### Schaum outlines vector analysis solution manual,Recent Posts

Ordinary derivatives of vectors. Space curves. Eliminating t, the equations become. having direction passing through the origin and outward from it. The field therefore appears as in Figure a where an appropriate scale is used. b Here each vector is equal to but opposite in direction to the corresponding one in a. The field there-fore appears as in Fig. In Fig. a the field has the appearance of a fluid emerging from a point source 0 and flowing in thedirections indicated. For this reason the field is called a source field and 0 is a source. b the field seems to be flowing toward 0, and the field is therefore called a sink field and 0is a sink. In three dimensions the corresponding interpretation is that a fluid is emerging radially from or pro-ceeding radially toward a line source or line sink.

The field therefore takes on the appearance of thatof a fluid emerging from source 0 and proceeding in all directions in space. This is a three dimension-al source field. Which of the following are scalars and which are vectors? a Kinetic energy, b electric field intensity, c entropy, d work, e centrifugal force, f temperature, g gravitational potential, h charge, i shear-ing stress, j frequency. a scalar, b vector, c scalar, d scalar, e vector, f scalar, g scalar, h scalar, i vector. j scalar An airplane travels miles due west and then miles north of west. Determine the resultant dis-. placement a graphically, b analytically. magnitude Find the resultant of the following displacements: A, 20 miles 30south of east; B, 50 miles due west;C, 40 miles northeast; D, 30 miles 60 south of west. An object P is acted upon by three coplanar forces as shown in Fig. Determine the force needed. If ABCDEF are the vertices of a regular hexagon, find the resultant of the forces represented by the vec-tors AB, AC, AD, AE and AF.

Two towns A and B are situated directly opposite each other on the banks of a river whose width is 8 miles. A man located at A wishes to reach town C which is 6 miles up-stream from and on the same side of the river as town B. Find the direction and speed ofthe wind. A lb weight is suspended from the center of a ropeas shown in the adjoining figure. Determine the ten-sion T in the rope. If A and B are given vectors representing the diagonals of a parallelogram, construct the parallelogram. Prove that the line joining the midpoints of two sides of a triangle is parallel to the third side and has onehalf of its magnitude. b Does the result hold if 0 is any point outside the triangle? Prove your result. In the adjoining figure, ABCD is a parallelogram with. P and Q the midpoints of sides BC and CD respec-tively. Prove that AP and AQ trisect diagonal BD atthe points E and F.

Prove that the medians of a triangle meet in a commonpoint which is a point of trisection of the medians. Show that there exists a triangle with sides which areequal and parallel to the medians of any given triangle. Let the position vectors of points P and Q relative to an origin 0 be given by p and q respectively. If R isa point which divides line PQ into segments which are in the ratio m : n show that the position vector of R. If r1, r2, A quadrilateral ABCD has masses of 1, 2, 3 and 4 units located respectively at its vertices A -1, -2, 2 ,B 3, 2, -1 , C 1, -2, 4 , and D 3, 1, 2. Find the coordinates of the centroid. Show that the equation of a plane which passes through three given points A, B, C not in the same straightline and having position vectors a, b, c relative to an origin 0, can be written. Determine PQ interms of i, j, k and find its magnitude. Find a the resultant of the forces, b the magnitude of the resultant.

a 2i-j b yr. a linearly dependent, b linearly independent. be different from zero. C1 C2 C3. Find a 0 1,-1,-2 , b 4 0,-3,1. a 36 b THE DOT OR SCALAR PRODUCT of two vectors A and B, denoted by A dot B , is de-fined as the product of the magnitudes of A and B and the cosine. The following laws are valid The mag-nitude of A x B is defined as the product of the magnitudes of. A and B and the sine of the angle 6 between them. In symbols,. TRIPLE PRODUCTS. Dot and cross multiplication of three vectors A, B and C may produce mean-ingful products of the form A B C, A- BxC and Ax BxC. The follow-. The product A BxC is sometimes called the scalar triple product or box product and may bedenoted by [ABC]. The product Ax BxC is called the vector triple product. In A B x C parentheses are sometimes omitted and we write A BxC see Problem How-ever, parentheses must be used in A x BxC see Problems 29 and The sets of vectors a, b, c and a', b', c' are called reciprocalsets or systems of vectors if.

