Engineering Mathematics keynotes, Summaries of Engineering Mathematics

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Engineering

Mathematics

Design Against Static Load

1. Basic Calculus ....................................................................................................................... 1 .1 – 1. 16

2. Ordinary Differential Equation ............................................................................................. 1 .17 – 1. 25

3. Vector Calculus ..................................................................................................................... 1.26 – 1. 34

4. Linear Algebra ...................................................................................................................... 1.35 – 1.

5. Probability and Statistics ...................................................................................................... 1.47 – 1. 58

6. Complex Calculus ................................................................................................................. 1.59 – 1.

ENGINEERING

MATHEMATICS

INDEX

Fig. 1.

(b) Concept of differentiability

A continuous function f ( x ) is said to be differentiable at x = a , if 𝑙𝑖𝑚

𝑥→𝑎

𝑓(𝑥)−𝑓(𝑎)

𝑥−𝑎

exists, that is, RHL and LHL exist at a point

under consideration in 𝑓 ′ (𝑥).

𝑥=𝑎

𝑥→𝑎

𝑓(𝑥)−𝑓(𝑎)

𝑥−𝑎

𝑓 ′ (𝑎) = 𝑡𝑎𝑛 𝜃, where 𝜃 is the angle made by the tangent to the curve at x=a with x – axis.

(c) Some Standard Derivatives

(i)

𝑑

𝑑𝑥

𝑛

𝑛− 1

(ii)

𝑑

𝑑𝑥

(iii)

𝑑

𝑑𝑥

(iv)

𝑑

𝑑𝑥

2

(v)

𝑑

𝑑𝑥

2

(vi)

𝑑

𝑑𝑥

(vii)

𝑑

𝑑𝑥

(viii)

𝑑

𝑑𝑥

− 1

1

√ 1 −𝑥

2

(ix)

𝑑

𝑑𝑥

− 1

− 1

√ 1 −𝑥

2

(x)

𝑑

𝑑𝑥

− 1

1

1 +𝑥

2

(xi)

𝑑

𝑑𝑥

− 1

− 1

1 +𝑥

2

(xii)

𝑑

𝑑𝑥

− 1

1

|𝑥|√𝑥

2

− 1

(xiii)

𝑑

𝑑𝑥

− 1

− 1

|𝑥|

√ 𝑥

2

− 1

; | x | > 1

(xiv)

𝑑

𝑑𝑥

𝑎

1

𝑥 𝑙𝑜𝑔 𝑒

𝑎

(xv)

𝑑

𝑑𝑥

𝑒

1

𝑥

(xvi)

𝑑

𝑑𝑥

𝑥

𝑥

𝑒

(xvii)

𝑑

𝑑𝑥

𝑥

𝑥

(xviii)

𝑑

𝑑𝑥

|𝑥|

𝑥

(xix)

𝑑

𝑑𝑥

𝑥

𝑥

𝑒

(xx)

𝑑

𝑑𝑥

(d) Product rule of differentiation

(i)

𝑑

𝑑𝑥

(ii) 𝑑(𝑢𝑣𝑤) = 𝑢𝑣𝑤 ′ + 𝑢𝑣 ′ 𝑤 + 𝑢 ′ 𝑣𝑤

(e) Quotient rule of differentiation

𝑑

𝑑𝑥

𝑓(𝑥)

𝑔(𝑥)

𝑔(𝑥).𝑓 ′ (𝑥)−𝑓(𝑥).𝑔 ′ (𝑥)

(𝑔

( 𝑥

) )

2

(f) Logarithmic differentiation:

Taking log might help in differentiation of a function. For example if

u

y = v

then we can take log both side and

differentiable to get

dy

dx

(g) Differentiation in parametric from :

If we write x and y in term of find variable ‘t’ that is x = f(t), y = (t), then

dy dy dt

dx dx dt

(h) Greatest Integer function / step function / integer part function

𝑓(𝑥) = [𝑥] = 𝑛, ∀ 𝑛 ≤ 𝑥 < 𝑛 + 1 where, 𝑛 ∈ 𝑍

𝑥→𝑎

[𝑥] = ∄ if a is an integer ( ∄ = do not exist)

