Summary (MTH301)

Revision

Equation of tangent plane: 0 0 0 0 0 0 0 0 0 0 0 0( , , )( ) ( , , )( ) ( , , )( ) 0f x y z x x f x y z y y f x y z z z

Critical Points: partial derivative -> fx=fy=0 -> get x,y are critical points

Extrema, Minima, Saddle Point: find fxx, fyy, fxy, and 2

.xx yy xyD f f f

0 00, ( , ) 0xxD D x y Relative min

0 00, ( , ) 0xxD D x y Relative max

0D Saddle point, 0D no conclusion

Lecture 23

Conversion formulas

Polar -> Cartesian Cartesian -> polar Rectangular -> cylindrical Rectangular -> spherical

cos

sin

x r

y r

2 2

1tan , tan

x y r

y y

x x

2 2 2

1tan /

/

x y z

y x

z

2 2

1tan /

r x y

y x

z z

Spherical->Rectangular Cylindrical->Rectangular Cylindrical -> Spherical Spherical -> Cylindrical

sin cos

sin sin

cos

x

y

z

cos

sin

x r

y r

z z

2 2

1tan ( / )

r z

Z

sin

cos

r

z r

Polar = ( , )r , Cartesian = ( , )x y , Spherical = ( , , )r , Cylindrical = ( , , )z , Rectangular = ( , , )x y z

General polar form of the line

( cos sin ) 0r A B C

Polar equation of circle

0 0

2 2 2

0 0 0

,      ( , ),          ( , )

2 cos( )

r a polar cordinates r P anarbitrary pont P r

r rr r a

Lecture 24

Sketching of curves in Polar coordinates

1-symmetry

Symmetry about initial line(x-axis): equation unchanged, replaced by .

Symmetry about y-axis: replaced by .

Symmetry about the pole: r Replaced by –r.

2-Position of the pole: check it passing through the pole or not.

3-Construct the sufficiently complete table of values.

Cardioids (heart shape) Lima cons

(1 cos )r a ------- left side (in 2nd & 3rd quadrant) sinr a b ------- upper side (in 1st & 2nd

quadrant )

(1 cos )r a ------- right side (in 1st & 4th quadrant) sinr a b ------- bottom side (in 3rd & 4th

quadrant)

(1 sin )r a ------- upper side (in 1st & 2nd quadrant) cosr a b ------- left side (in 2nd & 3rd quadrant)

(1 sin )r a ------- bottom side (in 3rd & 4th quadrant) cosr a b ------- right side (in 1st & 4th

quadrant)

Leminscate (8 shape) Spiral

2 2

2 2

2 2

2 2

cos 2

cos 2

sin 2

sin 2

r a

r a

r a

r a

r a , theta is in radian, a is positive

Rose curves

sin

cos

r a n

r a n

n equally spaced petals of loops if n is odd.

2n equally spaced petals of loops if n is even.

Lecture 25

Integral In polar coordinates

2

1

( )

( )( , ) ( , )

r r

r rA

f r dA f r drd

How to find limits of integration from sketch

Step 1. Since θ is held fixed for the first integration, draw a radial line from the origin through the region R at a fixed angle θ. This line crosses the boundary of R at most twice. The innermost point of intersection is one the curve r = r1(θ) and the outermost point is on the curve r = r2(θ). These intersections determine the r-limits of integration. Step 2. Imagine rotating a ray along the positive x-axis one revolution counterclockwise about the origin. The smallest angle at which this ray intersects the region R is θ = α and the largest angle is θ = β. This yields the θ-limits of the integration. See the examples on page 133

Changing Cartesian integrals into polar integrals

( , ) ( cos , sin )R R

f x y dxdy f r r rdrd

cos

sin

x r

y r

dxdy rdrd See the examples on page 135 & in lec 26

Lecture 27

Vector valued function & Parametric Equations in vector form

( )z z t k ( )y y t j ( )x x t i parametric form

( ) ( ( ), ( ), ( )) ( ) ( ) ( )r t x t y t z t x t i y t j z t k for three dimension

