vector calculus - wikidydwikidyd.iem.pw.edu.pl/attachments/ef(2f)vector_calculus/lecture1.pdf ·...

Electromagnetic Fields

Lecture 1

Vector Calculus

Electromagnetic Fields, Lecture 1, slide 2

What is field?

Mathematics: a map Rn →Rm which assigns each point a quantity (scalar or vector).

Physics:property of space – action (observed as a force acting on objects)

In this course:“Field” means property or space in which this property is observed or function which describes this property.


Vector calculus

Vector calculus (or vector analysis) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space R3

Euclidean vector is a geometric object that has both a magnitude (or length) and direction.

Vectors are fundamental in the physical sciences.


Why we use vectors

Under changes of coordinate systems they behave the same way points do.

Thus, vector notation of most of the equations used to describe physical fields' properties is independent on the coordinate system being used.

VECTORS ARE CONVENIENT!

[Physics] General covariance: the essential idea is that coordinates do not exist a priori in nature, but are only artifices used in describing nature, and hence should play no role in the formulation of fundamental physical laws.


Coordinate systemsA coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element.Using CS we can transform problems of geometry into problems concerning numbers (and then solve these problems by calculation).

Vectors allow nice and general presentation of ideas and general behaviour of field, but we, engineers, do need

numbers to quantitatively calculate, predict and design...

y

x

2.65

3.33

P(3.33,2.65)


Cartesian coordinate system

Three perpendicular planes are chosen and the three coordinates of a point are the signed distances to each of the planes

x – the signed distance from the yz planey – the signed distance from the xz planez – the signed distance from the xy plane


Cylindrical coordinate system

r – the radius is the Euclidean distance from the origin

φ – the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.

z – the height is the signed distance from the chosen plane

The origin, the plane and the reference direction on this plane are chosen.


Spherical coordinate system

r – the radius is the Euclidean distance from the origin

θ – the inclination (or polar angle) is the angle between the zenith direction and the line segment OP.

φ – the azimuth (or azimuthal angle) is the signed angle measured from the azimuth reference direction to the orthogonal projection of the line segment OP on the reference plane.

The origin, the plane and the azimuth reference direction are chosen. The zenith direction direction is perpendicular to the plane.


Coordinate transformationCartesian(x,y,z)

Cylindrical(ρ,φ,z)

Spherical(r,θ,φ)

Cartesian(x,y,z)


Spherical(r,θ,φ)

= x2 y2

=arctan y / x z=z

r= x2 y2z2

=arccos z /r

=

x=cos

y=sin

z=z

=2z2

=arctan y / x =arctan / z

x=r sin cos

y=r sinsin

z=r cos

=r sin

=

z=r cos teta

These are not all elements we must change! More follows....


Unit vectors transformationCartesian(x,y,z)


Spherical(r,θ,φ)

Cartesian

Cylindrical

Spherical

1=x1x

y1 y

1=−y1x

z1 y

1z=1z

1r=x1x y 1 yz1 z

r

1=x z1x y z1 y−21 z

r

1=−y1xx1y

1x=cos1−sin1

1 y=sin 1cos1

1z=1z

Still, these are not all elements we must change! More follows....

1x ,1 y ,1z

1r ,1 ,1

1 ,1 ,1z

1x=sin cos1rcos cos1−sin 1

1 y=sin sin 1rcoscos1cos1

1z=cos1r−sin 1

1r=

r1

zr1z

1=zr1−

r1 z

1=1

1=sin 1rcos1

1=1

1z=cos1r−sin 1


Representation of vectorsVectors are similar to points and thus their can be

represented the same way the points are.

y

x

2.65

3.33

P(3.33,2.65)

v=[3.33,2.65]v=[0,0,5.3]

A

Z


Basic operations

Scalar multiplication multiplication of a scalar (field) and a vector (field),

yielding a vector field:

w = a v

Vector addition (substraction) addition of two vectors (vector fields),

yielding a vector (field):

w = v + u

v

w=v + u

v u


Dot productMultiplication of two vectors (vector fields), yielding a scalar (field):

s = v ∙ u = |v||u| cos θ

θ

v

u

|v| cos θ

Properties: the dot product is commutative:

v ∙ u = u ∙ v

the dot product is distributive over vector addition:

w ∙ ( v + u ) = w ∙ u + w ∙ vUsing coordinates:

v = [ vi, v

j, v

k ], w = [ w

i, w

j, w

k ]

v ∙ u = v

iw

i+v

jw

j+v

kw

k


Cross productMultiplication of two vectors (vector fields), yielding a vector (field):

w = v × u = |v||u| sin θ 1n

θv

u

v×u

|v×u|Properties: the dot product is anticommutative:

v × u = - u × v

the dot product is distributive over vector addition:

w × ( v + u ) = w u + w × v

Using coordinates:v = [ v

i, v

j, v

k ], w = [ w

i, w

j, w

k ]

v × u = det [

i j kvi v j vkwi w j wk

]


Del – a special ,,vector”A convenient mathematical notation for three differential operators used in vector calculus:

∇=[ ∂∂ x,

∂∂ y,

∂∂ z ]

What is partial derivative?

f(x,y,z)=2xy+sin y + y e- z

∂ f∂ x

=2 y

∂ f∂ y

=cos ye−z

∂ f∂ z

=−y e−z

Del operator represented by nabla symbol


GradientThe gradient of a scalar field f is a vector field pointing in the direction of fastest increase of f.