L.H.L. = 𝑙𝑖𝑚

𝑥→𝑎

−

[𝑥] = 𝑎 − 1

R.H.L. = 𝑙𝑖𝑚

𝑥→𝑎

[𝑥] = 𝑎

(j) Some Standard Limits

(i) 𝑙𝑖𝑚

𝑥→ 0

𝑠𝑖𝑛 𝑥

𝑥

(ii) 𝑙𝑖𝑚

𝑥→ 0

𝑡𝑎𝑛 𝑥

𝑥

(iii) 𝑙𝑖𝑚

𝑥→ 0

1 −𝑐𝑜𝑠 𝑎𝑥

𝑥

2

𝑎

2

2

(iv) 𝑙𝑖𝑚

𝑥→ ∞

𝑠𝑖𝑛 𝑥

𝑥

(v) 𝑙𝑖𝑚

𝑥→ ∞

𝑐𝑜𝑠 𝑥

𝑥

(vi) 𝑙𝑖𝑚

𝑥→ 0

𝑏/𝑥

𝑎𝑏

(vii) 𝑙𝑖𝑚

𝑥→ ∞

𝑎

𝑥

𝑏𝑥

𝑎𝑏

(viii) 𝑙𝑖𝑚

𝑥→ 0

𝑎

𝑥

+𝑏

𝑥

2

1 /𝑥

(ix) 𝑙𝑖𝑚

𝑥→ 0

1

𝑥

• 2

𝑥

• 3

𝑥

+....+𝑛

𝑥

𝑛

1 /𝑥

𝑛

(x) 𝑙𝑖𝑚

𝑥→ 0

𝑎

𝑥

− 1

𝑥

𝑒

𝑥→ 0

𝑒

𝑥

− 1

𝑥

(xi) 𝑙𝑖𝑚

𝑥→ 0

1

𝑥

1.2 Mean Value Theorems

1.2.1 Lagrange’s Mean Value Theorem (LMVT):

If 𝑓(𝑥) is continuous in [a, b] and it is differentiable in (a, b) then ∃ at least one point ‘c’ such that c  (a, b) and

Here 𝑓 ′ (𝑐) slope of tangent to f ( x ) at x = c.

Tangent at x = c is parallel to the line connecting the points A and B

Fig.1. 3. LMVT

1.2.2 Rolle’s Mean Value Theorem

If 𝑓(𝑥) is continuous in [ a , b ] and differentiable in (a, b) and f ( a ) = f ( b ) then  at least one-point c  ( a , b ) such that 𝑓 ′ (𝑐) = 0.

Fig. 1. 4. Rolle’s mean value

1.2.3 Cauchy’s Mean Value Theorem

If f ( x ) and g ( x ) are continuous in [ a , b ] and differentiable in ( a , b ) then  at least one value of ‘c’ such that c  ( a , b ) and

𝑔 ′

( 𝐶

)

𝑓 ′

( 𝐶

)

𝑔

( 𝑏

) −𝑔

( 𝑎

)

𝑓

( 𝑏

) −𝑓

( 𝑎

)

1.3 Increasing and Decreasing Functions

1.3.1 Increasing Functions

A function f ( x ) is said to be increasing, if 𝑓

1

2

1

2

Or

A function f ( x ) is said to be increasing, if f ( x ) increases as x increases.

For a function 𝑓(𝑥) to be increasing at the point x=a, 𝑓

′

Example:

e

x

, log e

x → Monotonically increasing functions

sin x in (0, /2) → non-monotonic functions

1.3.2 Decreasing Functions

A function f ( x ) is said to be a decreasing function, if 𝑓

1

2

1

2

A function 𝑓

is said to be decreasing function, if 𝑓

decreases as x increases.

Example: 𝑒

−𝑥

→Monotonically decreasing function, sin 𝑥 in (

𝜋

2

1.4. Concept of Maxima and Minima

Let f ( x ) be a differentiable function, then to find the maximum (or) minimum of f ( x ).

(1) Find f ( x ) and equate to zero.

• Finding the expansion of e

x

about x = 0

𝑥

0

𝑥

0

𝑥

1

1!

2

1

2!

3

1

3!

𝑥

𝑥

1!

𝑥

2

2!

𝑥

3

3!

1.6 Integral Calculus

If F ( x ) is anti-derivative of f ( x ). That is, continuous and differentiable in ( a , b ), then we write ∫

𝑥=𝑏

𝑥=𝑎

𝐹(𝑎). Here f ( x ) is integrand

If 𝑓(𝑥) > 0 ∀𝑎 ≤ 𝑥 ≤ 𝑏, 𝑡ℎ𝑒𝑛 ∫

𝑏

𝑎

represents the shaded area in the given figure.

y = f ( ) x

x=a x=b

Fig.1. 6. Integration of continuous function

1.6.1 Mean Value Theorem of Integration

If f (x) is continuous in [ a , b ] and differentiable in ( a , b ) then ‘’ atleast one-point c ( a , b ) such that

