Topic 02 (VectorAnalysis)

Vector Analysis

EE 141 Lecture NotesTopic 2

Professor K. E. OughstunSchool of Engineering

College of Engineering & Mathematical SciencesUniversity of Vermont

2014

Motivation

First published in 1901.

Gradient of a Scalar Field

Let F (P) = F (r) = F (x , y , z) be a scalar function of position (orpoint function) with value F (P1) at the point P1 = (x , y , z) andvalue F (P2) at a neighboring point P2 = (x + dx , y + dy , z + dz).

The two points are separated by the differential distance vector

d ~̀= 1̂xdx + 1̂ydy + 1̂zdz (1)

where dx = 1̂x · d ~̀, dy = 1̂y · d ~̀, and dz = 1̂z · d ~̀.

Gradient of a Scalar Field

The Total Differential of F (P)

dF =∂F

∂xdx +

∂F

∂ydy +

∂F

∂zdz (2)

is then given by

dF =∂F

∂x1̂x · d ~̀+

∂F

∂y1̂y · d ~̀+

∂F

∂z1̂z · d ~̀

=

(1̂x∂F

∂x+ 1̂y

∂F

∂y+ 1̂z

∂F

∂z

)· d ~̀ (3)

The Gradient of the scalar field F (r) is then defined as the vector field

grad F (r) ≡ ∇F (r) ≡ 1̂x∂F

∂x+ 1̂y

∂F

∂y+ 1̂z

∂F

∂z(4)

so that Eq. (3) for the total differential may then be expressed as

dF = ∇F · d ~̀ (5)

Vector Differential Operator ∇

The vector differential operator ∇ or “del” is defined as

∇ ≡ 1̂x∂

∂x+ 1̂y

∂

∂y+ 1̂z

∂

∂z(6)

Notice that this is not a vector! It is an operator.

Explicit expressions for the various differential operations (gradient,divergence, curl, Laplacian) need to be derived in the variousorthogonal curvilinear coordinate systems that naturally arise inengineering problems with specific geometric symmetries, e.g.:

cylindrical polar coordinates with axial symmetry

spherical polar coordinates with point symmetry.

Directional Derivative

The directional derivative of the scalar field F (r) along the direction

specified by the unit vector 1̂` with d ~̀= 1̂`d` then follows from therelation (5), viz.

dF = ∇F · d ~̀= ∇F · 1̂`d`

asdF

d`= 1̂` · ∇F . (7)

It then follows that dF/d` has its maximum value when 1̂` is alongthe direction specified by ∇F . The streamlines of the vector field ∇Fare then orthogonal to the isotimic surfaces F (r) = constant.

The path integral of Eq. (5) then gives

F (P2)− F (P1) =

∫ P2

P1

∇F · d ~̀ (8)

independent of the path taken.

Properties of the Gradient Operator

For any two scalar point functions U(r) and V (r),

Distributive Law over Addition:

∇(U(r) + V (r)

)= ∇U(r) +∇V (r) (9)

Product Rule:

∇(U(r)V (r)

)= U(r)∇V (r) + V (r)∇U(r) (10)

Power Rule: For any n,

∇V n(r) = nV n−1(r)∇V (r) (11)

Problem 2: Derive each of these properties in rectangular coordinates.

Example: Gradient of the Plane Wave Propagation

Factor e jk·r

Consider the plane wave propagation factor U(r) = e jk·r with fixedwave vector k = 1̂xkx + 1̂yky + 1̂zkz , where r = 1̂xx + 1̂yy + 1̂zz isthe variable position vector. Then

∇e jk·r = ∇e j(kxx+kyy+kzz)

=

(1̂x

∂

∂x+ 1̂y

∂

∂y+ 1̂z

∂

∂z

)e j(kxx+kyy+kzz)

=(1̂x jkx + 1̂y jky + 1̂z jkz

)e j(kxx+kyy+kzz)

= jke jk·r

Note thatd

dte jωt = jωe jωt

Hence, just as one has one-dimensional Fourier analysis in ω-space,one also has three-dimensional Fourier analysis in k-space.

Divergence of a Vector Field

A vector field F(r) = F(x , y , z) may be described graphically by itsfield lines, flux lines, or streamlines.

