The document provides an overview of the contents and references for the Mathematics-I course. The contents include topics like ordinary differential equations, linear differential equations, mean value theorems, functions of several variables, curvature, evolutes, envelopes, curve tracing, integration, multiple integrals, series and sequences, vector differentiation and integration. It also lists several textbooks and references for further study.

1. MATHEMATICS-I

2. 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

3. 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

4. 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

5. UNIT-IV
CHAPTER-I:CURVATURE,
EVOLUTES AND ENVELOPES
CHAPTER-II:CURVE TRACING

6. UNIT HEADER
Name of the Course: B.Tech
Code No:07A1BS02
Year/Branch: I Year
CSE,IT,ECE,EEE,ME,CIVIL,AERO
Unit No: IV
No. of slides:19

7. UNIT INDEX
UNIT-IV
S. 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

8. 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.

9. 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

10. 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

11. 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

12. 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)

13. 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

14. 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

15. 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

16. 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.

17. 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.

18. 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.

19. 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.