Vector calculus1)Differential length, area and volume

2)Line, Surface and Volume Integral

Cartesian Coordinates

Differential quantities:

Differential distance:

Differential surface:

Differential Volume:

dzzdyydxxld ˆˆˆ

dxdyzsd

dxdzysd

dydzxsd

z

y

x

ˆ

ˆ

ˆ

dxdydzdv

Page 109

Cylindrical Coordinates:

Distance = r df

x

y

df

r

Differential Distances: ( dr, rdf, dz )

Cylindrical Coordinates:

Differential Distances: ( dρ, rdf, dz )

zadzadadld ˆˆˆ

zz addsd

adzdsd

adzdsd

ˆ

ˆ

ˆ

Differential Surfaces:

Differential Volume:

Spherical Coordinates:

Distance = r sinq df

x

y

df

r sinq

Differential Distances: ( dr, rdq, r sin q df )

r

f

P

x

z

y

q

Spherical Coordinates

Differential quantities:

Length:

Area:

Volume:

dRRddRR

dldldlRld R

sinˆˆˆ

ˆˆˆ

RdRddldlsd

dRdRdldlsd

ddRRdldlRsd

R

R

R

ˆˆ

sinˆˆ

sinˆˆ 2

ddRdRdv sin2

dRdl

Rddl

dRdlR

sin

Pages 113-115Back

Representation of differential length dl in coordinate systems:

zyx adzadyadxld ˆˆˆ

zr adzadradrld ˆˆˆ

adrardadrld r ˆsinˆˆ

rectangular

cylindrical

spherical

METRIC COEFFICIENTS

Example

• For the object on the right calculate:

• (a) The distance BC• (b) The distance CD• (c) The surface area ABCD• (d) The surface area ABO• (e) The surface area A OFD• (f) The volume ABDCFO

AREA INTEGRALS

• integration over 2 “delta” distances

dx

dy

Example:

x

y

2

6

3 7

AREA = 7

3

6

2

dxdy = 16

Note that: z = constant

In this course, area & surface integrals will be on similar types of surfaces e.g. r =constant or f = constant or q = constant et c….

Representation of differential surface element:

zadydxsd ˆ

Vector is NORMAL to surface

SURFACE NORMAL

DIFFERENTIALS FOR INTEGRALS

Example of Line differentials

or or

Example of Surface differentials

zadydxsd ˆ radzrdsd ˆ or

Example of Volume differentials dzdydxdv

xadxld ˆ

radrld ˆ

ardld ˆ

Coordinates System

Cartesian Coordinates (x, y, z)

Cylindrical Coordinates (r, φ, z)

Example

Solution

Example

Solution

Spherical Coordinates (R, θ, φ)

Example

Solution

Example

Solution