of reduction of order, illustrates a general procedure for repeated roots. Then the general solution is. Solve Problem 3. However, some of these terms appear in the complementary solution. Similarly, corresponding to Sex we would normally assume a trial solution dex. But since this term as well as dxex and dx2ex are in the complementary solution, we must use as trial solution dx3ex. Then we assume the general solution to be. However, one of these conditions is that the differential equation must be satisfied.

Thus we are at liberty to impose arbitrarily the second condition. Then we are led to the following equations for determining the functions Klt K2, Ks in the general solution. Since we are interested only in particular solutions,. Evaluate We can show by a method similar to that in Problem 3. Solve given that y — x is a solution. Comparing with Problem 3. This is typical in general when the roots of the indicial equation differ by any integer except zero. In other equations both cases would have to be considered, each leading to a series solution.

The general solution would then be obtained from these series on multiplying each by an arbitrary constant and adding. Thus there is a relationship between the constants Cj, e2, cs, c4 and cs, ce, c7, cg. To determine this relationship we must substitute the values of x and y obtained above in the original equations. To solve the resulting equations, we can then use the method of undetermined coefficients. A particle P of mass 2 gm moves on the x axis attracted toward origin O with a force numerically equal to Sx.

Thus by Newton's law, and. The graph of the motion is shown in Fig. The amplitude [maximum displacement from 0] is 10 cm.

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The period [time for a complete cycle] is ir sec. The motion is often called simple harmonic motion. Thus by Newton's law, or 2. The motion is non-oscillatory. The particle approaches 0 but never reaches it [see Fig. A 20 lb weight suspended from the end of a vertical spring stretches it 6 inches. Find the period and amplitude in each case. Let A and B [Fig. B is called the equilibrium position. Call y the displacement of W at any position C from the equilibrium position. Assume that y is positive in the downward direction. By Hooke's law, 20 lb stretches the spring.

Thus when W is at C, the tension in the spring is This illustrates the phenomenon of resonance and shows what can happen when the frequency of the applied force is equal to the natural frequency of the system. A rod AOB [Fig. A particle P of mass m is constrained to move along the rod. Assuming no frictional forces, find a a differential equation of motion of P, b the position of P at any time and c the condition under which P describes simple harmonic motion.

An inductor of 2 henries, resistor of 16 ohms and capacitor of. The first term is the steady-state current and the second, which becomes negligible as time increases, is called the transient current. Given the electric network of Fig. Find the currents in the various branches if the initial currents are zero. In traversing these loops we consider voltage drops as positive when we go against the current. A voltage rise is considered as the negative of a voltage drop. This is equivalent to Kirchhoff's first law. Illustrate by an example. Work Problem 3.

A particle moves on the x axis, attracted toward the origin 0 with a force proportional to its instantaneous distance from O. If the particle starts from rest at x — 0, find a a; as a function of t, b the amplitude, period and frequency after a long time. A 60 Ib weight hung on a vertical spring stretches it 2 inches. The weight is then pulled down 4 inches and released, a Find the position of the weight at any time if a damping force numerically equal to 15 times the instantaneous velocity is acting.

The charge on the capacitor in the network of Fig. In this chapter we assume that s is real. Later [Chapter 14] we shall find it convenient to take s as complex. Often, however, we shall omit this range of existence since in most instances it can easily be supplied when needed. For our present purposes, however, we need only the following properties: 4 The first is called a recursion formula for the gamma function.

Piecewise continuity. Another way of stating this is to say that a piecewise continuous function is one that has only a finite number of finite discontinuities. An example of a piecewise continuous function is shown in Fig. Exponential order. Using these we have the following theorem, Theorem It should be emphasized that these conditions are only sufficient [and not necessary], i. An interesting theorem which is related to Theorem is the following Theorem While it is clear that whenever a Laplace transform exists it is unique, the same is not true for an inverse Laplace transform.

It follows that. We can show that if two functions have the same Laplace transform, then they cannot differ from each other on any interval of positive length no matter how small. This is sometimes called Lerch's theorem. The theorem implies that if two functions have the same Laplace transform, then they are for all practical purposes the same and so in practice we can take the inverse Laplace transform as essentially unique.

In particular if two continuous functions have the same Laplace transform, they must be identical. The following theorems are fundamental. Then This can be extended as follows. Suppose also that f t ,f' t , It is possible to express various discontinuous functions in terms of the unit step function. In the following we shall consider some of the important results involving Laplace transforms and corresponding inverse Laplace transforms. In all cases we assume that f t satisfies the conditions of Theorem Similarly we can prove that it is associative and distributive [see Problem 4.