∫ 𝑓

( 𝑥

)

𝑏

𝑎

𝑑𝑥

( 𝑏−𝑎

)

Fig. 1. 7. Mean value of integration

1.7. Newton-Leibnitz Rule

If f ( x ) is continuously differentiable and ( x ), ( x ) are two functions for which the 1

st

derivative exists, then

𝜓(𝑥)

𝜙(𝑥)

Example:

𝑑

𝑑𝑥

𝑥

2

𝑥

2

2

1.8. Some Standard Integrals

𝑛

𝑥

𝑛+ 1

𝑛+ 1

1

𝑥

𝑒

𝑓 ′ (𝑥)

𝑓(𝑥)

𝑒

𝑠𝑖𝑛 𝑥

𝑐𝑜𝑠 𝑥

𝑒

𝑒

𝑐𝑜𝑠 𝑥

𝑠𝑖𝑛 𝑥

𝑒

𝑒

𝑠𝑒𝑐 𝑥(𝑠𝑒𝑐 𝑥+𝑡𝑎𝑛 𝑥)

(𝑠𝑒𝑐 𝑥+𝑡𝑎𝑛 𝑥)

𝑒

𝑒

𝑥

𝑎

𝑥

𝑙𝑜𝑔 𝑒

𝑎

1

𝑥.𝑙𝑜𝑔 𝑒

𝑎

𝑎

𝑥

𝑒

𝑥

1

2

2

𝑓 ′

( 𝑥

)

√𝑓(𝑥)

15. If f ( x ), g ( x ) are two functions. that are differentiable, then

𝑑𝑥 − ∫[𝑓 ′

]𝑑𝑥 + 𝐶

Fig.1. 8. Length of the curve

(b) The length of the arc of the curve x = f ( ) y between the points where y = a and y = b, is

2

b

a

dx

s dy

dy



(c) The length of the arc of the curve x = f t ( ), y = f t ( ) between the points where t = a and t = b, is

2 2

b

a

dx dy

s dt

dt dt



(d) The length of the arc of the curve r = f ( ), between the points where  =  and  = , is

2

2

dr

s r d

d





 

 

= +   

 

    

 



1.11 Surface Area of Solid generated by revolving a curve about a fixed axis.

Elemental Surface Area

𝑑𝐴 = 2 𝜋𝑦 × 𝑑𝑠 = 2 𝜋𝑦𝑑𝑠

 Total surface area = A = ∫

𝑥=𝑏

𝑥=𝑎

𝑑𝑦

𝑑𝑥

2

Fig.1. 9. Surface area

1.12 Volume of the solid

A. The volume of the solid obtained by revolving the curve y = f ( x ) between the lines x = a and x = b is given by

2

𝑥=𝑏

𝑥=𝑎

Fig. 1. 10. Volume of the solid

B. Revolution about the y-axis. Interchanging x and y in the above formula, we see that the volume of the solid generated

by the revolution, about y-axis, of the area, bounded by the curve

x = f ( ), y the y-axis and the abscissa y = a, y = b is

2

b

a

 x dy



.

1.13 Gamma Function

The integral ∫

−𝑥

𝑛− 1

∞

0

𝑑𝑥, (𝑛 > 0 ) is called Gamma function of n. It is denoted by Γ𝑛 = ∫

−𝑥

𝑛− 1

∞

0

Note :

/

0

sin cos

m n

m n

x xdx

m n





Where (x) is called the gamma function.

1.13.1 Properties of Gamma Function

(i) Γ𝑛 = (𝑛 − 1 )! (ii) Γ(𝑛 + 1 ) = (𝑛)!

(iii) Γ(𝑛 + 1 ) = 𝑛Γ𝑛 (iv) Γ (

1

2

1.14 Beta Function

The function  ( m , n ) = ∫ 𝑥

𝑚− 1

𝑛− 1

1

0

𝑑𝑥 ( m , n > 0) is called  function of m and n.

Example: 𝑓

3

2

2

3

3

2

2

3

3

3

2

2

3

3

is a homogenous function of degree ‘3’.

(d) Euler’s Theorem

If f ( x , y ) is a homogeneous function of degree ‘ n ’ then

(i) 𝑥.

