Schaum differential geometry pdf

Elementary manifolds. Tensor algebra. Applications of tensors to the equations of surface theory. Isometric mappings. Intrinsic geometry. Geodesic curvature. Geodesic coordinates. Geodesic polar coordinates.

Arcs of minimum length. Sur- faces with constant Gaussian curvature. Gauss-Bonnet theorem. Properties of curves and surfaces which depend only upon points close to a particular point of the figure are called local properties.

The study of local properties is called dif- ferential geometry in the small. Those properties which involve the entire geometric figure are called global properties. The study of global properties, in particular as they relate to local properties, is called differential geometry in the large.

Example 1. The radius of C is the radius of curvature of T at P. The radius of curvature is an example of a local property of the curve, for it depends only on the points on T near P. One-sidedness is an example of a global property of a figure, for it depends on the nature of the entire surface.

Observe that a small part of the surface surrounding an arbitrary point P is a regular two-sided surface, i. We first investigate local properties of curves and surfaces and then apply the results to problems of differential geometry in the large. The Dude. Sergio Lopez. Elias Costa Grivoyannis.

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Tania Majumder. Deepak Avarani. Daniel Laurence Salazar Itable. We are led, therefore, to the modified expression We now take yp to be the sum of 1 and 2 : Substituting 3 into the differential equation and simplifying, we obtain Equating coefficients of like terms, we have from which Equation 3 then gives and the general solution is Supplementary Problems In Problems In Problems Recall from Theorem 8.

This is permissible because we are seeking only one particular solution. This means that the system It is therefore more powerful than the method of undetermined coefficients, which is restricted to linear differential equations with constant coefficients and particular forms of j x.

Nonetheless, in those cases where both methods are applicable, the method of undetermined coefficients is usually the more efficient and, hence, preferable. As a practical matter, the integration of v' x may be impossible to perform. In such an event, other methods in particular, numerical techniques must be employed. This is a third-order equation with see Chapter 10 ; it follows from Eq.

Solve This is a third-order equation with see Chapter 10 ; it follows from Eq. Thus, Substituting these values into 1 , we obtain The general solution is therefore, This is a second-order equation for x t with It follows from Eq. The general solution is Solve In? We first write the differential equation in standard form, with unity as the coefficient of the highest derivative. The one exception is when ihc general solution is ihe homogeneous solution; lhai is, when the dilTerenlial equation under eon si derail on is iise.

The jicneral solution of the differential equation is given in Problem 1 I. I as Therefore. Solve The general solution of the differential equation is given in Problem Solve From Problem The solution to the initial-value problem is The system is in its equilibrium position when it is at rest.

The mass is set in motion by one or more of the following means: displacing the mass from its equilibrium position, providing it with an initial velocity, or subjecting it to an external force F i. A steel ball weighing Ib is suspended from a spring, whereupon the spring is stretched 2 ft from its natural length.

The applied force responsible for the 2-ft displacement is the weight of the ball, Ib. For convenience, we choose the downward direction as the positive direction and take the origin to be the center of gravity of the mass in the equilibrium position. We assume that the mass of the spring is negligible and can be neglected and that air resistance, when present, is proportional to the velocity of the mass. Note that the restoring force Fs always acts in a direction that will tend to return the system to the equilibrium position: if the mass is below the equilibrium position, then x is positive and -kx is negative; whereas if the mass is above the equilibrium position, then x is negative and -kx is positive.

The force of gravity does not explicitly appear in We automatically compensated for this force by measuring distance from the equilibrium position of the spring. If one wishes to exhibit gravity explicitly, then distance must be measured from the bottom end of the natural length of the spring.

The current 7 flowing through the circuit is measured in amperes and the charge q on the capacitor is measured in coulombs. Kirchhojfs loop law: The algebraic sum of the voltage drops in a simple closed electric circuit is zero. It is known that the voltage drops across a resistor, a capacitor, and an inductor are respectively RI, HC q, and L dlldt where q is the charge on the capacitor. The voltage drop across an emf is —E t.

Thus, from Kirchhoff s loop law, we have The relationship between q and 7 is Substituting these values into The second initial condition is obtained from Eq. Thus, An expression for the current can be gotten either by solving Eq. See Problems Such a body experiences two forces, a downward force due to gravity and a counter force governed by: Archimedes' principle: A body in liquid experiences a buoyant upward force equal to the weight of the liquid displaced by that body.

