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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