Function,. Fourier. Sine. Series. Half. Range. Expans ions. Fourier. Cosine. Integral ... Functions. Differentiation of Functions. (Associated with multiplication).
CALCULUS
y x-axis
Calculus
a
Integration-x-axis-
Fundamental Theorem
Integration is along x-axis
Relates integration and differentiation
b
Line integral can be given as sum of integration along x-axis and y-axis
Line Integral y
b a
Integration is on a curve in a planecan be broken as an integration along x-axis and an integration along y -axis.
x-axis
Green’s Theorem Relates line integral with double integral.
similar
y
Region
Double Integral Integral is on a region in x-y plane.
Analogous to line integrals Surface integral can be given as sum of double integrations along x-y-plane , x-z-plane and y-z plane.
x-axis
Surface in space
Stokes’s Theorem Relates line integral to surface integral. Can be seen as generalization of Green’s theorem. Becomes Green’s theorem in plane region.
Surface Integral Can be broken up into double integrals in x-y, x-z and y-z planes (analogous to line integral).
Triple Integral Integration is in a volume in xyz space.
Divergence Theorem Relates surface integral with triple integralAnalogous to Green’s theorem.
2
Scalar b
y-axis
(F dx F dy F dz) 1
b
or y-axis
2
a
b
Representation
x-axis
(F1x'F2 y'F3 z' )dt (in parametric form).
a
a
b
x-axis
Line Integration
Integration along x-axis
Vector
Line integral is integral on a plane and can be given as a sum of integrals both along x and y axis.
b
f ( x)dx
a
b
b
a
a
Using ‘t’ as a parameter
1
A
Ft
D
F
A D
Fn
B E
Q
P
x
0
a)
Ft
dy
dx x
0
b)
Consider the case of a force acting on a particle in the x-y plane. If Ft is the tangential force and Fn is the normal force at a point D then to move a small distance ds, Ftds is the work done .
Integration can give the work done by a variable force along the x-axis in the case of integration along x-axis. Here the work done is found by integration along curve AB. The total work done is same as sum of work (work is scalar) along both x and y-axis.
F ds Pdx Qdy t
AB
AB
b
a
a
Path independence, Theorems ds
y
b
( r )' dr / dt
2
Understanding Line Integral using work done B
The value of the integral can be given as a vector rather than a scalar.
F(r) dr F(r(t)) r' (t) dt
Fds (F dx F dy) y
3
a
y-axis
y-axis
b a
a
b
x-axis
a
a
b
b
b
a
a
Pu ( x ) dx Pv ( x ) dx
Line Integral b
f ( x)dx
a
Pdx Pu ( x ) dx Pv ( x ) dx b
Integration -x-axis b
b
x-axis
a
P
u ( x)
Pv ( x ) dx
a
b
Fds ( F1dx F2 dy) a
Green’s Theorem P R y dydx Pdx, Also we find
Q
Double Integral
R
R
d b f ( x, y )dxdy f ( x, y )dx dy c a
b
Double integral is carried out by using the fundamental theory of computation, by which (see Fig.)
b v( x) P P dydx dy dx y y a u( x)
P
v( x) u( x)
a
b
dx
P
u ( x)
a
Pv ( x ) dx
x dydx Qdy
Hence we get
R
Q P R x y dydx Pdx Qdy
Fourier Series, Transform
Fourier Fourier Series, 2p
Fourier Series, 2p, Complex form
Fourier-Series, Integrals, Transforms Fourier Series of any Period, 2L
Even Function, Fourier Cosine Series Odd Function, Fourier Sine Series
Application-ODE
Approximation
Half Range Expans ions
Fourier Integral
Fourier Cosine Integral
Fourier Sine Integral
Fourier Integral, Complex Form
Fourier Transform
Fourier Cosine Transform
Fourier Sine Transform
Discrete/Fast Fourier Transform,
Fourier Series of any Period, 2L
Fourier Series, 2p
Fourier Integral, L→∞
Fourier Integral, Complex Form
Fourier Transform Separate exponential And re-arrange
1) Write in Cos (wx-wv) terms 2) Eqn. in Sin (wx-wv) terms is 0. 3) Change to complex
To change scale first Write equation of Fourier series using ‘v’.
Euler formulas To derive Euler’s formulas use orthogonality condition of Trigonometric system.
Writing v=x For even function
Substituting
For sin function
We get
To derive substitute above value of f(x) and find
Similarly to derive substitute above value of f(x) and find
g(v)=f(x) gives
Fourier Cosine Fourier Sine Series For even function, Cosine Series For odd function, Sine Series
Fourier Series, 2p, Complex form
Fourier Cosine Fourier Sine Integral For even function, Cosine Integral For odd function, Sine Integral
Fourier Cosine Fourier Sine Transform 1) Re-arranging the eqn. for Cosine Integral 2) Writing v=x
A(w), B(w) slightly diff.
Substituting
Writing v=x
Discrete/Fast Fourier Transform,
LAPLACE TRANSFORM
Existence • Growth restriction • Piecewise continuous
Kernel of Transform
Laplace Transforms
Transform Inverse of Transform
Transforms of Diff/Integration
Diff/Integration of Transforms
Operation on Functions Operation on Transforms
Transform of Functions
Addition of Transforms / S-shifting(Associated with multiplicati on of function by e..)
Differentiation of Functions (Associated with multiplication)
Integration of Functions (associated with Division)
Differential Equations Given problem
S-space Subsidiary Eqn.