The flux density of F(r) passing through a differential element ofsurface ds = n̂ds whose directed orientation in space is specified bythe unit surface normal n̂, is defined as the amount of flux crossing aunit surface element of area, viz.

flux density ≡ F · n̂ds

ds= F · n̂ (12)


The total flux crossing a closed surface S with outward unit normalvector n̂ is then given by the surface integral

total flux =

∮SF · ds =

∮SF · n̂ds (13)

A positive value means that more flux leaves the region enclosedby the surface S than enters, indicating that S encloses a source.

A negative value means that more flux enters the regionenclosed by S than exits, indicating that S encloses a sink.


Consider determining the total outward flux of the vector field

F(r) = 1̂xFx(x , y , z) + 1̂yFy (x , y , z) + 1̂zFz(x , y , z) (14)

passing through the closed surface S comprised of the rectangularsurface elements Sj with outward unit normal vectors n̂j parallel tothe Cartesian coordinate axes.


The outward flux F1 through the surface element S1 with outwardunit normal vector n̂1 = −1̂x is given by

F1 =

∫∫S1

F · n̂ds

=

∫∫S1

(1̂xFx + 1̂yFy + 1̂zFz

)·(−1̂x

)dydz = −Fx(r1)∆y∆z ,

where Fx(r1) is the value of Fx at the center of surface element S1.

Similarly, the outward flux F2 through the surface element S2 withoutward unit normal vector n̂2 = 1̂x is given by

F2 =

∫∫S2

F · n̂ds

=

∫∫S2

(1̂xFx + 1̂yFy + 1̂zFz

)·(1̂x

)dydz = Fx(r2)∆y∆z ,

where Fx(r2) is the value of Fx at the center of surface element S2.


Because r2 = r1 + 1̂x∆x , then one has the Taylor series expansion

Fx(r2) = Fx(r1) +∂Fx

∂x∆x +O

{(∆x)2

}.

With this substitution, the expression for the outward flux F2 throughthe surface element S2 becomes

F2 =

[Fx(r1) +

∂Fx

∂x∆x

]∆y∆z ,

plus higher-order terms that are of the order of O{(∆x)2∆y∆z} as∆x → 0, ∆y → 0, ∆z → 0.

Notation: f (x) = O{

g(x)}

as x → x0 means that |f (x)/g(x)| ≤ Kas x → x0, where K > 0 is some positive constant.


The sum of the outward fluxes of the vector field F(r) through thetwo surface elements S1 and S2 is then given by

F1 + F2 =∂Fx

∂x∆x∆y∆z .

Similarly, the sum of the outward fluxes of F(r) through the twosurface elements S3 and S4 is given by

F3 + F4 =∂Fy

∂y∆x∆y∆z ,

and the sum of the outward fluxes of F(r) through the two surfaceelements S5 and S6 is given by

F5 + F6 =∂Fz

∂z∆x∆y∆z ,

plus higher-order terms of order O{(∆x)2∆y∆z}, O{∆x(∆y)2∆z}or O{∆x∆y(∆z)2}, respectively.


The total outward flux of the vector field F(r) through the closedsurface S is then given by

∑6j=1 Fj , so that∮

SF · ds =

(∂Fx

∂x+∂Fy

∂y+∂Fz

∂z

)∆V , (15)

where ∆V = ∆x∆y∆z is the element of volume enclosed by S.

The divergence of a vector field is then defined as the scalar field

divF(r) ≡ lim∆V→0

{1

∆V

∮SF · ds

}(16)

where ∆V is the volume element enclosed by the simply-connectedsurface S containing the point r, and where ds = n̂ds with outwardunit normal n̂ to the surface S.


Because ∇ ≡ 1̂x∂∂x

+ 1̂y∂∂y

+ 1̂z∂∂z

, then the divergence operation inrectangular coordinates is given by

divF(r) = ∇ · F(r) =∂Fx

∂x+∂Fy

∂y+∂Fz

∂z(17)

The vector field F(r) has a positive divergence at a point r if thenet flux out of an infinitesimally small surface S surroundingthat point is positive; ∆V then contains a flux source.

The vector field F(r) has a negative divergence at a point r ifthe net flux out of an infinitesimally small surface S surroundingthat point is negative; ∆V then contains a flux sink.

The vector field F(r) has a zero divergence at a point r if thenet flux out of an infinitesimally small surface S surroundingthat point vanishes; the field is then said to be divergenceless orsolenoidal at that point.

Properties of the Divergence Operator

Solenoidal Vector Field ⇐⇒ ∇ · F(r) = 0 ⇐⇒∮S F · ds = 0.