Equilibrium occurs when the buoyant force of the displaced liquid equals the force of gravity on the body.

Figure depicts the situation for a cylinder of radius r and height 77 where h units of cylinder height are submerged at equilibrium. At equilibrium, the volume of water displaced by the cylinder is 7tr2h, which provides a buoyant force of 7tr2hp that must equal the weight of the cylinder mg.

If the cylinder is raised out of the water by x t units, as shown in Fig. The downward or negative force on such a body remains mg but the buoyant or positive force is reduced to Jtr2[h - x t ]p.

It now follows from Newton's second law that Substituting For buoyancy problems defined by Eq. For electrical circuit problems, the independent variable x is replaced either by q in Eq. For damped motion, there are three separate cases to consider, depending on whether the roots of the associated characteristic equation see Chapter 9 are 1 real and distinct, 2 equal, or 3 complex conjugate. These cases are respectively classified as 1 overdamped, 2 critically damped, and 3 oscillatory damped or, in electrical problems, underdamped.

A steady-state motion or current is one that is not transient and does not become unbounded. Free undamped motion defined by Eq. Here c1, c2, and ft are constants with ft often referred to as circular frequency. The natural frequency j'is and it represents the number of complete oscillations per time unit undertaken by the solution. The period of the system of the time required to complete one oscillation is Equation The ball is started in motion with no initial velocity by displacing it 6 in above the equi- librium position.

The motion is free and undamped. Equation Find an expression for the motion of the mass, assuming no air resistance. The equation of motion is governed by Eq. Differentiating 2 , we obtain whereupon, and 2 simplifies to as the position of the mass at any time t. Determine the circular frequency, natural frequency, and period for the simple harmonic motion described in Problem Circular frequency: Natural frequency: Period: A kg mass is attached to a spring, stretching it 0.

The mass is started in motion from the equilibrium position with an initial velocity of 1 ml sec in the upward direction. Find the subsequent motion, if the force due to air resistance is i N. Find the subsequent motion of the mass if the force due to air resistance is -2ilb. Find the subsequent motion of the mass, if the force due to air resistance is -lilb. Show that the types of motions that result from free damped problems are completely determined by the quantity a2 — 4 km.

The corresponding motions are, respectively, overdamped, critically damped, and oscillatory damped. Since the real parts of both roots are always negative, the resulting motion in all three cases is transient. Find the subsequent motion of the mass if the force due to air resistance is iN. The equation of motion, These terms are the transient part of the solution.

Assuming no air resistance, find the subsequent motion of the weight. This phenomenon is called pure resonance. It is due to the forcing function F t having the same circular frequency as that of the associated free undamped system.

Write the steady-state motion found in Problem The steady-state displacement is given by 2 of Problem Substituting the given quantities into Eq. Hence, As in Problem Solve Problem Substituting the values given in Problem Therefore, and as before.

Note that although the current is completely transient, the charge on the capacitor is the sum of both transient and steady-state terms. An RCL circuit connected in series has a resistance of 5 ohms, an inductance of 0. Find an expression for the current flowing through this circuit if the initial current and the initial charge on the capacitor are both zero.

Substituting this value into 2 and simplifying, we obtain as before Determine the circular frequency, the natural frequency, and the period of the steady-state current found in Problem The current is given by 3 of Problem Write the steady-state current found in Problem Determine whether a cylinder of radius 4 in, height 10 in, and weight 15 Ib can float in a deep pool of water of weight density Let h denote the length in feet of the submerged portion of the cylinder at equilibrium.

Determine an expression for the motion of the cylinder described in Problem In the context of Fig. Determine whether a cylinder of diameter 10 cm, height 15 cm, and weight Let h denote the length in centimeters of the submerged portion of the cylinder at equilibrium. Let h denote the length of the submerged portion of the cylinder at equilibrium.

A solid cylinder partially submerged in water having weight density Determine the diameter of the cylinder if it weighs 2 Ib. We are given 0. The prism is set in motion by displacing it from its equilibrium position see Fig.

