DIFFERENTIAL GEOMETRY
CHAPTER I CURVES IN THREE DIMENSIONAL SPACE 1 CHAPTER II ELEMENTS OF VECTORIAL AND TENSORIAL ANALYSIS 13 CHAPTER III CURVILINEAR COORDINATES 40 CHAPTER IV GEOMETRY OF SURFACES 69 CHAPTER V CURVES ON SURFACES 101 REFERENCES 127 CONTENTS CHAPTER I CURVES IN THREE DIMENSIONAL SPACE 1 CURVES METHODS OF DE NING A CURVE REGULAR AND SINGULAR POINTS OF A CURVE LET E BE A THREE DIMENSIONAL EUCLIDEAN POINT SPACE THE STRICT MATHEMATICAL DE NITION OF SUCH A SPACE CAN BE FOUND IN 1 HOWEVER KNOWING THIS DE NITION IS NOT SO URGENT THE MATTER IS THAT E CAN BE UNDERSTOOD AS THE REGULAR THREE DIMENSIONAL SPACE THAT IN WHICH WE LIVE THE PROPERTIES OF THE SPACE E ARE STUDIED IN ELEMENTARY MATHEMATICS AND IN ANALYTICAL GEOMETRY ON THE BASE INTUITIVELY CLEAR VISUAL FORMS THE CONCEPT OF A LINE OR A CURVE IS ALSO RELATED TO SOME VISUAL FORM A CURVE IN THE SPACE E IS A SPATIALLY EXTENDED ONE DIMENSIONAL GEOMETRIC FORM THE ONE DIMENSIONALITY OF A CURVE REVEALS WHEN WE USE THE VECTORIAL PARAMETRIC METHOD OF DE NING IT R R T X1 T X2 T X3 T 1 1 WE HAVE ONE DEGREE OF FREEDOM WHEN CHOOSING A POINT ON THE CURVE 1 1 OUR CHOICE IS DETERMINED BY THE VALUE OF THE NUMERIC PARAMETER T TAKEN FROM SOME INTERVAL E G FROM THE UNIT INTERVAL 0 1 ON THE REAL AXIS R POINTS OF THE CURVE 1 1 ARE GIVEN BY THEIR RADIUS VECTORS1 R R T WHOSE COMPONENTS X1 T X2 T X3 T ARE FUNCTIONS OF THE PARAMETER T THE CONTINUITY OF THE CURVE 1 1 MEANS THAT THE FUNCTIONS X1 T X2 T X3 T SHOULD BE CONTINUOUS HOWEVER THIS CONDITION IS TOO WEAK AMONG CONTINUOUS CURVES THERE ARE SOME INSTANCES WHICH DO NOT AGREE WITH OUR INTUITIVE UNDERSTAND ING OF A CURVE IN THE COURSE OF MATHEMATICAL ANALYSIS THE PEANO CURVE IS OFTEN CONSIDERED AS AN EXAMPLE SEE 2 THIS IS A CONTINUOUS PARAMETRIC CURVE ON A PLANE SUCH THAT IT IS ENCLOSED WITHIN A UNIT SQUARE HAS NO SELF INTERSECTIONS AND PASSES THROUGH EACH POINT OF THIS SQUARE IN ORDER TO AVOID SUCH UNUSUAL CURVES THE FUNCTIONS XI T IN 1 1 ARE ASSUMED TO BE CONTINUOUSLY DI ERENTIABLE C1 CLASS FUNCTIONS OR AT LEAST PIECEWISE CONTINUOUSLY DI ERENTIABLE FUNCTIONS NOW LET S CONSIDER ANOTHER METHOD OF DE NING A CURVE AN ARBITRARY POINT OF THE SPACE E IS GIVEN BY THREE ARBITRARY PARAMETERS X1 X2 X3 ITS COORDINATES WE CAN RESTRICT THE DEGREE OF ARBITRARINESS BY CONSIDERING A SET OF POINTS WHOSE COORDINATES X1 X2 X3 SATISFY AN EQUATION OF THE FORM F X1 X2 X3 0 1 2 1 HERE WE ASSUME THAT SOME CARTESIAN COORDINATE SYSTEM IN E IS TAKEN 1 1 CURVES METHODS OF DEFINING A CURVE WHERE F IS SOME CONTINUOUSLY DI ERENTIABLE FUNCTION OF THREE VARIABLES IN A TYPICAL SITUATION FORMULA 1 2 STILL ADMITS TWO PARAMETRIC ARBITRARINESS CHOOSING ARBITRARILY TWO COORDINATES OF A POINT WE CAN DETERMINE ITS THIRD COORDINATE BY SOLVING THE EQUATION 1 2 THEREFORE 1 2 IS AN EQUATION OF A SURFACE IN THE INTERSECTION OF TWO SURFACES USUALLY A CURVE ARISES HENCE A SYSTEM OF TWO EQUATIONS OF THE FORM 1 2 DE NES A CURVE IN E F X1 X2 X3 0 G X1 X2 X3 0 1 3 IF A CURVE LIES ON A PLANE WE SAY THAT IT IS A PLANE CURVE FOR A PLANE CURVE ONE OF THE EQUATIONS 1 3 CAN BE