Vectorial Analysis

Copyright © 2012 por Ing. Mario Castillo.

Todos los derechos reservados. Ninguna parte de este libro puede ser reproducida o transmitida de cualquier forma o por cualquier medio, electrónico o mecánico, incluyendo fotocopia, grabación, o por cualquier sistema de almacenamiento y recuperación, sin permiso escrito del propietario del copyright.

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Contents

Prologue

Chapter I Vector And Scalars

Chapter II The Dot And Cross Product

Chapter III Vector And Scalars

Chapter IV Gradient

Chapter V Vector Integration

Chapter VI

Bibliography

PROLOGUE

THE SOLUTION FOR THE PROBLEMS PRESENTED IN THIS BOOK ARE SOLVED WITH ALGEBRA, ANALYTIC GEOMETRY, DIFFERENTIAL AND INTEGRAL CALCULUS GEOMETRY, MATLAB AND VECTOR ANALYSIS.

CHAPTER I ARE VECTOR AND SCALARS, WE SOLVE PROBLEMS OF PHYSICS (FORCES) AND DYNAMICS.

CHAPTER II DOT AND DOT PRODUCT, WE SOLVE PROBLEMS OF ADD, SUBTRACT, MULTIPLY AND VECTOR DIVISION IN THREE DIMENSIONS, ANGLES AND PLANES.

CHAPTER III DERIVATION OF VECTORS AND SCALARS, VELOCITY AND ACCELERATION AND ITS DIAGRAMS (VISUALIZATION). IN THREE DIMENSIONS.

CHAPTER IV GRADIENT OF ADD, SUBTRACT MULTIPLY AND DIVIDE FUNCTIONS. GRAPHICS IN TRIDIMENSIONAL FUNCTIONS, PERPENDICULAR VECTOR AT A SURFACE. ONE FUNCTION.

CHAPTER V. VECTOR INTEGRATION PROBLEMS OF VELOCITY AND ACCELERATION OF PARTICLES IN ONE PATH, FORCE BY DISTANCE. UNIT VECTORS.

CHAPTER VI. TRANSFORMATION OF COORDINATES SPHERICAL, CYLINDRICAL, PARABOLIC, ELLIPTIC UNIT VECTORS, VECTORS, ORTHOGONAL FUNCTIONS, THREE DIMENSION GRAPHICS.

THE AUTHOR OF THIS BOOK THANKS AT THE INSTRUCTORS OF THE ELECTRICAL ENGINEERING DEPARTMENT THE UNIVERSITY OF TEXAS PAN AMERICAN OF EDINBURG, DR HEINRICH FOLTZ AND DR BEN GHALIA MOUNIR BY ALL ITS ADVISES.

CHAPTER I

VECTOR AND SCALARS

1 WHICH OF THE FOLLOWING ARE SCALARS AND WHICH ARE VECTORS?

2 AN AIRPLANE TRAVELS 200 MILES DUE WEST AND THEN 150 MILES 60 DEGREES NORTH OF WEST. DETERMINE THE RESULTANTE DISPLACEMENT.

R1 = 200;

ANG1 = pi;

R2 = 150;

ANG2 = 2*pi/3;

Rx = (R1*cos (ANG1) + R2*cos (ANG2));

Ry = R1*sin (ANG1) + R2*sin (ANG2);

R = (Rx^2 + Ry^2) ^.5;

fprintf (‘R %f /n', R);

fprintf (‘Rx %f /n', Rx);

fprintf (‘Ry %f /n', Ry);

theta = (180/pi)*(atan(Ry/Rx));

fprintf('theta %f /n', theta);

RESULT R = 304.138127 /n Rx= -275.000000 /n Ry = 129.903811 /n

Theta = -25.284996 /n

3 FIND THE RESULTANT OF THE FOLLOWING DISPLACEMENTS: A. 20 MILES 30º SOUTH OF EAST; B. 50 MILES DUE WEST; C. 40 MILES NORTHEAST; D. 30 MILES 60 DEGREES SOUTH H OF WEST.

R1 = 20;

ANG1 = - pi/6;

R2 = 50;

ANG2 = - pi;

R3 = 40;

ANG3 = pi/4;

R4 = 30;

ANG4 = - 2*pi/3;

Rx = R1*cos(ANG1) + R2*cos(ANG2) + R3*cos(ANG3) + R4*cos(ANG4);

Ry = R1*sin(ANG1) + R2*sin(ANG2) + R3*sin(ANG3) + R4*sin(ANG4);

R = ( Rx^2 + Ry^2)^.5;

fprintf('R %f /n', R);

fprintf('Rx %f /n', Rx);

fprintf('Ry %f /n', Ry);

theta = (180/pi)*(atan(Ry/Rx));

SOLUTION R 20.866494 /nRx -19.395221 /nRy -7.696491 /n

4 SHOW GRAPHICALLY THAT - (A - B) = - A + B

5 F ABCDEF ARE THE VERTICES OF A REGULAR HEXAGON, FIND THE RESULTANT OF THE FORCES REPRESENTED BY THE VECTORS AB, AC, AD, AE AND AF.

Fx = 150 + 200(cos 30) = 150 + 173.2 = 323.2lb

Fy = 100 - 200(sin (30)) = 100 - 100 = 0 lb

6 IF A AND B ARE GIVEN VECTORS SHOW THAT (A) | A + B | <= | A | + B

                                                                ( B ) | A - B | < = | A | - | B |

= Ax + Ay, B = Bx + By

|A + B| = |Ax + Bx + Ay + By| <= | Ax + Ay| +