( ) ( ( ), ( )) ( ) ( )r t x t y t x t i y t j in 2D

Cross product

i×i=j×j=k×k=0

i×j=k j×i=-k

j×k=I k×j=-i

k×i=j i×k=-j

Dot Product

i.i = j.j = k.k =1

i.j = j.i = 0

j.k = j.k = 0

k.i = i.k = 0

Lecture 28

Limits of vector valued function

0 0 0 0lim ( ) (lim ( ) lim ( ) lim ( ) )t t t t

r t x t i y t j z t k

Continuity

0( )r t is defined, 0

lim ( )t

r t

exists, 0

0lim ( ) ( )t

r t r t

Derivative

'

0

( ) ( )( ) lim

h

dt r t h r tr t

dx h

' ' ' '( ) ( ) ( ) ( )r t x t i y t j z t k

Tangent Vectors and tangent line

0 0( ) '( )r r t tr t

Theorem

( ). '( ) 0r t r t

Integral of vector valued function

( ) ( ) ( ) ( )

b b b b

a a a a

r t dt x t dt i y t dt j z t dt k

( ) ( ) ( )

bb

a

a

r t dt Rt R b R a

Properties of Derivative & Antiderivative

Lecture 29

Arc Length formula & its theorem

2 2 2 2 2

,

b b

a a

dx dy dx dy dzL dt L dt

dt dt dt dt dt

Lecture 30

Exact Differential

( , ),

,   

      differenctial

z zz f x y dz dx dy

x y

z zP Q

x y

P Qexact

x x

Integration of Exact Differential

   

       

b b

a a

dz Pdx Qdy

z Pdx also z Qdy

Area enclosed by the closed curve

1 2

1 2

y=f(x), y=F(x)

A ( ) , A ( )

A A

b b

a a

f x dx F x dx

A ydx

Lecture 31

Line Integral

( )

( )    

tAB AB

C

Fds Pdx Qdy Rdw

I Pdx Qdy Rdw General form

Properties of Line Integral

1. ( )

2.

3. , dx=0

0

4. , dy=0

0,

5.

tC C

t tAB BA

CC C

CC C

C AB KA KB

F ds Pdx Qdy Rdw

F ds F ds

x k

Pdx I Qdy

y k

Qdy I Pdx

I I I I

Lecture 32

Line Integral w.r.t arc length

2

1

2

( , ) ( , ) 1x

c x

dxf x y dx f x y dx

dt

Parametric Equation

2

1

2 2

( , ) ( , )x

c x

dx dyI f x y ds f x y dt

dt dt

Lecture 33

Exact Differential in 3 dimension

,    ,    z z z

P Q Rx y w

( , , ),z z z

z f x y w dz dx dy dwx y w

,    ,    P Q P R R Q

y x w x y w

Green’s Theorem ( )R C

P Qdxdy Pdx Qdy

y x

Lecture 34

Gradient of a scalar function

del operator i j kx y z

grad i j k i j kx y z x y z

(Div)Divergence of a vector function

1 2 3A a i a j a k ,

1 2 3. ( )divA A i j k a i a j a kx y z

,

31 2.aa a

Ax y z

The grad operator acts on scalar and gives vector.

The div operator acts on vector and gives scalar.

(Curl) Curl of a vector function

1 2 3A a i a j a k curl A

1 2 3( )A i j k a i a j a kx y z

3 32 1 2 1

1 2 3

i j k

a aa a a acurl A i j k

x y z y z z x x y

a a a

Lecture 35

Wallis sine formula

n is even = {2,4,6,8,10,12………}

2

0

1 3 5 7 9 5 3 1sin . . . . ............. . . .

2 4 6 8 6 4 2 2

n n n n n nxdx

n n n n n

n Is odd = {1,3,5,7,9,11,13,……}

2

0

1 3 5 7 9 6 4 2cos . . . . ............. . .

2 4 6 8 7 5 3

n n n n n nxdx

n n n n n

2 22 2

0 0

3 32 2

0 0

4 42 2

0 0

5 52 2

0 0

6 62 2

0 0

7 72 2

0 0

8

1cos . sin

2 2

2cos sin

3

3 1cos . . sin

4 2 2

4 2cos . sin

5 3

5 3 1cos . . . sin

6 4 2 2

6 4 2cos . . sin

7 5 3

7 5 3cos . . .