Determine the differential equation governing the subsequent motion of this prism. For the prism depicted in Fig. By Archimedes' principle, this buoyant force at equilibrium must equal the weight of the prism mg; hence, We arbitrarily take the upward direction to be the positive x-direction.

If the prism is raised out of the water by x t units, as shown in Fig. It now follows from Newton's second law that Substituting 1 into this last equation, we simplify it to Fig. A lb weight is suspended from a spring and stretches it 2 in from its natural length. Find the spring constant.

A mass of 0. It is then set into motion by stretching the spring 2 in from its equilibrium position and releasing the mass from rest.

Find the position of the weight at any time t if there is no external force and no air resistance. Find the position of the mass at any time t if there is no external force and no air resistance. A lb weight is attached to a spring, stretching it 8 ft from its natural length. Find the subsequent motion of the weight, if the medium offers negligible resistance.

Determine a the circular frequency, b the natural frequency, and c the period for the vibrations described in Problem Find the solution to Eq. A -slug mass is hung onto a spring, whereupon the spring is stretched 6 in from its natural length. Find the subsequent motion of the mass, if the force due to air resistance is —2x Ib. A -j-slug mass is attached to a spring so that the spring is stretched 2 ft from its natural length.

The mass is started in motion with no initial velocity by displacing it yft in the upward direction. Find the subsequent motion of the mass, if the medium offers a resistance of —4x Ib. The mass is set into motion by displacing it 6 in below its equilibrium position with no initial velocity. Find the subsequent motion of the mass, if the force due to the medium is —4x Ib.

Find the subsequent motion of the mass if the surrounding medium offers a resistance of -4iN. Find the subsequent motion of the mass, if the force due to air resistance is —4x Ib.

A lb weight is attached to a spring whereupon the spring is stretched 1. Find the subsequent motion of the weight if the surrounding medium offers a negligible resistance. A lb weight is attached to a spring whereupon the spring is stretched 2 ft and allowed to come to rest. Find the subsequent motion of the weight if the surrounding medium offers a resistance of —2x Ib.

Write the steady-state portion of the motion found in Problem Find the subsequent motion of the mass if the surrounding medium offers a resistance of —3x N.

Assuming no initial current and no initial charge on the capacitor, find expressions for the current flowing through the circuit and the charge on the capacitor at any time t. Assuming no initial current and no initial charge on the capacitor, find an expression for the current flowing through the circuit at any time t. Determine the steady-state current in the circuit described in Problem An RCL circuit connected in series with a resistance of 16 ohms, a capacitor of 0.

Assuming no initial current and no initial charge on the capacitor, find an expression for the charge on the capacitor at any time t. Determine the steady-state charge on the capacitor in the circuit described in Problem Find the subsequent steady-state current in the circuit. Initial conditions are not needed. Find the steady-state current in the circuit. Hint Initial conditions are not needed. Determine the equilibrium position of a cylinder of radius 3 in, height 20 in, and weight 57rlb that is floating with its axis vertical in a deep pool of water of weight density Find an expression for the motion of the cylinder described in Problem Write the harmonic motion of the cylinder described in Problem Determine the equilibrium position of a cylinder of radius 2 ft, height 4 ft, and weight Ib that is floating with its axis vertical in a deep pool of water of weight density Determine the equilibrium position of a cylinder of radius 30 cm, height cm, and weight 2.

Find the general solution to Eq. Hint: Use the results of Problem The box is set into motion by displacing it x0 units from its equilibrium position and giving it an initial velocity of v0. Determine the differential equation governing the subsequent motion of the box.

Determine a the period of oscillations for the motion described in Problem II all the elements are numbers. Ihen the matrix is called a constant matrix.

Matrices will prove to be very helpful in several ways. For example, we can recast higher-order differential equations into a sjslem of first-order differential equations using matrices see Chapter In particular, the first matrix is a constant matrix, whereas the last two are not.

A matrix is square if it has the same number of rows and columns. The third matrix given in Example I5. I is a vector. That is, Matrix addition is both associative and commutalue.

Matrix multiplication is associative and distributes over addition; in general, however, it is not commutative. Theorem Cayley—Hamilton theorem. Any square matrix satisfies its own characteristic equation. That is, if then Solved Problems Find 3A - B for the matrices given in Problem Find 2A - B 2 for the matrices given in Problem But Therefore, the cancellation law is not valid for matrix multiplication.

