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Maths Required for Physics, Summaries of Physics

University of Pretoria Physics

This Document has all the maths you need to understand to cope with physics

Typology: Summaries

2017/2018

Uploaded on 03/04/2018

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01 Math Intro Transparencies.doc: 1.1 Mathematical symbols and constants

1

06/02/2017 13:36:00

01 Math Intro Transparencies.doc: 1.1 Mathematical symbols and constants

1

06/02/2017 13:36:00

1 Mathematical

introduction 1 Wiskundige

inleiding

Raffaello Sanzio (1483-1520): Scuola di Atene

“If I have seen farther than others it is by standing on the shoulders of giants.”

– Isaac Newton

Discover Summaries of Physics University of Pretoria

Partial preview of the text

Download Maths Required for Physics and more Summaries Physics in PDF only on Docsity!

01 Math Intro Transparencies.doc: 1.1 Mathematical symbols and constants 1 06/02/2017 13:36:

1 Mathematical

introduction

1 Wiskundige

inleiding

Raffaello Sanzio (1483-1520): Scuola di Atene

“If I have seen farther than others it is by standing on the shoulders of giants.”

Isaac Newton

01 Math Intro Transparencies.doc: 1.1 Mathematical symbols and constants 2 06/02/2017 13:36:

1.1 Mathematical symbols and

constants

1.1 Wiskundige simbole en

konstantes

Mathematical symbols and constants Wiskundige simbole en konstantes

is equal to = is gelyk aan is not equal to (^) ≠ is ongelyk aan is approximately equal to (^) ≈ is ongeveer gelyk aan is identical to (or the definition of) (^) ≡ is identies gelyk aan (of die definisie van) is of the same order of magnitude as (^) ∼ is van dieselfde grooteorde as is proportional to (^) ∝ is regeweredig aan is greater than > is groter as is much greater than >> is baie groter as is less than < is kleiner as is much less than << is baie kleiner as is greater than or equal to (^) ≥ is groter as of gelyk aan is less than or equal to (^) ≤ is kleiner as of gelyk aan because (^)  omdat therefore (^) ∴ daarom this implies (^) ⇒ dit impliseer of daaruit volg tends to (^) → stewe na infinitely large (^) ∞ oneindig groot

01 Math Intro Transparencies.doc: 1.1 Wiskundige simbole en konstantes 4 06/02/2017 13:36:

The Greek alphabet: Die Griekse alfabet:

Α α alpha /^ alfa^ Ν ν nu

Β β beta^ Ξ ξ xi

Γ γ gamma^ Ο ο omicron /^ omikron

∆ δ delta^ Π π pi

Ε ε epsilon^ Ρ ρ rho

Ζ ζ zeta^ Σ σ sigma

Η η eta^ Τ τ tau

Θ θ theta^ Υ υ upsilon

Ι ι iota^ Φ φ phi

Κ κ kappa^ Χ χ chi

Λ λ lambda^ Ψ ψ psi

Μ μ mu^ Ω ω omega

01 Math Intro Transparencies.doc: 1.2` Exponents, logarithms and factorisation 5 06/02/2017 13:36:

1.2` Exponents, logarithms and

factorisation

1.2 Eksponente, logaritmes en

faktorisering

Rules for exponents: Reëls vir eksponente

( )

( ) pq p q

q q^ p q p

p

p q p q

p p p

n n

n

a a

a a a

ab a b

a

a a

a

n

n a a a a a

÷ =

× =

= ×

= × × × ×

−

(If isaninteger)

times ...

0

1 /

01 Math Intro Transparencies.doc: 1.2 Eksponente, logaritmes en faktorisering 7 06/02/2017 13:36:

Rules for logarithms: Reëls vir logaritmes:

1

log log

log log log 1 0 log (1/ ) log log ( ) log log log ( / ) log log log (log ) (log ) (log ) (log )

n a a n a (^) n a a a a a a a a a a a b a b b

P n P

P P

P P

P Q P Q

P P b P a

× = +

= × = ÷

In the above equations, P , Q , a and b are positive numbers, and a , b ≠ 1.

In die vergelykings hierbo, is P , Q , a en b positiewe getalle, met a , b ≠ 1.

Factorisation: Faktorisering:

a

b b ac ax bx c a x x x x x

a b a b a b

( )( ) where

2 2

− ± −

- = − − =

− + ±

01 Math Intro Transparencies.doc: 1.3 Radian measure 8 06/02/2017 13:36:

1.3 Radian measure 1.3 Radiaalmaat

Measurement of rotation: Meting van rotasie:

angle hoek reference line verwysingslyn vertex hoekpunt

O

B

A

01 Math Intro Transparencies.doc: 1.4 Trigonometric relationships 10 06/02/2017 13:36:

1.4 Trigonometric relationships 1.4 Trigonometriese verbande

r

x'

y'

O

y

x

P(x', y')

r

y ′

sinθ = ,

r

x ′

cos θ = ,

x

y ′

tanθ =

Graphs of trigonometric functions: Grafieke van trigonometriese funksies:

y = sin θ y = cos θ

y

θ

- 2 π - π π^2 π -

01 Math Intro Transparencies.doc: 1.4 Trigonometriese verbande 11 06/02/2017 13:36:

Identities: Identiteite:

cos( ) cos cos sin sin

sin( ) sin cos cos sin

sin cos 1

cos

sin tan 2 2

Inverse functions: Inverse funksies:

θ =arcsin( y ′/ r ) or θ = sin −^1 ( y ′/ r )

θ =arccos( x ′/ r ) or θ = cos −^1 ( x ′/ r )

θ =arctan( y ′/ x ′ ) or θ = tan −^1 ( y ′/ x ′)

sin (sin )

a = a

01 Math Intro Transparencies.doc: 1. 4 Trigonometriese verbande 13 06/02/2017 13:36:

01 Math Intro Transparencies.doc: 1.4 Trigonometriese verbande 13 06/02/2017 13:36:

Approximations for small angles Benaderings vir klein hoeke

a

r

t s

A

D

B C

For small angles ( θ << 1 ) we have t ≈ s and r ≈ a. So

we can write:

Vir klein hoeke ( θ << 1 ) is t ≈ s en r ≈ a. Ons kan

dus skryf:

r

s r

t sin

r

s a

t tan

cos = ≈ = 1 r

r r

a

Note: The approximations for sin and tan work only

if θ is in radians!