Although the above theorems are often useful in finding inverse Laplace transforms, perhaps the most important single elementary method for our purposes is the method of partial fractions. For illustrations of the method see Problems 4. The method of Laplace transforms is particularly useful for solving linear differential equations with constant coefficients and associated initial conditions.

To accomplish this we take the Laplace transform of the given differential equation [or equations in the case of a system], making use of the initial conditions. This leads to an algebraic equation [or system of algebraic equations] in the Laplace transform of the required solution. By solving for this Laplace transform and then taking the inverse, the required solution is obtained.

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For illustrations see Problems 4. Since formulation of many physical problems leads to linear differential equations with initial conditions, the Laplace transform method is particularly suited for obtaining their solutions. For applications to various fields see Problems 4. There exists a direct method for finding inverse Laplace transforms, called the complex inversion formula.

This makes use of the theory of complex variables and is considered in Chapter We must show that if c1 e2 are any constants and f i t , f 2 t any functions whose Laplace transforms exist, then We have. Since this integral does not converge, the Laplace transform does not exist. In parts a , b , c we have omitted the range of existence which can easily be supplied.

Prove a Theorem , and 6 Theorem , page Using this and the result 1 , we have. Thus by Problem 4. We have, since jf is a linear operator [Problem 4. Prove Theorem , page , for a n - 1, 6 any positive integer n. The given equation is called an integral equation since the unknown function occurs under the integral. Another method. This method affords some simplification of procedure. Then Since.

Solve Taking the Laplace transform of the given differential equation and using the initial conditions, which can be written. A mass m [Fig. An external force F t acts on the mass as well as a resistive force proportional to the instantaneous velocity. Then by Newton's law,. From equations 1 and 2 of Problem 3. Find the Laplace transforms of each of the following:. Solve a Problem 3. Use Laplace transforms to find the charge and current at any time in a series circuit having an inductance L, capacitance C, resistance R and e.

Treat all cases assuming that the initial charge and current are zero. There are quantities in physics characterized by both magnitude and direction, such as displacement, velocity, force and acceleration. To describe such quantities, we introduce the concept of a vector as a directed line segment PQ from one point P called the initial point to another point Q called the terminal point.

The magnitude or Fig. Other quantities in physics are characterized by magnitude only, such as mass, length and temperature. Such quantities are often called scalars to distinguish them from vectors, but it must be emphasized that apart from units such as feet, degrees, etc. We can thus denote them by ordinary letters as usual.

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The operations of addition, subtraction and multiplication familiar in the algebra of numbers are, with suitable definition, capable of extension to an algebra of vectors. The following definitions are fundamental. Two vectors A and B are equal if they have the same magnitude and direction regardless of their initial points. The sum or resultant of vectors A and B of Fig. The definition here is equivalent to the parallelogram law for vector addition as indicated in Fig. Extensions to sums of more than two vectors are immediate.

For example, Fig. This has a magnitude of zero but its direction is not defined. Note that in these laws only multiplication of a vector by one or more scalars is defined. On pages and we define products of vectors. We use right-handed rectangular coordinate systems unless otherwise specified. In general,. Let Ai, A2, A3 be the rectangular coordinates of the terminal point of vector A with initial point at O. The vectors Aii, A 2 j and Ask are called the rectangular component vectors, or simply component vectors, of A in the x, y and z directions respectively.

The following laws are valid: 1. The magnitude of A x B is defined as the product of the magnitudes of A and B and the sine of the angle between them. In Symbols. Bz B8 6. Ci C2 Cs 3. The product A x B x C is called the vector triple product. However, parentheses must be used in A x B x C see Problem 5. This is often expressed by stating that in a scalar triple product the dot and the cross can be interchanged without affecting the result [see Problem 5.

The function concept is easily extended. We sometimes say that a vector function A x, y, z defines a vector field since it associates a vector with each point of a region. The following statements show the analogy which exists. This is equivalent to the statement lim A u — A uo. The derivative of A u is defined as 7. Derivatives of products obey rules similar to those for scalar functions. However, when cross products are involved the order may be important.

Some examples are:. If r is the vector joining the origin O of a coordinate system and the point x,y,z , then specification of the vector function r u defines x, y and z as functions of u. As u changes, the terminal point of r describes a space curve see e Fig. If the parameter u is the arc length s measured from some fixed point on the curve, then 9. These concepts have important applications in mechanics. The divergence of A is defined by div A. A point P in Fig. We call ui,uz,u3 the curvilinear coordinates of the point. From 18 , we have If ei, 62, e3 are mutually perpendicular at any point P, the curvilinear coordinates are called orthogonal.

