Chapter 1
Basic Mathematics
Applied in Physics
QUADRATIC EQUATION
−b ± b 2 − 4ac
Roots of ax2 + bx + c = 0 are x =
2a
b c
Sum of roots x1 + x2 = − ; Product of roots x1x2 =
a a
BINOMIAL APPROXIMATION
If x << 1, then (1 + x)n ≈ 1 + nx & (1 – x)n ≈ 1 – nx
LOGARITHM
log mn = log m + log n
log m/n = log m – log n
log mn = n log m
logem = 2.303 log10m
log 2 = 0.3010
COMPONENDO AND DIVIDENDO LAW
p a p+q a+b
=If = then
q b p−q a −b
ARITHMETIC PROGRESSION-AP FORMULA
a, a + d, a + 2d, a + 3d, …, a + (n – 1)d, here d = common difference
n
Sum of n terms Sn = [2a + (n – 1)d]
2
1
, NOTES
n(n + 1)
(i) 1 + 2 + 3 + 4 + 5 … + n =
2
n(n + 1)(2n + 1)
(ii) 12 + 22 + 32 + … + n2 =
6
GEOMETRICAL PROGRESSION-GP FORMULA
a, ar, ar2, … here, r = common ratio
a(1 − r n )
Sum of n terms Sn =
1− r
a
Sum of ∞ terms S∞ =
1− r
TRIGONOMETRY
• 2p radian = 360° ⇒ 1 rad = 57.3°
1
• cosec q =
sin θ
1
• sec q =
cos θ
1
• cot q =
tan θ
• sin2q + cos2q = 1
• 1 + tan2q = sec2q
• 1 + cot2q = cosec2q
• sin (A ± B) = sin A cos B ± cos A sin B
• cos (A ± B) = cos A cos B ∓ sin A sin B
tan A ± tan B
• tan (A ± B) =
1 ∓ tan A tan B
• sin 2A = 2 sin A cos A
• cos 2A = cos2 A – sin2 A = 1 – 2 sin2A = 2 cos2 A – 1
2 tan A
• tan 2A =
1 − tan 2 A
sine law
sin A sin B sin C
= =
a b c
Hand Book (Physics) 2
, cosine law
b2 + c2 − a 2 c2 + a 2 − b2 a 2 + b2 − c2
= cos A = , cos B = , cos C
2bc 2ca 2ab
A
A
c b
B C
B a C
Approximation for small q
• sin q ≈ q
• cos q ≈ 1
• tan q ≈ q ≈ sin q
Differentiation
n dy
• y =x → = nx n −1
dx
dy 1
• y = lnx → =
dx x
dy
• y = sin x → = cos x
dx
dy
• y= cos x → = − sin x
dx
dy
• y= eαx +β → =αeαx +β
dx
dy dv du
• y = uv → =u + v (Pr oduct rule)
dx dx dx
dy df (g(x)) d(g(x))
y f (g(x)) → =
• = × (Chain rule)
dx dg(x) dx
dy
• y = k = constant ⇒ =0
dx
du dv
v −u
u dy dx dx
• y= ⇒ = (Division Rule)
v dx v2
3 Basic Mathematics Applied in Physics
,Integration
x n +1
• ∫x=
n
dx + C, n ≠ −1
n +1
1
• ∫ x=
dx
nx + C
• ∫ sin xdx =
− cos x + C
• ∫ cos xdx
= sin x + C
1 αx +β
∫ e=
αx +β
• dx e +C
α
(αx + β) n +1
• ∫ (α=
x + β) n dx +C
α(n + 1)
Maxima and Minima of a Function y = f(x)
dy d2 y
• For maximum value = 0 & 2 = − ve
dx dx
dy d2 y
• For minimum value = 0 & 2 = + ve
dx dx
Average of a Varying Quantity
x2 x2
If y = f(x) then < y=
>= y
∫=
x1
ydx ∫
x1
ydx
x2
∫ dx x
x1
2 − x1
Formulae for Calculation of Area
• Area of a square = (side)2
• Area of rectangle = length × breadth
• Area of a triangle = 1/2 × base × height
• Area of a trapezoid = 1/2 × (distance between parallel sides)
× (sum of parallel sides)
• Area enclosed by a circle = p r2 (r = radius)
• Surface area of a sphere = 4p r2 (r = radius)
• Area of a parallelogram = base × height
Hand Book (Physics) 4
, • Area of curved surface of cylinder = 2p rl (r = radius and l = length)
• Area of whole surface of cylinder = 2pr (r + l) (l = length)
• Area of ellipse = p ab (a & b are semi major and semi minor axis
repsectively).