Find Ax if Find Find J A dt for A as given in Problem Find the eigenvalues of A? Verify the Cayley-Hamilton theorem for the matrix of Problem Find a AB and b BA.

Find A2. Find A7. FindB 2. Find a CD and b DC. Find a Ax and b xA. Find AC. Find the characteristic equation and eigenvalues of A. Find the characteristic equation and eigenvalues of B. Find the characteristic equation and eigenvalues of 3A. Find the characteristic equation and the eigenvalues of C. Determine the multiplicity of each eigenvalue. Find the characteristic equation and the eigenvalues of D.

Find for A as given in Problem Find Adt for A as given in Problem The infinite series However, it follows with some effort from Theorem I. Thus: Theorem Furthermore, if X; is an eigeinalue of multiplicity k. When com- puting the various derivatives in Method of computation: For each eigenvalue A,, of A? When this is done for each eigenvalue, the set of all equations so obtained can be solved for a0, «i, These values are then substituted into Eq.

From Eq. Substituting these values successively into Eq. Substituting these values successively into It follows from Theorem From Eqs.

It now follows from Theorem Now, according to Eq. Consider the following second- order differential equation: We see that! The method of reduction is as follows. Step 1. Rewrite Thus, where and Step 2. Define n new variables the same number as the order of the original differential equation ; Xi t , x2 t , Express dxnldt in terms of the new variables.

Proceed by first differentiating the last equation of Equations Define Then the initial conditions This last equa- tion is an immediate consequence of Eqs. The procedure is nearly identical to the method for reducing a single equation to matrix form; only Step 2 changes. With a system of equations, Step 2 is generalized so that new variables are defined for each of the unknown functions in the set.

Put the initial-value problem into the form of System Proceeding as in Problem Convert the differential equation into the matrix equation x? The given differential equation has no prescribed initial conditions, so Step 5 is omitted. Put the following system into the form of System Put the following system into matrix form: We proceed exactly as in Problems Supplementary Problems Reduce each of the following systems to a first-order matrix system.

In this chapter, and in the two succeeding chapters, we introduce several qualitative approaches in dealing with differential equations. Observe that in a particular problem, f x , y may be independent of x, of y, or of x and y. A line element is a short line segment that begins at the point x, y and has a slope specified by A collection of line elements is a direction field.

The graphs of solutions to If the left side of Eq. When they are simple to draw, isoclines yield many line elements at once which is useful for constructing direction fields. To obtain a graphical approximation to the solution curve of Eqs. Denote the terminal point of this second line element as x2, y2.

Follow with a third line element constructed at x2, y2 and continue it a short distance. The process proceeds iteratively and concludes when enough of the solution curve has been drawn to meet the needs of those concerned with the problem.

This formula is often written as where as required by Eq. In general, the smaller the step-size, the more accurate the approximate solution becomes at the price of more work to obtain that solution. Thus, the final choice of h may be a compromise between accuracy and effort.

If h is chosen too large, then the approximate solution may not resemble the real solution at all, a condition known as numerical instability. To avoid numerical instability, Euler's method is repeated, each time with a step-size one half its previous value, until two successive approximations are close enough to each other to satisfy the needs of the solver. Continuing in this manner we generate the more complete direction field shown in Fig. To avoid confusion between line elements asso- ciated with the differential equation and axis markings, we deleted the axes in Fig.

The origin is at the center of the graph. Describe the isoclines associated with the differential equation defined in Problem For the differential equation in Problem Some of these line elements are also drawn in Fig.

Draw two solution curves to the differential equation given in Problem A direction field for this equation is given by Fig. Two solution curves are shown in Fig. Observe that each solution curve follows the flow of the line elements in the direction field. Continuing in this manner, we generate Fig. At each point, we graph a short line segment emanating from the point at the specified angle from the horizontal.

To avoid confusion between line elements associated with the differential equation and axis markings, we deleted the axes in Fig. Draw three solution curves to the differential equation given in Problem Three solution curves are shown in Fig. These and other isoclines with some of their associated line elements are drawn in Fig. Draw four solution curves to the differential equation given in Problem Four solution curves are drawn in Fig. Note that the differential equation is solved easily by direct integration.