𝜕𝑓

𝜕𝑥

𝜕𝑓

𝜕𝑦

(ii) 𝑥

2

𝜕

2

𝑓

𝜕𝑥

2

𝜕

2

𝑓

𝜕𝑥𝜕𝑦

2

𝜕

2

𝑓

𝜕𝑦

2

If f ( x , y ) = g ( x , y ) + h ( x , y ) + ( x , y ) where g ( x , y ), h ( x , y ) and ( x , y ) are homogenous functions of degrees m, n and

p respectively, then

𝜕𝑓

𝜕𝑥

𝜕𝑓

𝜕𝑦

2

𝜕

2

𝑓

𝜕𝑥

2

𝜕

2

𝑓

𝜕𝑥𝜕𝑦

2

𝜕

2

𝑓

𝜕𝑦

2

(e) Total derivative:

(i) If u = f(x, y) and if x = (t), y = v(t) then..

du u dx u dy

dt x dt y dt

(ii) If u be a function of x and y, where y is a function of x, then.

du u u dy

dx x y dx

(iii) If u = f(x, y) and

1 1 2

x = f ( , t t )

and

2 1 2

y = f ( t , t ),

then

1 1 1

u u x u y

t x t y t

and

2 2 2

u u x u y

t x t y t

(iv) If x and y are connected by an equation of the form f(x, y) = 0, then

dy f x

dx f y

(f) Concept of Maxima and Minima in Two Variables

If f ( x , y ) is a two-variable differentiable function, then to find the maxima (or) minima.

Step-1: Calculate 𝑝 =

𝜕𝑓

𝜕𝑥

and 𝑞 =

𝜕𝑓

𝜕𝑦

and equate p = 0, q = 0

Let ( x 0

, y 0

) be a stationary point.

Step- 2 : Calculate r , s , t where 𝑟 =

𝜕

2

𝑓

𝜕𝑥

2

(𝑥

0

,𝑦

0

)

𝜕

2

𝑓

𝜕𝑥.𝜕𝑦

(𝑥

0

,𝑦

0

)

𝜕

2

𝑓

𝜕𝑦

2

(𝑥

0

,𝑦

0

)

Case (i): If 𝑟𝑡 − 𝑠

2

0 and r > 0, then the function f ( x , y ) has minimum at ( x 0

, y 0

) and the minimum value is f ( x 0

, y 0

Case (ii): If 𝑟𝑡 − 𝑠

2

0 and r < 0, then the function f ( x , y ) has maximum at ( x 0

, y 0

) and the maximum value is

f ( x 0

, y 0

Case (iii): If 𝑟𝑡 − 𝑠

2

< 0 ; then we cannot comment on the existence of maxima and minima.

Such stationary points where 𝑟𝑡 − 𝑠

2

= 0 are called saddle points.

(g) Concept of Constraint Maxima and Minima (Method of Lagrange’s unidentified multipliers).

If f ( x , y , z ) is a continuous and differentiable function, such that the variables x , y and z are related/constrained by the

equation ( x , y , z ) = C then to calculate the extreme value of f ( x , y , z ) using Lagrange’s Method of unidentified multipliers.

Step-1: Form the function F ( x , y , z ) = f ( x , y , z ) + {( x , y , z ) – C}, where 𝜆 is a multiplier.

Step-2: Calculate

𝜕𝐹

𝜕𝑥

𝜕𝐹

𝜕𝑦

and

𝜕𝐹

𝜕𝑧

and equate them to zero

Step- 3 : Equate the values of  from the above 3 equations and obtain the relation between the variables x , y and z.

Step- 4 : Substitute the relation between x , y and z in ( x , y , z ) = C and get the values of x , y , z. Let they be ( x 0

, y 0

, z 0

Step-5: Calculate f ( x 0

, y 0

, z 0

The value f ( x 0

, y 0

, z 0

) is the extreme value of f ( x , y , z ).

(h) Multiple Integrals

If f (x, y ) is continuous and differentiable at every point within a region ‘ R ’ bounded by some curves is given by

𝑅

If there is a double integral,

𝑦=𝜓(𝑥)

𝑦=𝜙(𝑥)

𝑥=𝑏

𝑥=𝑎

[Let ( x ) > ( x )]

Then I = area of the region bounded by the curves, y = ( x ); y = ( x ), x = a and x = b if f ( x , y ) = 1

Value of x – co-ordinate of centroid of the region bounded by y = ( x ); y = ( x ); x = a , x = b if f ( x , y ) = x

(i) Change of Orders of Integration

𝑦=𝜓(𝑥)

𝑦=𝜙(𝑥)

𝑥=𝑏

𝑥=𝑎

𝑥= ℎ (𝑦)

𝑥=𝑔(𝑦)

𝑦=𝑑

𝑦=𝑐

In change of order of Integration, the order of the Integrating variables changes and the limits as well.