Prove that the projection of A on B is equal to A b, whereb is a unit vector in the direction of B. Through the initial and terminal points of A pass planes per- Ependicular to B at G and H respectively as in the adjacent figure;then. Evaluate each of the following. We first have to show that the vectors form a triangle. Let a, P. y be the angles which A makes with the positive x, y, z axes respectively. The cosines of a, 3, and y are called the direction cosines of A. See Prob. Prove that the diagonals of a rhombus are perpendicular.

Hence OQ is perpendicular to RP. Then C is perpendicular to A. Refer to Fig- a below. Let r be the position vector of point P, and Q the terminal point of B. In Problem 18 find the distance from the origin to the plane. The distance from the origin to the plane is the projection of B on A. a Fig. a above. Then D has the same magnitude as C but is opposite in direction, i. The commutative law for cross products is not valid. Thisis equivalent to multiplying vector B by A and rotatingthe resultant vector through 90 to the positionshown in the adjoining diagram. Similarly, A x C is the vector obtained by multi-plying C by A and rotating the resultant vector through90 to the position shown. Resolve B into two component vectors, one perpen-dicular to A and the other parallel to A, and denote themby B1 and B respectively.

Thus the magnitude of A x B 1 is AB sin B, the same asthe magnitude of A X B. Also, the direction of A x B1 isthe same as the direction of A x B. Multiplying by -1, using Prob. Note that the order of factors in cross products is important. The usual laws of algebra apply only if prop-er order is maintained. Another Method. Note that this is equivalent to the theorem: If two rows of. Find the area of the triangle having vertices at P 1, 3, 2 , Q 2, -1, 1 , R -1, 2, 3. A x B is a vector perpendicular to the plane of A and B. Prove the law of sines for plane triangles. Let a, b and c represent the sides of triangle ABC. Mul-tiplying by a x, b x and c x in succession, we find. Consider a tetrahedron with faces Fl, F2 , F3 , F4. Let V1, V2, V3 , V4 be vectors whose magnitudes arerespectively equal to the areas of Fl , F2 , F3, F4 andwhose directions are perpendicular to these facesin the outward direction.

This result can be generalized to closed polyhedra and in the limiting case to any closed surface. Because of the application presented here it is sometimes convenient to assign a direction to area and. Find an expression for the moment of a force F about a point P. The moment M of F about P is in magnitude equal to F times the perpendicular distance from P to the. to the plane of r and F, then when the force F acts the screwwill move in the direction of r x F. A rigid body rotates about an axis through point 0 withangular speed w. Also, vmust be perpendicular to both w and r and is such that r, 4 andv form a right-handed system.

The vector Ca is called the angular velocity. Let n be a unit normal to parallelogram 1,having the direction of B x C, and let h be theheight of the terminal point of A above the par-allelogram 1. By a theorem of determinants which states that interchange of two rows of a determinant changes itssign, we have. In such case there cannot be. any ambiguity since the only possible interpretations are A B x C and A B x C. The latter howeverhas no meaning since the cross product of a scalar with a vector is undefined. Note that A A. B x C can have no meaning other than A B x C. If A, B and C are coplanar the volume of the parallelepiped formed by them is zero. Then by Problem. points Pi x1, yi, z1 , P2 x2, y2, z2 and P3 x3,y3, z3. Find an equation for the plane passing through P1,P2 and P3. Find an equation for the plane determined by the points P1 2, -1, 1 , P2 3, 2, -1 and P3 ,-1, 3, 2.

Let r be the position vector of any point in the plane of P. Then the vectors r - a, b -a andc -a are coplanar, so that by Problem Bj-C2AsB3 j. the associative law for vector cross products is notvalid for all vectors A, B, C. Let PQR be a spherical triangle whose sides p, q, r are arcs of great circles. Prove thatsin P. Suppose that the sphere see figure below has unit radius, and let unit vectors A, B and C be drawnfrom the center 0 of the sphere to P, Q and R respectively. The results can also be seen by noting, for example, that a has. From a and b we see that the sets of vectors a, b, c and a', b', c' are reciprocal vectors.