Observe that solution curves have different shapes depending on whether they are above both of these isoclines, between them, or below them. A representative solution curve of each type is drawn in Fig. Give a geometric derivation of Euler's method. Give an analytic derivation of Euler's method. Let Y x represent the true solution. Then, using the definition of the derivative, we have If A. Thus, which is Euler's method. As before, Then, using Eq.

Note that more accurate results are obtained when smaller values of h are used. If we plot xn, yn for integer values of n between 0 and 10, inclusively, and then connect successive points with straight line segments, we would generate a graph almost indistinguishable from Fig. Find y 0. Then, using Eq. We proceed exactly as in Problem The results of these computations are given in Table The calculations are found in Table Supplementary Problems Direction fields are provided in Problems Sketch some of the solution curves.

See Fig. Describe the isoclines for the equation in Problem Find y 1. It is interesting to note that often the only required operations are addition, subtraction, multiplication, division and functional evaluations. In this chapter, we consider only first-order initial-value problems of the form Generalizations to higher-order problems are given in Chapter Remarks made in Chapter 18 on the step-size remain valid for all the numerical methods presented below.

The approximate solution at xn will be designated by y xn , or simply yn. The true solution at xn will be denoted by either Y xn or Yn. Note that once yn is known, Eq. In general, the corrector depends on the predicted value. It then follows from Eq. The first of these values is given by the initial condition in Eq. The other three starting values are gotten by the Runge-Kutta method.

In other words, if the true solution of an initial-value problem is a polynomial of degree n or less, then the approximate solution and the true solution will be identical for a method of order n. In general, the higher the order, the more accurate the method. Euler's method, Eq. Then using Eqs. Compare it to Table From Then using Find y l. Since the true solution is a second-degree polynomial and the modified Euler's method is a second-order method, this agreement is expected.

Compare it with Table Thus, Then, using Eqs. Note that ys is significantly different from pys and y'4 is significantly different from py'4. When significant changes occur, they are often the result of numerical instability, which can be remedied by a smaller step-size. Sometimes, however, significant differences arise because of a singularity in the solution.

Figure is a direction field for this differential equation. The cusp between 1. The analytic solution to the differential equation is given in Problem 4. The values of y0, yi, y2, y? Using Eqs. Then, using Eqs. In this chapter we investigate several numerical techniques dealing with such sjslems.

System In particular, The derivatives associated with the predicted values are obtained similarly, by replacing y and z in Eq. As in Chapter 19, four sets of starting values are required for the Adams-Bashforth-Moulton method. The first set comes directly from the initial conditions; the other three sets are obtained from the Runge-Kutta method. We thus obtain the first-order system The given differential equation can be rewritten as or We thus obtain the first-order system Then, using Use the Runge-Kutta method to solve 3.

It follows from Problem Use the Adams-Bashforth-Moulton method to solve 3. Formulate the Adams-Bashforth-Moulton method for System Formulate Milne's method for System All starting values and their derivatives are identical to those given in Problem Using the formulas given in Problem Obtain appropriate starting values from Table Formulate the modified Euler's method for System Formulate the Runge-Kutta method for System Convergence occurs when she l i m i t exists.

When evaluating the integral in Eq. The Laplace transforms for a number of elementary functions are calculated in Problems They are equally applicable for functions of any independent variable and are generated by replacing the variable x in the above equations by any variable of interest. The books do not aim to provide all of the mathematical foundations upon which the Internet is based.

Instead, these cover only a partial panorama and the key principles. Volume 1 explores Internet engineering, while the supporting mathematics is covered in Volume 2. The chapters on mathematics complement those on the engineering episodes, and an effort has been made to make this work succinct, yet self-contained. Elements of information theory, algebraic coding theory, cryptography, Internet traffic, dynamics and control of Internet congestion, and queueing theory are discussed.

In addition, stochastic networks, graph-theoretic algorithms, application of game theory to the Internet, Internet economics, data mining and knowledge discovery, and quantum computation, communication, and cryptography are also discussed.

In order to study the structure and function of the Internet, only a basic knowledge of number theory, abstract algebra, matrices and determinants, graph theory, geometry, analysis, optimization theory, probability theory, and stochastic processes, is required. These mathematical disciplines are defined and developed in the books to the extent that is needed to develop and justify their application to Internet engineering.