• Surface area of a cube = 6(side)2
Volume of Geometrical Figures
4 3
• Volume of a sphere = p r (r = radius)
3
• Volume of a cylinder = p r2 l (r = radius and l = length)
1
• Volume of a cone = p r2 h (r = radius and h = height)
3
Some Basic Plots
• Straight Line
y = mx + c (where m is the slope of the line and c is the y intercept)
y
c
x
0
• Parabola
y = ax2
5 Basic Mathematics Applied in Physics
,• Exponential function
y = ex
y
(0, 1)
x
10 10
8 8 x
1
y = 2x 6 6 y= 2
4 4
2 2
x x
–4 –3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3 4
KEY TIPS
• To convert an angle from degree to radian, we should multiply it
by p/180° and to convert an angle from radian to degree, we should
multiply it by 180°/p.
• By help of differentiation, if y is given, we can find dy/dx and by help
of integration, if dy/dx is given, we can find y.
• The maximum and minimum values of function A cos q + B sin q are
A 2 + B2 and − A 2 + B2 respectively.
• (a + b)2 = a2 + b2 + 2ab
• (a – b)2 = a2 + b2 – 2ab
• (a + b) (a – b) = a2 – b2
• (a + b)3 = a3 + b3 + 3ab (a + b)
• (a – b)3 = a3 – b3 – 3ab (a – b)
qqq
Hand Book (Physics) 6
, Chapter 2
Vectors
VECTOR QUANTITIES
A physical quantity which requires magnitude and a particular direction for
its complete expression.
B Head
Arrow shows Direction
AB
Length shows Magnitude
A
Tail
Triangle Law of Vector Addition R = A + B
R= A 2 + B2 + 2ABcos θ
R
Bsin θ B B sin q
tan α =
A + Bcos θ a q
θ θ A B cos q
If A = B then R = 2A cos &a=
2 2
Rmax = A + B for q = 0°; Rmin = A – B for q = 180°
Parallelogram Law of Vector Addition
If two vectors are represented by two adjacent sides of a parallelogram which
are directed away from their common point then their sum (i.e. resultant
vector) is given by the diagonal of the paralellogram passing away through
that common point.
7
Basic Mathematics
Applied in Physics
QUADRATIC EQUATION
−b ± b 2 − 4ac
Roots of ax2 + bx + c = 0 are x =
2a
b c
Sum of roots x1 + x2 = − ; Product of roots x1x2 =
a a
BINOMIAL APPROXIMATION
If x << 1, then (1 + x)n ≈ 1 + nx & (1 – x)n ≈ 1 – nx
LOGARITHM
log mn = log m + log n
log m/n = log m – log n
log mn = n log m
logem = 2.303 log10m
log 2 = 0.3010
COMPONENDO AND DIVIDENDO LAW
p a p+q a+b
=If = then
q b p−q a −b
ARITHMETIC PROGRESSION-AP FORMULA
a, a + d, a + 2d, a + 3d, …, a + (n – 1)d, here d = common difference
n
Sum of n terms Sn = [2a + (n – 1)d]
2
1
, NOTES
n(n + 1)
(i) 1 + 2 + 3 + 4 + 5 … + n =
2
n(n + 1)(2n + 1)
(ii) 12 + 22 + 32 + … + n2 =
6
GEOMETRICAL PROGRESSION-GP FORMULA
a, ar, ar2, … here, r = common ratio
a(1 − r n )
Sum of n terms Sn =
1− r
a
Sum of ∞ terms S∞ =
1− r
TRIGONOMETRY
• 2p radian = 360° ⇒ 1 rad = 57.3°
1
• cosec q =
sin θ
1
• sec q =
cos θ
1
• cot q =
tan θ
• sin2q + cos2q = 1
• 1 + tan2q = sec2q
• 1 + cot2q = cosec2q
• sin (A ± B) = sin A cos B ± cos A sin B
• cos (A ± B) = cos A cos B ∓ sin A sin B
tan A ± tan B
• tan (A ± B) =
1 ∓ tan A tan B
• sin 2A = 2 sin A cos A
• cos 2A = cos2 A – sin2 A = 1 – 2 sin2A = 2 cos2 A – 1
2 tan A
• tan 2A =
1 − tan 2 A
sine law
sin A sin B sin C
= =
a b c
Hand Book (Physics) 2
, cosine law
b2 + c2 − a 2 c2 + a 2 − b2 a 2 + b2 − c2
= cos A = , cos B = , cos C
2bc 2ca 2ab
A
A
c b
B C
B a C
Approximation for small q
• sin q ≈ q
• cos q ≈ 1