d By Problem 43, if a, b and c are non-coplanar a b x c 0. Then from part c it follows thata b' x c X 0 , so that a', b' and c are also non-coplanar. Find the acute angles which the line joining the points 1,-3,2 and 3,-5,1 makes with the coordinate. Find the direction cosines of the line joining the points 3,2,-4 and 1,-1,2. Determine the angles. of the triangle. Show that the parallelo-. gram is a rhombus and determine the length of its sides and its angles. Find the work done in moving an object along a straight line from 3,2,-1 to 2,-1,4 in a force field given. Let F be a constant vector force field. Show that the work done in moving an object around any closed pol-. a Find an equation of a plane perpendicular to a given vector A and distant p from the origin.

b Express the equation of a in rectangular coordinates. Let r1 and r2 be unit vectors in the xy plane making angles a and R with the positive x-axis. b By considering r1. r2 prove the trigonometric formulas. Let a be the position vector of a given point x1, y1, z1 , and r the position vector of any point x, y, z. a Sphere, center at x1, y1, z1 and radius 3. b Plane perpendicular to a and passing through its terminal point. a Find an equation for the plane passing through Q and perpendicular to line PQ. b What is the distance from the point -1,1,1 to the plane? Find the area of a triangle with vertices at 3,-1,2 , 1,-1,-3 and 4,-3,1. Find the moment of F about the point. Find the. Let points P. Find the distance from P to the plane OQR. Find the shortest distance from 6,-4,4 to the line joining 2,1,2 and 3,-1,4. Given points P 2,1,3 , Q 1,2,1 , R -1,-2,-2 and S 1,-4,0 , find the shortest distance between lines PQ and. Prove that the perpendiculars from the vertices of a triangle to the opposite sides extended if necessary.

meet in a point the orthocenter of the triangle. Prove that the perpendicular bisectors of the sides of a triangle meet in a point the circumcenter of the tri-. Prove the law of cosines for. with analogous formulas for cos q and cos r obtained by cyclic permutation of the letters. Prove that the only right-handed self-reciprocal sets of vectors are the unit vectors i, j , k. Prove that there is one and only one set of vectors reciprocal to a given set of non-coplanar vectors a, b, c. Let R u be a vector depending on a single scalar variable u. Since dR is itself a vector depending on u, we can consider its derivative with respect to u. Ifthis derivative exists it is denoted by a R.

In like manner higher order derivatives are described. SPACE CURVES. If in particular R u is the position vector r u joining the origin 0 of a coordinatesystem and any point x, y, z , then. the di-rection of Ar see adjacent figure. represents the velocity v withwhich the terminal point of r describes the curve. Similarly, dalong the curve. Equivalently, 6 u is continu-. Equivalently, R u functions R1 u , R2 u and R3 u are continuous at u or if Alu ois continuous at u if for each positive number e we can find some positive number 8 such that. A scalar or vector function of u is called differentiable of order n if its nth derivative exists. function which is differentiable is necessarily continuous but the converse is not true.

Unless other-wise stated we assume that all functions considered are differentiable to any order needed in a par-ticular discussion. If A, B and C are differentiable vector functions of a scalar u, and0 is a differentiable scalar function of u, then. are the partial derivatives of A with respect to y and z respectively if these limits exist. The remarks on continuity and differentiability for functions of one variable can be extended to. functions of two or more variables. Similar defi-nitions hold for vector functions. For functions of two or more variables we use the term differentiable to mean that the functionhas continuous first partial derivatives. The term is used by others in a slightly weaker sense. Higher derivatives can be defined as in the calculus. order of differentiation does not matter. Rules for partial differentiation of vectors are similar to those used in elementary calculus for.

If C is a space curvedefined by the function r u , then we have seen that du is a vector in. the direction of the tangent to C. If the scalar u is taken as the arc length s measured from some fixedpoint on C, then -d-r- is a unit tangent vector to C and is denoted by T see diagram below. rate at which T changes with respect to s is a mea-sure of the curvature of C and is given by dT. Download Schaum s Outline of Vector Analysis 2ed Book in PDF, Epub and Kindle. Schaum s Outline of Complex Variables 2ed. Author : Murray Spiegel,Seymour Lipschutz,John Schiller,Dennis Spellman Publsiher : McGraw Hill Professional Total Pages : Release : Genre : Study Aids ISBN : GET BOOK. Download Schaum s Outline of Complex Variables 2ed Book in PDF, Epub and Kindle.