∇ f x , y , z=[ ∂ f∂ x,∂ f∂ y,∂ f∂ z ]

f x , y =x2 y2

∇ f x , y =[2 x ,2 y ]


Gradient example

Field of a dipole (+q,-q): 3D surface, equilines and gradient (direction only) (Example from http://www.gnuplot.info/demo/vector.html)


Line integral of a vector field

∫Lu r ⋅d r=∫a

bu r t ⋅r ' t dt

where is a bijective parametrization of Lr : [a ,b ] L

|dr| u

ur

u

∫Lu r ⋅d r≈∑i

Liur i1

23

i... ...


Gradient theorem

∫L∇ f⋅d r=f e−f b

b

e

b

L

We use this theorem when trying to integratea vector field u along a line. If u is a gradientof a scalar field f, then the integral is path-independent.


Flux of a vector field

=∬Su⋅ndS

u

n

uun


DivergenceThe divergence of a vector field u is a scalar field that measures source or sink of u at a given point.

∇⋅u=limr0

∯S r u⋅ndS

∣V r ∣u= [ux x , y , z , uy x , y , z ,uz x , y , z ]

∇⋅u=∂ux x , y , z

∂ x

∂uy x , y , z

∂ y

∂uz x , y , z

∂ z

u=[ x3

3,y2

2 ]

∇⋅u=x2 y


Divergence theoremAlso knows as Gauss' theorem, Ostrogradsky's theorem or Gauss-Ostrogradsky theorem.

∭V∇⋅udV=∯∂V

u⋅ndS

Volume integral of the divergence of a vector field u is equal to the outward flux of u through the volume boundary (surface).

V

S

n

n

n n


CurlThe curl of a vector field u is a vector field that measures rotation of u at a given point.

∇×u=limr0

∯S rn×udS

∣V r ∣ u= [ux x , y , z , uy x , y , z ,uz x , y , z ]

∇×u=∣1x 1 y 1z∂

∂ x∂

∂ y∂

∂ zux uy uz

∣u=[ y

3

3,x2

2,0]

∇×u= [0,0, x− y2 ]


Curl – interpretation

u=[ y3

3,x2

2,0]

∇×u= [0,0, x− y2 ]


Stokes' theoremStrictly speaking this is Kelvin-Stokes theorem (a special case of more general Stokes' theorem).

∬S∇×u dS=∮

∂Sud r

Flux of the curl of a vector field u through a surface is equal to the line integral of u along the surface boundary (closed line).

uu

dS

∂S

S


Green's theoremLet's start with curl of a 2D vector, written in 3D as u=[ L, M, 0 ]:

Calculating a flux of this curl through any 2D surface dS=[dx×dy]:

According to the Stokes' theorem

Green's theorem:

∇×u= ∂0∂ y−

∂M∂ z i ∂L∂ z −∂0

∂ x j ∂M∂ x −∂ L∂ y k

∬s∇×u d S=∬s

∇×u⋅k dS=∬S ∂M∂ x −∂ L∂ y dS

∬S∇×u dS=∮

∂Sud r

∬S ∂M∂ x −∂ L∂ y dS=∮∂ S

u⋅d r=∮∂ S

[ L,M ,0]⋅[dx ,dy , dz ]=∮∂ S

LdxM dy

∮∂SLdxM dy =∬S ∂M∂ x −

∂L∂ y dS


Second order derivatives

The most important second order differential operator is Laplacian:

which may be defined for vector field as:

f=∇⋅∇ f=∇2 f

In math we have several possibilities:for scalar field:

and

for vector field:

and and

∇⋅∇ f ∇×∇ f≡0

∇ ∇⋅u ∇×∇×u ∇⋅∇×u≡0

∇2u=∇∇⋅u−∇×∇×u


Identities

∇×∇ f≡0

Curl of the gradient of any scalar field is always zero vector:

This identity allows us express any curl free vector field by a scalar field.

Divergence of the curl of any vector field is always zero:

This identity allows us express any divergence free vector field by another vector field.

∇⋅∇×u≡0


Conservative field

u=∇ f

A vector field u is said to be conservative if it is a gradient of scalar field:

As we have seen earlier, curl of a conservative field must be zero:

From Stokes' theorem it follows that:

∇×u≡0

∮∂Sud r=∬S

∇×udS=0

∮Cud r=∮C

∇ f d r=0

∫L∇ f⋅d r=f e−f b

∫L'∇ f⋅d r=f e−f b

e

b

L

L'


Solenoidal field

∇⋅u=0

A vector field u is said to be solenoidal if its divergence is zero:

As we have seen earlier, a solenoidal field can be expressed as the curl of another vector field:

From divergence theorem it follows that flux of solenoidal field through anyclosed surface must be zero:

∇⋅∇×w≡0 u=∇×w

∯∂Vud S=∭V

∇⋅u dV=0


PseudovectorIt was said earlier, that vectors transforms with coordinate system changes like points do. Strictly speaking we can observe two slightly different behavior of “vectors” under mirroring:

Electric field E of single positive charge q.E is a vector field.

Magnetic field B of a single wire .B is a pseudovector field.


TensorTensors can bee seen as an extension of series: scalar, vector, … In general they can be considered as a multidimensional array of numbers or functions.

In field theory they have many uses, but here we shall need them to express material properties of some non-trivial media.

T=[ t xx t xyt yx t yy ]

vector calculus - wikidydwikidyd.iem.pw.edu.pl/attachments/ef(2f)vector_calculus/lecture1.pdf ·...

Documents