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CONTENTS Ordinary Differential Equations of First Order and First Degree Linear Differential Equations of Second and Higher Order Mean Value Theorems Functions of Several Variables Curvature, Evolutes and Envelopes Curve Tracing Applications of Integration Multiple Integrals Series and Sequences Vector Differentiation and Vector Operators Vector Integration Vector Integral Theorems Laplace transforms
TEXT BOOKS A text book of Engineering Mathematics, Vol-I T.K.V.Iyengar, B.Krishna Gandhi and Others, S.Chand & Company A text book of Engineering Mathematics, C.Sankaraiah, V.G.S.Book Links A text book of Engineering Mathematics, Shahnaz A Bathul, Right Publishers A text book of Engineering Mathematics, P.Nageshwara Rao, Y.Narasimhulu & N.Prabhakar Rao, Deepthi Publications
REFERENCES A text book of Engineering Mathematics, B.V.Raman, Tata Mc Graw Hill Advanced Engineering Mathematics, Irvin Kreyszig, Wiley India Pvt. Ltd. A text Book of Engineering Mathematics, Thamson Book collection
UNIT-IV CHAPTER-I:CURVATURE,EVOLUTES AND ENVELOPESCHAPTER-II:CURVE TRACING
UNIT HEADER Name of the Course: B.Tech Code No:07A1BS02 Year/Branch: I YearCSE,IT,ECE,EEE,ME,CIVIL,AERO Unit No: IV No. of slides:19
UNIT INDEX UNIT-IVS. No. Module Lecture PPT Slide No. No. 1 Introduction, Curvature, L1-3 8-10 Radius of curvature 2 Centre of curvature, L4-9 11-15 Circle of curvature, Evolute and envelopes 3 Curve tracing in L10-12 16-19 cartesian form, polar form and parametric form
Lecture-1 CURVATURE Curvature is a concept introduced to quantify the bending of curves at any point. Note: The curvature at any point of the circle is equal to the reciprocal of its radius.The curvature of the circle decreases as the radius increases. Theorem: The curvature of a circle at any point on it is a constant.
Lecture-2 RADIUS OF CURVATURE The reciprocal of the curvature at any point of a curve is defined to be the radius of curvature at that point. Note: The radius of curvature of a circle of radius r at any point is r. Example 1: The radius of curvature at any point on the curve xy=c2 is (x2+y2)3/2/2xy Example 2: The radius of curvature at (3a/2,3a/2) of the curve x3+y3=3axy is 3√2a/16
Lecture-3 Formulae for RADIUS OF CURVATURE In cartesian form ρ=[1+(y′)2]3/2/y″ In polar form ρ=(r2+r12)3/2/(r2+2r12 –rr2) Example 1:r=a(1-Cosθ). Here ρ=4/3a Sinθ/2 Example 2:x=a(θ+Sinθ),y=a(1-Cosθ) at π/2. Here ρ=2√2a Example 3:x=a(Cost+tSint),y=a(Sint-tCost). Here ρ=at
Lecture-4 CENTRE OF CURVATURE The centre of curvature at any point P on a curve is the point which lies on the positive direction of the normal at P and is at a distance ρ from it. The centre of curvature at any point of a curve lies on the side towards which side the curve is concave. Example: Centre of curvature at (a/4,a/4) of the curve √x+√y=√a is a2/2
Lecture-5 Formula for CENTRE OF CURVATURE X=x – [y1(1+y12)]/y2 Y=y +[(1+y12)]/y2 Example 1:x3+y3=2 at (1,1). Here X=1/2, Y=1/2 Example 2:x=a(θ-Sinθ),y=a(1-Cosθ). Here X=a(θ+Sinθ), Y=-a(1-Cosθ)
Lecture-6 CIRCLE OF CURVATURE The circle of curvature at any point of a curve is the circle with centre at the centre of curvature at P and radius equal to the radius of curvature at the point. If (X,Y) be the centre and ρ be the radius of curvature, then the equation of the circle of curvature at the given point (x, y) is given by (x-X)2+(y-Y)2=ρ2 Example: y=x3+2x2+x+1 at (0,1).Here the circle of curvature is x2+y2+x-3y+2=0
Lecture-7 EVOLUTE The locus of the centre of curvature C of a variable point P on a curve is called the evolute of the curve. The curve itself is called Involute of the evolute. Example 1: x2/a2-y2/b2=1. Here the evolute is (ax)2/3-(by)2/3=(a2+b2)2/3 Example 2: x=a Cosθ,y=b Sinθ. Here the evolute is (ax)2/3+ (by)2/3=(a2-b2)2/3
Lecture-8 ENVELOPE Let f(x,y,c) be a function of three variables x,y,c. A curve which touches each member of a given family of curves is called envelope of that family. Example 1: y=mx+a/m where m is parameter. Here envelope is y2=4ax Example 2: (x/a)Cosθ+(y/b)Sinθ=1 where θ is parameter and a,b are constants. Here envelope is x2/a2+y2/b2=1
Lecture-9 CURVE TRACING The elementary method of drawing the graph of a plane curve is to tabulate the x and y values that satisfy the equation of the given curve, plot the points and join them by means of a smooth curve. A systematic and a more elegant method is the analytical method of studying the characteristics of the curve such as its symmetry, region, intersection with axes, tangents, asymptotes, concavity, etc., and then drawing its graph.
Lecture-10 TRACING CARTESIAN CURVES The general method of tracing a curve y=f(x) is as follows:1)Symmetry: Find out whether the given curve is symmetrical about x-axis,y-axis or any line.2)Origin:The curve passes through origin or not.3)Tangents: If the curve passes through the origin then the equations of the tangent at the origin are obtained by equating to zero, the lowest degree term in the equation of the curve.4)Asymptotes: Find all possible asymptotes.5) Points: Find the points by putting x=0,y=0.6)Rising and Falling: Find the region in which curve is rising and falling.7) Concavity: Find the curve is whether concavity upwards or downwards.
Lecture-11 TRACING POLAR CURVES 1)Symmetry: Find whether the curve is symmetrical about any line. 2)Region: Find the limits for r and θ. 3)Asymptotes: Find whether the curve has any asymptotes. 4)Points: Giving different values to θ.Find the corresponding values of r and determine the points where the tangent coincides with the radius vector or perpendicular to the radius vector.
Lecture-12 TRACING IN PARAMETRIC FORM 1)Symmetry:Check whether curve is symmetrical about any line or not. 2)Origin:Curve passes through origin or not. 3)Limitations of the curve:Find limits of given equation. 4)Region:Find the region of the given curves. 5)Tangents: Find tangents.