It is clear that when the Jacobian is identically zero there is no parallelepiped. In such case there is a functional relationship between x, y and z, i. See Fig. Note that corresponding results can be obtained for polar coordinates in the plane by omitting z dependence. Show that the addition of vectors is associative, i. Extensions of the results of Problems 5. Prove that the line joining the midpoint of two sides of a triangle is parallel to the third side and has half its length.

From Fig. Determine the vector having initial point P xi,yi,zi and terminal point Q xz,yz,Zi and find its magnitude. The position The position r! Prove that the projection of A on B is equal to A-b, where b is a unit vector in the direction of B. Through the initial and terminal points of A pass planes perpendicular to B at G and H respectively as in the adjacent Fig.

Let a be a unit vector in the direction of A; then [see Fig. By Problem 5. The ordinary laws of algebra are valid for dot products. Then D has the same magnitude as C but is opposite in direction, i. Multiplying by —1, using Problem 5. Note that the order of factors in cross products is important. The usual laws of algebra apply only if proper order is maintained. Prove that the area of a parallelogram with sides A and B is A x B. Let n be a unit normal to parallelogram I, having the direction of B X C, and let h be the height of the terminal point of A above the parallelogram I.

Find an equation for the plane passing through Pi,Pz and Pa. We assume that P1 P2 and P3 do not lie in the same straight line; hence they determine a plane. Find an equation for the plane passing through the points Pi 3,1, — 2 , P2 -l,2,4 , P, 2,-l,l. The quantities p and K are respectively the radius of curvature and curvature of the space curve.

But is perpendicular to ar and therefore to the surface. In rectangular form this is 6 If r is the vector drawn from O in Pig. In rectangular form this becomes.

If F x, y, z is defined at any point on a space curve C and s is the arclength to x, y, z from some given point on C, show that. Find ds2 in a cylindrical and 6 spherical coordinates and determine the scale factors. The volume element in orthogonal curvilinear coordinates MI uz, ua is dV. Then a.

The two equations in general define the dependent variables u and v as implicit functions of the independent variables x and y. Hence we obtain! Note that it is possible to devise mnemonic rules for writing at once the required partial derivatives in terms of Jacobians. A man travels 25 miles northeast, 15 miles due east and 10 miles due south. By using an appropriate scale determine graphically a how far and 6 in what direction he is from his starting position. Is it possible to determine the answer analytically? Prove that the medians of a triangle intersect at a point which is a trisection point of each median.

A triangle has vertices at A Z, 3,1 , B —l, 1,2 , C l, —2,3. Find a the length of the median drawn from B to side AC and 6 the acute angle which this median makes with side BC. Find the shortest distance from the point 3,2,1 to the plane determined by 1,1,0 , 3, —1,1 , -1,0,2. Find the magnitude of the a velocity and 6 acceleration at any time t.

Prove that. These are called the Frenet-Serret formulas. If U, V, A, B have continuous partial derivatives prove that: a 5. Let E and H be two vectors assumed to have continuous partial derivatives of second order at least with respect to position and time. Suppose further that E and H satisfy the equations U. Equations 1 are a special case of Maxwell's equations.

The result 2 led Maxwell to the conclusion that light was an electromagnetic phenomena. The constant c is the velocity of light. These results are useful in thermodynamics where P, V, T correspond to pressure, volume and temperature of a physical system. If this limit exists it is denoted by. Then 3 can be written 4. The result 4 indicates how a double integral can be evaluated by expressing it in terms of two single integrals called iterated integrals. If the double integral exists, 4 and 5 will in general yield the same value. In writing a double integral, either of the forms 4 or 5 , whichever is appropriate, may be used.

We call one form an interchange of the order of integration with respect to the other form. The above results are easily generalized to closed regions in three dimensions. For example, consider a function F x,y,z defined in a closed three dimensional region "tR.. If this limit exists we denote it by 7.

The integration can also be performed in any other order to give an equivalent result. Extensions to higher dimensions are also possible. We then have. The results 9 and 11 correspond to change of variables for double and triple integrals. Generalizations to higher dimensions are easily made. Let P x,y and Q x,y be singlevalued functions defined at all points of C. Subdivide C into n parts by choosing n — 1 points on it given by xi,yi , x2,y2 ,.. Form the sum n. The limit does exist if P and Q are continuous or piecewise continuous at all points of C.

In an exactly analogous manner one may define a line integral along a curve C in three dimensional space as Other types of line integrals, depending on particular curves, can be defined. For example, we can express the line integral 15 in the form If at each point x, y, z we associate a force F acting on an object i. Combinations of the above methods may be used in the evaluation. Similar methods are used for evaluating line integrals along space curves. For example:. If a plane region has the property that any closed curve in it can be continuously shrunk to a point without leaving the region, then the region is called simply-connected, otherwise it is called multiply-connected [see Problem 6.