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qllCove-age of all course fundamentals for vectoranalysis, with an introduction to tensor analysis. tiara I tan. r'i tr N a. hington f Au. nIrrd Mrkik ii Um Milan. Copyright Q by McGraw-Hill, Inc. All Rights Reserved. Printed in theUnited States of America. No part of this publication may be reproduced,stored in a retrieval system, or transmitted, in any form or by any means,electronic, mechanical. or otherwise. without the priorwritten permission of the publisher. ISBN X. PrefaceVector analysis, which had its beginnings in the middle of the 19th century, has in recent.

years become an essential part of the mathematical background required of engineers, phy-sicists, mathematicians and other scientists. This requirement is far from accidental, for notonly does vector analysis provide a concise notation for presenting equations arising frommathematical formulations of physical and geometrical problems but it is also a natural aidin forming mental pictures of physical and geometrical ideas. In short, it might very well beconsidered a most rewarding language and mode of thought for the physical sciences. This book is designed to be used either as a textbook for a formal course in vectoranalysis or as a very useful supplement to all current standard texts. It should also be ofconsiderable value to those taking courses in physics, mechanics, electromagnetic theory,aerodynamics or any of the numerous other fields in which vector methods are employed.

Each chapter begins with a clear statement of pertinent definitions, principles andtheorems together with illustrative and other descriptive material. This is followed bygraded sets of solved and supplementary problems. The solved problems serve to illustrateand amplify the theory, bring into sharp focus those fine points without which the studentcontinually feels himself on unsafe ground, and provide the repetition of basic principlesso vital to effective teaching. Numerous proofs of theorems and derivations of formulasare included among the solved problems. The large number of supplementary problemswith answers serve as a complete review of the material of each chapter. Topics covered include the algebra and the differential and integral calculus of vec-tors, Stokes' theorem, the divergence theorem and other integral theorems together withmany applications drawn from various fields.

Added features are the chapters on curvilin-ear coordinates and tensor analysis which should prove extremely useful in the study ofadvanced engineering, physics and mathematics. Considerably more material has been included here than can be covered in most firstcourses. This has been done to make the book more flexible, to provide a more useful bookof reference, and to stimulate further interest in the topics. The author gratefully acknowledges his indebtedness to Mr. Henry Hayden for typo-graphical layout and art work for the figures. The realism of these figures adds greatly tothe effectiveness of presentation in a subject where spatial visualizations play such an im-portant role.

VECTORS AND SCALARS 1Vectors. Vector algebra. Laws of vector algebra. Unit vectors. Rectangular unitvectors. Components of a vector. Scalar fields. Vector fields. THE DOT AND CROSS PRODUCT 16Dot or scalar products. Cross or vector products. Triple products. Reciprocal sets ofvectors. Space curves. Continuity and differentiability. Differen-tiation formulas. Partial derivatives of vectors Differentials of vectors. Formulas involving del. Line integrals. Surface integrals. Volume integrals. The divergence theorem of Gauss. Stokes' theorem. Green's theorem in the plane. Re-lated integral theorems. Integral operator form for del. Orthogonal curvilinear coordinates.

Unit vectors incurvilinear systems. Arc length and volume elements. Gradient, divergence and curl. Special orthogonal coordinate systems. Cylindrical coordinates. Spherical coordinates. Parabolic cylindrical coordinates. Paraboloidal coordinates. Elliptic cylindrical coordinates. Prolate spheroidal coordinates. Oblate spheroidal coordinates. Ellipsoidal coordinates. Bipolar coordinates. TENSOR ANALYSIS Physical laws. Spaces of N dimensions. Coordinate transformations. The summationconvention. Contravariant and covariant vectors. Contravariant, covariant and mixedtensors. The Kronecker delta. Tensors of rank greater than two. Scalars or invariants. Tensor fields. Symmetric and skew-symmetric tensors. Fundamental operations withtensors. Matrix algebra. The line element and metric tensor. Conjugate orreciprocal tensors. Associated tensors. Length of a vector. Angle between vectors. Christoffel's symbols. Transformation laws of Christoffel's symbols. Covariant derivatives.

Permutation symbols and tensors. Tensor form of gradient,divergence and curl. The intrinsic or absolute derivative. Relative and absolute tensors. INDEX Graphically a vector is represented by an arrow OP Fig. l de-fining the direction, the magnitude of the vector being indicated bythe length of the arrow. The tail end 0 of the arrow is called theorigin or initial point of the vector, and the head P is called theterminal point or terminus. Analytically a vector is represented by a letter with an arrowover it, as A in Fig.