NB: Hierdie benaderings vir sin en tan werk slegs as

θ in radiale gemeet word!

01 Math Intro Transparencies.doc: 1.4 Trigonometriese verbande 14 06/02/2017 13:36:

Geometric formulae Meetkundige formules

Rectangle, length a and width b Area = ab

b

a

Reghoek met lengte a en breedte b Oppervlakte = ab

Parallelogram, base b and height h

Area = bh = ab sin θ

h

b

a

Parallelogram met basis b en hoogte h

Oppervlakte = bh = ab sin θ

Triangle, base b and height h

Area = 21 bh = 21 bc sin θ

a

b

c

h

Driehoek met basis b en hoogte h

Oppervlakte = 21 bh = 21 bc sin θ

01 Math Intro Transparencies.doc: 1.4 Trigonometriese verbande 16 06/02/2017 13:36:

Annulus, inner radius r , outer radius R Area =π( R^2 − r^2 )

=π( R + r )( R − r )

Ring met binnestraal r , buitestraal R Oppervlakte =π( R^2 − r^2 )

=π( R + r )( R − r )

Narrow annulus, radius r and width ∆ r

Area ≈ 2 π r∆r

r

∆ r Smal ring met straal^ r^ en wydte^ ∆ r

Oppervlakte ≈ 2 π r∆r

Rectangular parallelepiped, sides a , b and c Area = 2 ( ab + bc + ca )

Volume = abc a^

b

c

Reghoekige parallelepipedum met sye a , b en c Oppervlakte = 2 ( ab + bc + ca ) Volume = abc

01 Math Intro Transparencies.doc: 1.4 Trigonometriese verbande 17 06/02/2017 13:36:

Solid right circular cylinder, radius r

Area = 2 π r 2 + 2 π rh

Volume = π r^2 h

Soliede regte sirkelsilinder met straal r en hoogte h

Oppervlakte = 2 π r 2 + 2 π rh

Volume = π r^2 h

Hollow right circular cylinder, inner radius r and outer radius R

Area = 2 π ( R^2 − r^2 )+ 2 π h ( R + r )

Material vol. =π( R^2 − r^2 ) h

Hol regte sirkelsilinder met binnestraal r en buitestraal R

Oppervlakte = 2 π ( R^2 − r^2 )+ 2 π h ( R + r )

Materiaal vol. =π( R^2 − r^2 ) h

Thin walled, hollow right circular cylinder

Area ≈ 4 π rh

Material vol. ≈ 2 π rh ∆ r

Dunwandige hol regte sirkelsilinder

Maths Required for Physics, Summaries of Physics

Related documents

Partial preview of the text

Download Maths Required for Physics and more Summaries Physics in PDF only on Docsity!

1 Mathematical

introduction

1 Wiskundige

inleiding

1.1 Mathematical symbols and

constants

1.1 Wiskundige simbole en

konstantes

Mathematical symbols and constants Wiskundige simbole en konstantes

The Greek alphabet: Die Griekse alfabet:

Α α alpha /^ alfa^ Ν ν nu

Β β beta^ Ξ ξ xi

Γ γ gamma^ Ο ο omicron /^ omikron

∆ δ delta^ Π π pi

Ε ε epsilon^ Ρ ρ rho

Ζ ζ zeta^ Σ σ sigma

Η η eta^ Τ τ tau

Θ θ theta^ Υ υ upsilon

Ι ι iota^ Φ φ phi

Κ κ kappa^ Χ χ chi

Λ λ lambda^ Ψ ψ psi

Μ μ mu^ Ω ω omega

1.2` Exponents, logarithms and

factorisation

1.2 Eksponente, logaritmes en

faktorisering

Rules for exponents: Reëls vir eksponente

÷ =

× =

= ×

= × × × ×

Rules for logarithms: Reëls vir logaritmes:

P P

P Q P Q

P Q P Q

× = +

= × = ÷

Factorisation: Faktorisering:

1.3 Radian measure 1.3 Radiaalmaat

Measurement of rotation: Meting van rotasie:

O

B

A

1.4 Trigonometric relationships 1.4 Trigonometriese verbande

sinθ = ,

cos θ = ,

tanθ =

Graphs of trigonometric functions: Grafieke van trigonometriese funksies:

Identities: Identiteite:

Inverse functions: Inverse funksies:

θ =arcsin( y ′/ r ) or θ = sin −^1 ( y ′/ r )

θ =arccos( x ′/ r ) or θ = cos −^1 ( x ′/ r )

θ =arctan( y ′/ x ′ ) or θ = tan −^1 ( y ′/ x ′)

sin (sin )

a = a

Approximations for small angles Benaderings vir klein hoeke

a

r

t s

A

D

B C

For small angles ( θ << 1 ) we have t ≈ s and r ≈ a. So

Vir klein hoeke ( θ << 1 ) is t ≈ s en r ≈ a. Ons kan

if θ is in radians!

θ in radiale gemeet word!

Geometric formulae Meetkundige formules

b

a

Area = bh = ab sin θ

h

b

a

Oppervlakte = bh = ab sin θ

Area = 21 bh = 21 bc sin θ

a