• tan q ≈ q ≈ sin q
Differentiation
n dy
• y =x → = nx n −1
dx
dy 1
• y = lnx → =
dx x
dy
• y = sin x → = cos x
dx
dy
• y= cos x → = − sin x
dx
dy
• y= eαx +β → =αeαx +β
dx
dy dv du
• y = uv → =u + v (Pr oduct rule)
dx dx dx
dy df (g(x)) d(g(x))
y f (g(x)) → =
• = × (Chain rule)
dx dg(x) dx
dy
• y = k = constant ⇒ =0
dx
du dv
v −u
u dy dx dx
• y= ⇒ = (Division Rule)
v dx v2
3 Basic Mathematics Applied in Physics
,Integration
x n +1
• ∫x=
n
dx + C, n ≠ −1
n +1
1
• ∫ x=
dx
nx + C
• ∫ sin xdx =
− cos x + C
• ∫ cos xdx
= sin x + C
1 αx +β
∫ e=
αx +β
• dx e +C
α
(αx + β) n +1
• ∫ (α=
x + β) n dx +C
α(n + 1)
Maxima and Minima of a Function y = f(x)
dy d2 y
• For maximum value = 0 & 2 = − ve
dx dx
dy d2 y
• For minimum value = 0 & 2 = + ve
dx dx
Average of a Varying Quantity
x2 x2
If y = f(x) then < y=
>= y
∫=
x1
ydx ∫
x1
ydx
x2
∫ dx x
x1
2 − x1
Formulae for Calculation of Area
• Area of a square = (side)2
• Area of rectangle = length × breadth
• Area of a triangle = 1/2 × base × height
• Area of a trapezoid = 1/2 × (distance between parallel sides)
× (sum of parallel sides)
• Area enclosed by a circle = p r2 (r = radius)
• Surface area of a sphere = 4p r2 (r = radius)
• Area of a parallelogram = base × height
Hand Book (Physics) 4
, • Area of curved surface of cylinder = 2p rl (r = radius and l = length)
• Area of whole surface of cylinder = 2pr (r + l) (l = length)
• Area of ellipse = p ab (a & b are semi major and semi minor axis
repsectively).
• Surface area of a cube = 6(side)2
Volume of Geometrical Figures
4 3
• Volume of a sphere = p r (r = radius)
3
• Volume of a cylinder = p r2 l (r = radius and l = length)
1
• Volume of a cone = p r2 h (r = radius and h = height)
3
Some Basic Plots
• Straight Line
y = mx + c (where m is the slope of the line and c is the y intercept)
y
c
x
0
• Parabola
y = ax2
5 Basic Mathematics Applied in Physics
,• Exponential function
y = ex
y
(0, 1)
x
10 10
8 8 x
1
y = 2x 6 6 y= 2
4 4
2 2
x x
–4 –3 –2 –1 0 1 2 3 –3 –2 –1 0 1 2 3 4
KEY TIPS
• To convert an angle from degree to radian, we should multiply it
by p/180° and to convert an angle from radian to degree, we should
multiply it by 180°/p.
• By help of differentiation, if y is given, we can find dy/dx and by help
of integration, if dy/dx is given, we can find y.
• The maximum and minimum values of function A cos q + B sin q are
A 2 + B2 and − A 2 + B2 respectively.
• (a + b)2 = a2 + b2 + 2ab
• (a – b)2 = a2 + b2 – 2ab
• (a + b) (a – b) = a2 – b2
• (a + b)3 = a3 + b3 + 3ab (a + b)
• (a – b)3 = a3 – b3 – 3ab (a – b)
qqq
Hand Book (Physics) 6
, Chapter 2
Vectors
VECTOR QUANTITIES
A physical quantity which requires magnitude and a particular direction for
its complete expression.
B Head
Arrow shows Direction
AB
Length shows Magnitude
A
Tail
Triangle Law of Vector Addition R = A + B
R= A 2 + B2 + 2ABcos θ
R
Bsin θ B B sin q
tan α =
A + Bcos θ a q
θ θ A B cos q
If A = B then R = 2A cos &a=
2 2
Rmax = A + B for q = 0°; Rmin = A – B for q = 180°
Parallelogram Law of Vector Addition
If two vectors are represented by two adjacent sides of a parallelogram which
are directed away from their common point then their sum (i.e. resultant
vector) is given by the diagonal of the paralellogram passing away through
that common point.
7