REPLACED BY THE EQUATION OF A PLANE A X1 B X2 C X3 D 0 SUPPOSE THAT A CURVE IS GIVEN BY THE EQUATIONS 1 3 LET S CHOOSE ONE OF THE VARIABLES X1 X2 OR X3 FOR A PARAMETER E G WE CAN TAKE X1 T TO MAKE CERTAIN THEN WRITING THE SYSTEM OF THE EQUATIONS 1 3 AS F T X2 X3 0 G T X2 X3 0 AND SOLVING THEM WITH RESPECT TO X2 AND X3 WE GET TWO FUNCTIONS X2 T AND X3 T HENCE THE SAME CURVE CAN BE GIVEN IN VECTORIAL PARAMETRIC FORM R R T T X2 T X3 T CONVERSELY ASSUME THAT A CURVE IS INITIALLY GIVEN IN VECTORIAL PARAMETRIC FORM BY MEANS OF VECTOR FUNCTIONS 1 1 THEN USING THE FUNCTIONS X1 T X2 T X3 T WE CONSTRUCT THE FOLLOWING TWO SYSTEMS OF EQUATIONS X1 X1 T 0 X2 X2 T 0 X1 X1 T 0 X3 X3 T 0 1 4 EXCLUDING THE PARAMETER T FROM THE RST SYSTEM OF EQUATIONS 1 4 WE OBTAIN SOME FUNCTIONAL RELATION FOR TWO VARIABLE X1 AND X2 WE CAN WRITE IT AS F X1 X2 0 SIMILARLY THE SECOND SYSTEM REDUCES TO THE EQUATION G X1 X3 0 BOTH THESE EQUATIONS CONSTITUTE A SYSTEM WHICH IS A SPECIAL INSTANCE OF 1 3 F X1 X2 0 G X1 X3 0 THIS MEANS THAT THE VECTORIAL PARAMETRIC REPRESENTATION OF A CURVE CAN BE TRANS FORMED TO THE FORM OF A SYSTEM OF EQUATIONS 1 3 NONE OF THE ABOVE TWO METHODS OF DE NING A CURVE IN E IS ABSOLUTELY PREFERABLE IN SOME CASES THE RST METHOD IS BETTER IN OTHER CASES THE SECOND ONE IS USED HOWEVER FOR CONSTRUCTING THE THEORY OF CURVES THE VECTORIAL PARAMETRIC METHOD IS MORE SUITABLE SUPPOSE THAT WE HAVE A PARAMETRIC CURVE OF THE SMOOTHNESS CLASS C1 THIS IS A CURVE WITH THE COORDINATE FUNCTIONS X1 T X2 T X3 T BEING COPYRIGHT C SHARIPOV R A 1996 2004 2 CHAPTER I CURVES IN THREE DIMENSIONAL SPACE CONTINUOUSLY DI ERENTIABLE LET S CHOOSE TWO DI ERENT VALUES OF THE PARAMETER T AND T T T WHERE T IS AN INCREMENT OF THE PARAMETER LET A AND B BE TWO POINTS ON THE CURVE CORRESPONDING TO THAT TWO VALUES OF THE PARAMETER T WE DRAW THE STRAIGHT LINE PASSING THROUGH THESE POINTS A AND B THIS IS A SECANT FOR THE CURVE DIRECTING VECTORS OF THIS SECANT ARE COLLINEAR TO THE VECTOR AB WE CHOOSE ONE OF THEM A AB T R T T R T T 1 5 TENDING T TO ZERO WE ND THAT THE POINT B MOVES TOWARD THE POINT A THEN THE SECANT TENDS TO ITS LIMIT POSITION AND BECOMES THE TANGENT LINE OF THE CURVE AT THE POINT A THEREFORE LIMIT VALUE OF THE VECTOR 1 5 IS A TANGENT VECTOR OF THE CURVE AT THE POINT A T LIM T A DR T DT R T 1 6 THE COMPONENTS OF THE TANGENT VECTOR 1 6 ARE EVALUATED BY DI ERENTIATING THE COMPONENTS OF THE RADIUS VECTOR R T WITH RESPECT TO THE VARIABLE T THE TANGENT VECTOR R T DETERMINES THE DIRECTION OF THE INSTANTANEOUS DISPLACE MENT OF THE POINT R T FOR THE GIVEN VALUE OF THE PARAMETER T THOSE POINTS AT WHICH THE DERIVATIVE R T VANISHES ARE SPECIAL ONES THEY ARE STOPPING POINTS UPON STOPPING THE POINT CAN BEGIN MOVING IN QUITE DI ERENT DIRECTION FOR EXAMPLE LET S CONSIDER THE FOLLOWING TWO PLANE CURVES R T T2 T3 R T T4 T3 1 7 AT T 0 BOTH CURVES 1 7 PASS THROUGH THE ORIGIN AND THE TANGENT VECTORS OF BOTH CURVES AT THE ORIGIN ARE EQUAL TO ZERO HOWEVER THE BEHAVIOR OF THESE CURVES NEAR THE ORIGIN IS QUITE DI ERENT THE RST CURVE HAS A BEAK LIKE FRACTURE AT THE ORIGIN 3