For any two vector functions F(r) and G(r) and scalar function φ(r),

Distributive Law over Vector Addition:

∇ ·(F(r) + G(r)

)= ∇ · F(r) +∇ · G(r) (18)

Product Rule for Scalar Multiplication:

∇ ·(φ(r)F(r)

)= φ(r)∇ · F(r) + F(r) · ∇φ(r) (19)

Problem 3: Derive each of these properties in rectangular coordinates.

Problem 4: Show that ∇ · r = 3, where r ≡ 1̂xx + 1̂yy + 1̂zz is theposition vector.

Divergence Theorem

From Eq. (15), for each volume element ∆Vj with closed surface SJof a simply connected region V =

∑j Vj ,(

∇ · F(r))∆Vj =

∮SjF(r) · ds.

Summing over all of the volume elements ∆Vj in V and taking thelimit as ∆Vj → 0, this becomes∑

j

(∇ · F(r)

)∆Vj =

∑j

∮SjF(r) · ds =

∮SF(r) · ds,

where S is the surface enclosing V . In this limit as ∆Vj → 0, thesummation on the left goes over to the volume integral over V ,resulting in the Divergence Theorem∫∫∫

V

(∇ · F(r)

)d3r =

∮SF(r) · ds (20)

Example: Divergence of the Vector Plane Wave

Propagation Factor Ae jk·r

Consider the vector plane wave propagation factor Ae jk·r with fixedwave vector k = 1̂xkx + 1̂yky + 1̂zkz and variable position vector

r = 1̂xx + 1̂yy + 1̂zz . Here A is a constant vector which describesthe fixed orientation of the vector wave field in space (Polarization).Then

∇ · Ae jk·r =∂

∂x

(Axe jk·r)+

∂

∂y

(Aye jk·r)+

∂

∂z

(Aze jk·r)

= jkxAxe jk·r + jkyAye jk·r + jkzAze jk·r

= jk · Ae jk·r,

or, using the product rule (19) for scalar multiplication and the factthat ∇ · A = 0,

∇ · Ae jk·r = (∇ · A) e jk·r + A · ∇e jk·r = jk · Ae jk·r.

Circulation of a Vector Field around a Closed

Contour

The circulation of a vector field F(r) around a closed contour C isdefined by the contour integral

Circulation ≡∮CF(r) · d ~̀ (21)

where d ~̀ denotes the differential element of arc length tangent to thecontour C at the point r.

For example, the circulation of a uniform vector field F(r) = K iszero, as ∮

CK · d ~̀= K ·

∮C

d ~̀= 0

for any closed contour C.

Green’s Theorem for Regular Regions

Consider evaluating the surface integral∫∫ ∂u(x ,y)

∂ydxdy over a plane

region in the xy -plane that is bounded below and above by theregular arcs AB and CD, respectively, and on the sides by straightlines parallel to the y -axis, as illustrated.

This surface integral may be directly evaluated by integrating firstwith respect to y between the two points P ′(x , y) and P ′′(x , y) onthe lower and upper curves, respectively, and then with respect to xbetween the extreme limits a and b, as follows:


∫∫∂u(x , y)

∂ydxdy =

∫ b

a

[u(x , y ′′)− u(x , y ′)

]dx

=

∫ C

D

udx −∫ B

A

udx = −∫ D

C

udx −∫ B

A

udx .

Along the straight line segments BC and DA, x is constant so that∫ C

B

udx =

∫ D

A

udx = 0.

Adding these two path integrals to the above expression then gives∫∫∂u(x , y)

∂ydxdy = −

∮u(x , y)dx ,

the closed contour integral on the right hand side being taken in thecounterclockwise direction.


If a plane region S can be partitioned into a finite number of parts,each of the type just considered, then an equation of the above typeis valid for each of them. It then follows that∫∫

S

∂u(x , y)

∂ydxdy = −

∮C

u(x , y)dx , (22)

where C is the entire boundary of S.In a similar manner, if the plane region S can be partitioned into afinite number of subregions, each bounded on the sides by regulararcs with respect to the y -axis and above and below by straight linesegments parallel to the x-axis, then∫∫

S

∂v(x , y)

∂xdxdy =

∮C

v(x , y)dy . (23)


The sum of Eqs. (22) and (23) then gives Green’s Theorem forRegular Regions∫∫

S

(∂v

∂x− ∂u

∂y

)dxdy =

∮C

(udx + vdy

)(24)

Notice that if the positive unit normal vector n̂ to the plane surfaceS in the xy -plane is taken along the 1̂z direction, then the positivesense of integration around the contour C enclosing the region S isdetermined by the right-hand rule.

