Vector Calculus

COORDINATE SYSTEMS

• RECTANGULAR or Cartesian

• CYLINDRICAL• SPHERICAL

Choice is based on symmetry of problem

Examples:

Sheets - RECTANGULAR

Wires/Cables - CYLINDRICAL

Spheres - SPHERICAL

Vector Calculus

Cylindrical Symmetry Spherical Symmetry

Cartesian Coordinates

P (x, y, z)

x

y

z

P(x,y,z)

Rectangular CoordinatesOr

z

y

x

A vector A in Cartesian coordinates can be written as

),,( zyx AAA or zzyyxx aAaAaA

where ax,ay and az are unit vectors along x, y and z-directions.

Cylindrical Coordinates

P (ρ, Φ, z)

x= ρ cos Φ, y=ρ sin Φ, z=z

z

Φ

z

ρx

y

P(ρ, Φ, z)

z

20

0

A vector A in Cylindrical coordinates can be written as

),,( zAAA orzzaAaAaA

where aρ,aΦ and az are unit vectors along ρ, Φ and z-directions.

zzx

yyx ,tan, 122

The relationships between (ax,ay, az) and (aρ,aΦ, az)are

zz

y

x

aa

aaa

aaa

cossin

sincos

zz

yx

yx

aa

aaa

aaa

cossin

sincosor

zzyxyx aAaAAaAAA )cossin()sincos(

Then the relationships between (Ax,Ay, Az) and (Aρ, AΦ, Az)are

zz

yx

yx

AA

AAA

AAA

cossin

sincos

z

y

x

z A

A

A

A

A

A

100

0cossin

0sincos

In matrix form we can write

Spherical Coordinates

P (r, θ, Φ)

x=r sin θ cos Φ, y=r sin θ sin Φ, Z=r cos θ

20

0

0

r

A vector A in Spherical coordinates can be written as

),,( AAAr or aAaAaA rr

where ar, aθ, and aΦ are unit vectors along r, θ, and Φ-directions.

θ

Φ

r

z

yx

P(r, θ, Φ)

x

y

z

yxzyxr 1

221222 tan,tan,

The relationships between (ax,ay, az) and (ar,aθ,aΦ)are

aaa

aaaa

aaaa

rz

ry

rx

sincos

cossincossinsin

sincoscoscossin

yx

zyx

zyxr

aaa

aaaa

aaaa

cossin

sinsincoscoscos

cossinsincossin

or

Then the relationships between (Ax,Ay, Az) and (Ar, Aθ,and AΦ)are

aAA

aAAA

aAAAA

yx

zyx

rzyx

)cossin(

)sinsincoscoscos(

)cossinsincossin(

z

y

xr

A

A

A

A

A

A

0cossin

sinsincoscoscos

cossinsincossin

In matrix form we can write

cossin

sinsincoscoscos

cossinsincossin

yx

zyx

zyxr

AAA

AAAA

AAAA

Cartesian CoordinatesP(x, y, z)

Spherical CoordinatesP(r, θ, Φ)

Cylindrical CoordinatesP(ρ, Φ, z)

x

y

zP(x,y,z)

Φ

z

rx y

z

P(ρ, Φ, z)

θ

Φ

r

z

yx

P(r, θ, Φ)

Differential Length, Area and Volume

Differential displacement

zyx dzadyadxadl

Differential area

zyx dxdyadxdzadydzadS

Differential VolumedxdydzdV

Cartesian Coordinates

Cylindrical Coordinates

ρρ

ρ

ρ

ρρ

ρ

ρ

ρρ

ρ

Differential Length, Area and Volume

Differential displacement

zdzaadaddl

Differential area

zadddzaddzaddS

Differential Volume

dzdddV

Cylindrical Coordinates

Spherical Coordinates

Differential Length, Area and Volume

Differential displacement

adrarddradl r sin

Differential area

ardrdadrdraddrdS r sinsin2

Differential Volume

ddrdrdV sin2

Spherical Coordinates

Line, Surface and Volume Integrals

Line Integral

L

dlA.

Surface Integral

Volume Integral

S

dSA.

dvpV

v

Gradient, Divergence and Curl

• Gradient of a scalar function is a vector quantity.

• Divergence of a vector is a scalar quantity.

• Curl of a vector is a vector quantity.

• The Laplacian of a scalar A

f Vector

A.

The Del Operator

A

A2

Del Operator

Cartesian Coordinates

zyx az

ay

ax

Cylindrical Coordinates

Spherical Coordinates

zazaa

1

a

ra

rar r

sin

11

The gradient of a scalar field V is a vector

that represents both the magnitude and

the direction of the maximum space rate

of increase of V.

Gradient of a Scalar

zyx az

Va

y

Va

x

VV

zaz

Va

Va

VV

1

a

V

ra

V

ra

r

VV r

sin

11

The divergence of A at a given point P is

the outward flux per unit volume as the

volume shrinks about P.

Divergence of a Vector

v

dSA

AdivA S

v

.

lim.0

z

A

y

A

x

AA

.

z

AAAA z

1)(

1.

The curl of A is an axial vector whose magnitude is the maximum circulation of A per unit area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum.

Curl of a Vector

nL

sa

S

dlA

AcurlA

max

0

.

lim

Where ΔS is the area bounded by the curve L and an is the unit vector normal to the surface ΔS

zyx

zyx

AAAzyx

aaa

A

z

z

AAAz

aaa

A

1

ArrAA

r

arraa

rA

r

r

sin

sin

sin

12

Cartesian Coordinates Cylindrical Coordinates

Spherical Coordinates

The divergence theorem states that the

total outward flux of a vector field A

through the closed surface S is the same

as the volume integral of the divergence

of A.

Divergence or Gauss’ Theorem

V

AdvdSA ..

L S

dSAdlA ).(.

Stokes’ TheoremStokes’s theorem states that the circulation of a

vector field A around a closed path L is equal to

the surface integral of the curl of A over the open

surface S bounded by L, provided A and

are continuous on S

A