Solve for y(t)
Solve for Y in S-space
t-space
Unit step Function/tshifting (Associated with multiplication of Transform by e..)
Dirac Delta function
Multiplication of Transforms/ Convolution
Differentiation of Transforms (Associated with multiplication)
Integration of Transforms (Associated with Division)
Kernel of Transform
Transform
Laplace Transforms
Inverse of Transform Transforms of Diff/Integration
Diff/Integration of Transforms
Operation on Functions (Trans of Funcs, Diff, Integration, Diff Eqn., Unit Step/Dirac Delta) Operation on Transforms, Add, Multiply, Diff., Integrate Addition of Transforms/ S-shifting-
Transform of Functions
(Associated with multiplication of function by e..)
Differentiation of Functions
Unit step Function/t-shifting
Integration of Functions
(Associated with multiplication)
(Associated with multiplication of Transform by e..)
(associated with Division)
Dirac Delta function
Multiplication of Transforms/ Convolution
Differentiation of Transforms (Associated with multiplication)
Integration of Transforms (Associated with Division)
Unit Step
For f(t)=1
Area=1
1
Transform
Cos wt
Sin wt
Cosh at (rem---with s)
u(t-a)
1
s-shifting
Function
1/k
u(t-0)=u(t)
Differential Equations
Sinh at
Input is r(t), Output is y Substituting above equations For Differentiation we get
m
t-space
S-space
Given problem
Subsidiary Eqn.
Solve for y(t)
Solve for Y in Sspace
k
m
m
a
f(t) =msin t
a+k
a
a
Then Y=RQ a
Special linear ODE’s
f(t-a)u(t-a)= f(t)u(t-a)= msin t u(t-a) msin(t-a) u(t-a)
t-shifting
For Non-homogenous Linear ODE’s If
Dirac-Delta
Kernel of Transform
Laplace Transform
(Examples from Kreyszig)
Transform Inverse of Transform
On Functions
Transform of Functions
Addition/ S-shifting
Transform of Functions
Addition
Differentiation of Functions
On Transforms
Integration of Functions
Differentiation
Dirac Delta/Impulse Differential Eqn
From Functions Find Transform
Indicates Integration
By differentiation formula
Unit step/tshifting
Indicates Shifting
By Integration formula
From (1) and (2)
Write f(t) in terms of unit step Use
Solving Inverses
Inverse of g(t)=1
Differentiation/ Integration of Transforms Since
Finding Transforms
Hence
Inverse Transforms ‘s’ and ‘t’ shift
t-shift
S-shifting
Multiplication of Transforms/ Convolution
Finding Inverses a) Given F(s) we find F’(s). SinceL tf(t)=-F’(s) we can find f(t)
s-shift a=3
Differential Eqn.
From s-shifting of Cos and Sin formulas,
a=-1, w=20
a1,w=20
Differential Equations
Differential Eqn. Same as normal Diff. eqn. except Input is unit step Function, f(t) obtained in steps as
For y(0)=y’(0)=0 Y=RQ
Knowing ‘q’ and ‘r’ we can solve in an alternate manner.
Special linear ODE’s With variable Coefficient
Example-Laguerre’s ODE
T-space
ProblemGiven-Diff. Eqn., Initial Values y”-y=t, y(0)=1, y’(0)=1
S-space Write Eqn. in ‘s’ terms
Re-arrange
Solution - S-space
Solution - t-space Integral Eqn.
Certain Integral Equations Can be solved by convolution
Can be written as
Since we know Transform of ty” and ty’ we can find Eqn. in s-space
COMPLEX ANALYSIS
Harmonic Conjugates From u one can get v and vice versa
Complex Laplace’s Eqn. Differentiating 1st Cauchy Riemann eqn. in x and 2nd in y we get one laplace eqn.
Limits, Diff. in Complex plane. Function
Hyp.
In real we approach along the axis-in complex we approach in a plane
Definition of Complex integral Integral is on complex plane (has some similarity with line integral). The integral can be written in terms of u and v.
Cauchy-Riemann
Cauchy-Integral Theorem
Path-Independence
Approaching limit Dx and Dy →0 and Dy and Dx →0 way and equating gives CauchyRiemann.
For analytic, simply connectedwriting the integral in terms of u and v, applying Green’s theorem and then Cauchy Riemann the integral is shown to be 0.
For analytic, simply connected integration from A to B and B to A is 0 (by Cauchy Integral Theorem ) –hence it can be shown that integration from A to B along two diff. paths give same result.
Derivatives Exp.
Trig.
Path-independence Dependence If function is not analytic integral it is path dependent. Thus integration of 1/(z-z0 ) along a unit circle |z|=1 is 2pi.
Log.
We can start from Exp. From exp. Eqns.we can get Trig. Eqns Trig. Eqns. are inter-related with. Hyp. Exp. are also related to Log.
Cauchy-Integral Formula The integral of f(z)/(z-z0 ) can be written as integral of f(z0)/(z-z0) and (f(z)f(z0)) /(z-z0 ) . The second integral is shown to be 0. Hence the answer to the integral of f(z)/(z-z0) is 2pif(z0) (see path depen-)
Power Series Representation
Liouville’s Theorem
Integration is carried about a circle C of radius ‘r’. M is greater than absolute value of f(z). Substituting values we get n!M/rn greater than the absolute value of fn (z0 ).
For an entire function if |f(z0 )|