Unless otherwise specifically stated, this sign convention determinedby the right-hand rule is adopted here.

George Green (1793 – 1841)

Curl of a Vector Field

In Cartesian coordinates, the curl of a continuous vector field

F(r) = 1̂xu(x , y , z) + 1̂yv(x , y , z) + 1̂zw(x , y , z)

with continuous first partial derivatives is defined as

∇× F(r) =

(1̂x

∂

∂x+ 1̂y

∂

∂y+ 1̂z

∂

∂z

)×(1̂xu(x , y , z) + 1̂yv(x , y , z) + 1̂zw(x , y , z)

)=

∣∣∣∣∣∣1̂x 1̂y 1̂z

∂/∂x ∂/∂y ∂/∂zu(x , y , z) v(x , y , z) w(x , y , z)

∣∣∣∣∣∣= 1̂x

(∂w

∂y− ∂v

∂z

)+ 1̂y

(∂u

∂z− ∂w

∂x

)+ 1̂z

(∂v

∂x− ∂u

∂y

).

(25)

Curl of a Vector Field, Green’s Theorem

With this definition, the integrand appearing on the left-hand side ofEq. (24) is seen to be the z-component of the curl of F(r) and theintegrand appearing on the right-hand side of that equation is seen tobe F · dr in the xy -plane. Consequently, Green’s theorem for regularregions in the plane (24) is equivalent to∫∫

S∇× F · d~a =

∮CF · dr (26)

with d~a = 1̂zdxdy and dr = 1̂xdx + 1̂ydy taken to be tangent to thecontour C, the positive direction of integration about the contour Cbeing determined by the right-hand rule. Analogous expressions areobtained in the xz- and yz-planes.

Curl of a Vector Field, Stokes’ Theorem

The generalization of Green’s theorem (26) to R3 then leads toStokes’ Theorem ∫∫

S∇× F · n̂d2r =

∮CF · dr (27)

where n̂ denotes the unit normal vector to the surface S, the positivedirection of integration about the contour C being determined by theright-hand rule with respect to n̂.

G. G. Stokes, 1st Baronet (1819–1903)

Curl of a Vector Field

Application of Stokes’ theorem to a regular surface element ∆Scontaining the point P in its interior gives∫∫

∆S(∇× F(r)) · n̂d2r︸︷︷︸(∇×F(r))·n̂∆S

=

∮CF(r) · dr,

where, by the mean value theorem for integrals, (∇× F(r)) · n̂ issome intermediate value between the maximum and minimum valuesof the quantity (∇× F(r)) · n̂ on ∆S. Hence, in the limit as ∆Sshrinks to the point P ,

(∇× F(P)) · n̂ = lim∆S→0

1

∆S

∮CF(r) · dr (28)

This Integral Definition of the Curl shows that the normal componentof the curl of a vector field at a point P is given by the circulationper unit area about that point in the plane orthogonal to n̂.

Properties of the Curl Operator

Irrotational Vector Field ⇐⇒ ∇× F(r) = 0 ⇐⇒∮C F · d ~̀= 0.

For any two vector functions F(r) and G(r) and scalar function φ(r),Distributive Law over Vector Addition:

∇×(F(r) + G(r)

)= ∇× F(r) +∇× G(r) (29)

Product Rule for Scalar Multiplication:

∇×(φ(r)F(r)

)= φ(r)∇× F(r) +∇φ(r)× F(r) (30)

Product Rule for Vector Multiplication

∇× (F× G) = G · ∇F− F · ∇G + F(∇ · G)− G(∇ · F) (31)

The curl of the gradient of a scalar field vanishes

∇× (∇φ) = 0 (32)

The divergence of the curl of a vector field vanishes

∇ · (∇× F) = 0 (33)

Additional Vector Differentiation Properties

Problem 5: Derive Properties (32) and (33).

Problem 6: Show that ∇× r = 0,where r = 1̂xx + 1̂yy + 1̂zz .

Problem 7: Show that

F(r) · ∇r = F(r) (34)

For any two vector functions F(r) and G(r),

∇(F · G) = F · ∇G + G · ∇F + F× (∇× G) + G× (∇× F) (35)

Notice that, for example

F · ∇G = (F · ∇)G,

where

F · ∇ = Fx∂

∂x+ Fy

∂

∂y+ Fz

∂

∂z(36)

is the projection of the vector differential operator ∇ onto vector F.