Note : When limits are constants, the order of integration does not matter,

y d y d x b x b

y c x a x a y c

f x y dxdy f x y dydx

= = = =

= = = =

   

1.17 Jacobian of the Transformation

(i) The Jacobian of the transformation, ( )

1 2

x = f u v , , y = f ( , ) u v is defined as,

u v

u v

x x x y

J

u v y y

ORDINARY DIFFERENTIAL

EQUATION

2.1 Differential Equation

The equation involving differential coefficients is called a Differential Equation (DE).

2

𝑑𝑦

𝑑𝑥

2

𝜕

2

𝑇

𝜕𝑥

2

𝜕𝑇

𝜕𝑥

2

𝜕

2

𝑢

𝜕𝑥

2

2

𝜕

2

𝑢

𝜕𝑦

2

2.1.1 Ordinary Differential Equations (ODE)

The DEs involving only one independent variable is called ordinary differential equation.

Example:

2

𝑑𝑦

𝑑𝑥

2

−𝑥

𝑑𝑦

𝑑𝑥

2

𝑥

2.1.2 Partial Differential Equations

The DEs involving two (or) more independent variables are called Partial Differential Equations (PDEs).

Example:

𝜕

2

𝑢

𝜕𝑥

2

2

𝜕

2

𝑢

𝜕𝑡

2

𝜕

2

𝑢

𝜕𝑥

2

𝜕

2

𝑢

𝜕𝑦

2

1

𝐾

𝜕𝑢

𝜕𝑡

𝜕

2

𝑢

𝜕𝑥

2

𝜕

2

𝑢

𝜕𝑦

2

2.1.3 Order of a Differential Equation

The order of the highest derivative that occurs in a DE is called order of a DE.

Example:

𝑑

2

𝑦

𝑑𝑥

2

𝑑𝑦

𝑑𝑥

3

− 𝑦 = 0 → Order = 2

𝑑𝑦

𝑑𝑥

𝑑

2

𝑦

𝑑𝑥

2

𝑑

3

𝑦

𝑑𝑥

3

2

𝑥

→ Order = 3

𝜕

2

𝑢

𝜕𝑥

2

1

𝐶

2

𝜕

2

𝑢

𝜕𝑡

2

→ Order = 2

𝜕

2

𝑢

𝜕𝑥

2

1

𝛼

𝜕𝑢

𝜕𝑡

→ Order = 2

2.1.4 Degree of a Differential Equation

The Degree of the highest order derivative that occurs in a DE, when the DE is free from fractional (or) radical powers.

Example:

(1) The Degree of the DE (

𝑑

2

𝑦

𝑑𝑥

2

1

𝑑𝑦

𝑑𝑥

3

− 3 𝑦 = 0 is 1.

(2) The Degree of the DE (

𝑑

2

𝑦

𝑑𝑥

2

1

𝑑𝑦

𝑑𝑥

3

• 4 𝑦 = 0 is 2

𝑑

2

𝑦

𝑑𝑥

2

2

𝑑𝑦

𝑑𝑥

3

2

𝑑

2

𝑦

𝑑𝑥

2

2

𝑑𝑦

𝑑𝑥

3

(3) It is not possible every time that we can find the degree of a given differential equation. The degree of any differential

equation can be found, when it is in the form of a polynomial; otherwise, the degree cannot be defined.

Example degree of the DE is not defined

2 2

2 2

cos 5

d y d y

x

dx dx

2.2. Formation of Differential Equations

If a solution y = f ( x ) contains n arbitrary constants in it, then differentiate y for n times and calculate 𝑦 ′ , 𝑦", 𝑦 ′′′ ,.... 𝑦

(𝑛)

So, from

the ( n + 1) equations available, try to eliminate the arbitrary constants in y = f ( x )

• The different equation formed for the solution, 𝑦 = 𝐶 1

𝐾 1

𝑥

2

𝐾 2

𝑥

where C 1

, C

2

are arbitrary constants is

𝑑

2

𝑦

𝑑𝑥

2

1

2

𝑑𝑦

𝑑𝑥

1

2

• If the solution is 𝑦 = 𝐶

1

𝐾

1

𝑥

2

𝐾

2

𝑥

3

𝐾

3

𝑥

where C 1

, C

2

, C

3

are arbitrary constants, then the DE is 𝑦′′′ −

1

2

3

1

2

2

3

3

1

1

2

3

2.2.1 First Order DE

The general form of a 1

st

order DE is given by

𝑑𝑦

𝑑𝑥

If

𝑑𝑦

𝑑𝑥

𝑀(𝑥,𝑦)

𝑁(𝑥,𝑦)