8 6 4

xdx xdx

xdx xdx

xdx xdx

xdx xdx

xdx xdx

xdx xdx

xdx

82 2

0 0

9 92 2

0 0

10 102 2

0 0

11 112 2

0 0

12 122 2

0 0

13

1. sin

2 2

8 6 4 2cos . . . sin

9 7 5 3

9 7 5 3 1cos . . . . . sin

10 8 6 4 2 2

10 8 6 4 2cos . . . . sin

11 9 7 5 3

11 9 7 5 3 1cos . . . . . . sin

12 10 8 6 4 2 2

cos

xdx

xdx xdx

xdx xdx

xdx xdx

xdx xdx

xdx

132 2

0 0

14 142 2

0 0

15 152 2

0 0

16 162

0 0

12 10 8 6 4 2. . . . . sin

13 11 9 7 5 3

13 11 9 7 5 3 1cos . . . . . . . sin

14 12 10 8 6 4 2 2

14 12 10 8 6 4 2cos . . . . . . sin

15 13 11 9 7 5 3

15 13 11 9 7 5 3 1cos . . . . . . . . sin

16 14 12 10 8 6 4 2 2

xdx

xdx xdx

xdx xdx

xdx xdx

2

17 172 2

0 0

18 182 2

0 0

16 14 12 10 8 6 4 2cos . . . . . . . sin

17 15 13 11 9 7 5 3

17 15 13 11 9 7 5 3 1cos . . . . . . . . . sin

18 16 14 12 10 8 6 4 2 2

xdx xdx

xdx xdx

Lecture 36

Scalar field

Line integral ( )CV r dr       ( ),   ( ),   ( )dr dxi dyj dzk x x u y y u z z u

Vector field

1 2 3. ( )                    C C

F dr Fdx Fdy Fdz F Fi F i F i dr dxi dyj dzk

Lecture 37

Surface area integral by Scalar field

Surface area integral by Vector field

Lecture 38

Conservative Vector field

1 1( ) ( ). . 0

C AB C ABF dr F dr

Check whether or not a vector field is conservative

( )    . 0       (b)   curlF=0       F=gradVa F dr anyone of these condition is applied as is convenient.

Divergence (Guass’s) theorem: .V S

divFdV F dS

Stokes Theorem: . .S CcurlF dS F dr

Lecture 39

Periodic Function:If function values repeat at regular intervals of the independent variable. Regular interval b/w

repetitions is the period of the oscillation ( ) ( )f x p f x : P is a period

Useful integrals and their summary

Lecture 40-41

Fourier series

Periodic function of period 2

0

1

( ) { cos sin }n n

n

f x A a nx b nx

A0 , an, bn

Expanded form 0 1 2 1 2( ) cos cos2 ..... sin sin 2 ...... sin .....nf x A a x a x b x b x b nx

Fourier coefficients

Dirichlet Conditions

a) The function f(x) must be defined and single valued.

b) F(x) must be continuous or have a finite number of finite discontinuous within a periodic table.

c) F(x) must be piecewise continuous in periodic interval.

If these conditions are satisfied, the series converges to f(x), if 1x x is a point of continuity.

Effects of harmonic (on page no 204)

Odd function Even Function

2

( ) ( )

( )

( 9) 81 (9)

( 12) 144 (12)

f x f x

f x x

f f

f f

3

( ) ( )

( )

( 2) 8 (2)

( 3) 27 (3)

f x f x

f x x

f f

f f

Product of Odd & Even Function (Rules)

E = even, O = odd, * = multiply, (-) = Minus, (+) = plus N = neither

E * E = E ---- + * + = +, O * O = E ------ (-) * (-) = (+), O * E = O ----- (-) * (+) = (-), O * N = N, N * O = N

Two useful facts emerge from odd & even functions

Theorem 1

If f(x) is defined over the interval x and f(x) is even, then the Fourier series f(x) contains cosine terms only.

Included in this is a0 which may be regarded as an cos nx with n=0

Theorem 2

If f(x) is an Odd function defined over the interval x , then the Fourier series for f(x) contains sine terms only

Lecture 42

Sum of Fourier series at a point of discontinuity

Half-Range Series

Lecture 43

Function with period T

0

1

0

1

1( ) { cos sin }

2

           

1( ) sin( )      1,2,3,4.......

2

n n

x

n

x

f t a a n t b n t

which can be written as

f t A B n t n

Fourier coefficients

Half-Range Series

Half-wave rectifier (read form book on page 220)

Lecture 44-45

Inverse Transform