Example: Curl of the Vector Plane Wave

Propagation Factor Ae jk·r

Consider the vector plane wave propagation factor Ae jk·r with fixedwave vector k = 1̂xkx + 1̂yky + 1̂zkz and A a constant vector. Then

∇× Ae jk·r = 1̂x

[Az

∂

∂y

(e jk·r)− Ay

∂

∂z

(e jk·r)]

+1̂y

[Ax

∂

∂z

(e jk·r)− Az

∂

∂x

(e jk·r)]

+1̂z

[Ay

∂

∂x

(e jk·r)− Ax

∂

∂y

(e jk·r)]

= j[1̂x(Azky − Aykz) + 1̂y (Axkz − Azkx)

+1̂z(Aykx − Axky )]e jk·r = jk× Ae jk·r,

or, using the product rule (30) with ∇× A = 0,

∇× Ae jk·r = (∇× A) e jk·r +∇e jk·r × A = jk× Ae jk·r.

The Laplacian Operator

Consider a scalar field φ(r) = φ(x , y , z) with gradient

∇φ(r) = 1̂x∂φ(r)

∂x+ 1̂y

∂φ(r)

∂y+ 1̂z

∂φ(r)

∂z,

which in turn is a vector field with divergence

∇ ·(∇φ(r)

)=

(1̂x

∂

∂x+ 1̂y

∂

∂y+ 1̂z

∂

∂z

)·(1̂x∂φ

∂x+ 1̂y

∂φ

∂y+ 1̂z

∂φ

∂z

)=

∂2φ

∂x2+∂2φ

∂y 2+∂2φ

∂z2.

The Laplacian operator is then defined as

∇2 ≡ ∇ · ∇ =∂2

∂x2+

∂2

∂y 2+

∂2

∂z2(37)

The Laplacian & CurlCurl Operators

The Laplacian of a vector field F(r) = 1̂xFx(r) + 1̂yFy (r) + 1̂zFz(r) isgiven by

∇2F(r) = 1̂x∇2Fx + 1̂y∇2Fy + 1̂z∇2Fz . (38)

Through direct substitution, it is found that the curlcurl operator isgiven by

∇× (∇× F) = ∇(∇ · F)−∇2F (39)

when operating on a continuous vector field F with continuous first-and second-order partial derivatives.

Problem 8: Verify Eq. (39) for the curlcurl operator in Cartesiancoordinates.

Pierre-Simon Laplace (1749 – 1827)

Green’s Integral Identities

Begin with the divergence theorem [see Eq. (20)]∮SF · ds =

∫∫∫V

(∇ · F

)d3r .

With F(r) = φ(r)∇ψ(r), one obtains Green’s First Integral Identity∮S

(φ∇ψ

)· ds =

∫∫∫V

(φ∇2ψ +∇φ · ∇ψ

)d3r (40)

Interchanging φ and ψ in this equation gives∮S

(ψ∇φ

)· ds =

∫∫∫V

(ψ∇2φ +∇ψ · ∇φ

)d3r .

Subtracting this result from the expression given in Eq. (40) yieldsGreen’s Second Integral Identity∮

S

(φ∇ψ − ψ∇φ

)· ds =

∫∫∫V

(φ∇2ψ − ψ∇2φ

)d3r (41)

Problem 9

Beginning with the divergence theorem (∮S F · n̂d2r =

∫V ∇ · Fd3r),

where the closed surface S forms the complete boundary of theregion V with outward unit normal vector n̂, show that, for anysufficiently continuous vector field G = G(r) and scalar field f = f (r):∮

Sf (r)n̂d2r =

∫V

(∇f (r)

)d3r , (42)∮

Sn̂× G(r)d2r =

∫V

(∇× G(r)

)d3r . (43)

Problem 10

Beginning with the Stokes’ theorem (∮C F · dr =

∫S(∇× F) · n̂d2r),

where the closed contour C forms the complete boundary of thesurface S with unit normal vector n̂ taken in the positive direction,show that, for any sufficiently continuous vector field G = G(r) andscalar field f = f (r):∮

Cdr × G(r) =

∫S

(n̂×∇

)× G(r)d2r , (44)∮

Cf (r)dr =

∫S

(n̂×∇f (r)

)d2r . (45)

Topic 02 (VectorAnalysis)

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