As the parameter t varies from ti to tz, the plane curve is described in a certain sense or direction. For curves in the xy plane, we arbitrarily describe this direction as positive or negative according as a person traversing the curve in this direction with his head pointing in the positive z direction has the region enclosed by the curve always toward his left or right respectively.

If we look down upon a simple closed curve in the xy plane, this amounts to saying that traversal of the curve in the counterclockwise direction is taken as positive while traversal in the clockwise direction is taken as negative. This theorem is also true for regions bounded by two or more closed curves i.

See Problem 6. The results in Theorem can be extended to line integrals in space. Thus we have Theorem A necessary a n d sufficient condition f. The results can be expressed concisely in terms of vectors. If A represents a force field F which acts on an object, the result is equivalent to the statement that the work done in moving the object from one point to another is independent of the path joining the two points if and only if Such a force field is often called conservative.

Form the sum The results 33 or 34 can be used to evaluate In the above we have assumed that S is such that any line parallel to the z axis intersects S in only one point. In case S is not of this type, we can usually subdivide S into surfaces Si,S2, Then the surface integral over S is defined as the sum of the surface integrals over Si, S 2 , In some cases it is better to project S on to the yz or xz planes.

For such cases 30 can be evaluated by appropriately modifying 33 and Let S be a closed surface bounding a region of volume V. Choose the outward drawn normal to the surface as the -positive normal and assume that a, p, y are the angles which this normal makes with the positive x, y and z axes respectively. Then if At, A2 and A3 are continuous and have continuous partial derivatives in the region In words this theorem, called the divergence theorem or Green's theorem in space, states that the surface integral of the normal component of a vector A taken over a closed surface is equal to the integral of the divergence of A taken over the volume enclosed by the surface.

Let S be an open, two-sided surface bounded by a closed non-intersecting curve C simple closed curve. Consider a directed line normal to S as positive if it is on one side of S, and negative if it is on the other side of S. The choice of which side is positive is arbitrary but should be decided upon in advance.

Then if Ai,A2, A3 are single-valued, continuous, and have continuous first partial derivatives in a region of space including S, we have Note that if, as a special case the result The integration with respect to y keeping x constant from y — 1 to y — x2 corresponds formally to summing in a vertical column see Fig. Although the above integration has been accomplished in the order z, y, x, any other order is clearly possible and the final answer should be the same.

Note that the value for y could have been predicted because of symmetry. Justify equation 21 , page , for changing variables in a double integral. An investigation reveals that this limit is. The vector r from the origin O to point P is.

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The transformation equations in cylindrical coordinates are The Jacobian of the transformation is. The volume is most easily found by using cylindrical coordinates. The integration can also be performed in other orders to yield the same result. Then the integral over this part of the path is. Find the work done in moving a particle once around an ellipse C in the xy plane, if the ellipse has center at the origin with semi-major and semi-minor axes 4' and 3 respectively, as indicated in Fig.

Then the line integral equals. In traversing C we have chosen the counterclockwise direction indicated in Fig. We call this the positive direction, or say that C has been traversed in the positive sense. If C were traversed in the clockwise negative direction the value of the integral would be — 96ir.

Prove Green's theorem in the plane if C is a closed curve which has the property that any straight line parallel to the coordinate axes cuts C in at most two points. The positive direction in traversing C is as shown in Pig. Extend the proof of Green's theorem in the plane given in Problem 6. Consider a closed curve C such as shown in the adjoining Fig. A region which is not simply-connected is called multiply-connected. We have shown here that Green's theorem in the plane applies to simply-connected regions bounded by closed curves.

In Problem 6. For more complicated simply-connected regions it may be necessary to construct more lines, such as ST, to establish the theorem. It is seen that the positive directions are those indicated in the adjoining figure. In order to establish the theorem, construct a line, such as AD, called a cross-cut, connecting the exterior and interior boundaries.

Prove that a necessary and sufficient condition that. If r is the boundary of T, then by Green's theorem. Let P and Q be defined as in Problem 6. Then the required integral equals. Note that in this evaluation the arbitrary constant c can be omitted. If y is the angle between the normal line to any point x, y, z of a surface S and the positive z axis, prove that secy. Give a physical interpretation in each case. The required integral is equal to U. Physically this could represent the surface area of S, or the mass of 5 assuming unit density.

Find the surface area of a hemisphere of radius a cut off by a cylinder having this radius as diameter. Equations for the hemisphere and cylinder see Fig. Two methods of evaluation are possible. Using polar coordinates. Note that the above integrals are improper and should actually be treated by appropriate limiting procedures. The numerator and denominator can be obtained from the results of Problem 6.