EE3005: Electromagnetic
Theory
Vector Calculus
Integration and differentiation of vectors
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Vector Differential Calculus
• Vector Differential includes
• Gradient (∇𝑉): The derivative of a scalar function resulting in a vector
function
• Divergence (∇. 𝐀): The derivative of a vector function resulting in a scalar
function
• Curl (∇ × 𝐀): The derivative of a vector function resulting in another vector
function
• Scalar Laplacian(∇2 𝑉): Divergence of the gradient of a scalar function
• Vector Laplacian: Scalar Laplacian of each component o fa vector function
Divergence of the gradient of a scalar function
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The Gradient of a Scalar Field
EE3005 by Dr. Arshad Hassan Figure: Gradient of a scalar.
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• The gradient can be obtained by evaluating the difference in the field dV between points P1 and
P2 of Figure
• Where V1, V2, and V3 are contours on which V(x,y,z) is constant. From calculus,
dV/d𝑙 is a maximum when 𝜃 = 0, that EE3005 by Dr. Arshad Hassan Figure: Gradient of a scalar.
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is, when d𝑙 is in the direction of G. arshad.hassan@nu.edu.pk
Cont…
• Hence,
• Max. rate of change, dV/dn is the normal derivative. Thus G has its
magnitude and direction as those of the maximum rate of change of
V. By definition, G is the gradient of V. Therefore,
• In Cartesian coordinates, the del operator can be written as,
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Cont….
• For cylindrical coordinates,
• For Spherical coordinate,
• Reading Assignment: Derivation of these formulas is your reading assignment
• The following computation formulas, should be noted:
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Properties o scalar Field V
• The following fundamental properties of the gradient of a scalar field V
1. The magnitude of ∇𝑉 equals the maximum rate of change in V per unit
distance.
2. ∇𝑉 points in the direction of the maximum rate of change in V.
3. ∇𝑉 at any point is perpendicular to the constant V surface that passes
through that point (see points P and Q in Figure)
4. The projection (or component) of ∇𝑉 in the direction of a unit vector a is
∇𝑉 • a and is called the directional derivative of V along a. This is the rate
of change of V in the direction of a. For example, dV/dl is the directional
derivative of V along P1P2 in Figure. Thus the gradient of a scalar
function V provides us with both the direction in which V changes most
rapidly and the magnitude of the maximum directional derivative of V.
5. If A = ∇𝑉, V is said to be the scalar potential of A.
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Reading Assignment
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Practice Problem#1
• Find the gradient of the following scalar field:
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Practice problem
• Determine the rate of change of the scalar field 𝑓 𝑥, 𝑦, 𝑧 = 𝑥𝑦 +
2𝑧 2 at [1,1,1] in the direction of the vector 𝑎𝑥 − 2𝑎𝑦 + 𝑎𝑧
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DIVERGENCE OF A VECTOR AND DIVERGENCE
THEOREM
• The divergence of A at a given point P is the outward flux per unit volume as the
volume shrinks about P.
• Hence,
Figure: Illustration of the divergence of a vector field at P; (a) positive divergence, (b)
negative divergence, (c) zero divergence.
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Cont…
• The divergence of A at point P(xo, yo, zo) in a Cartesian system is given
by
• For cylindrical coordinates,
• For Spherical coordinate,
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Properties of the divergence of a vector field
• The following properties of the divergence of a vector field:
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Divergence Theorem or Gauss Theorm
• 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.
• This is called the divergence theorem, otherwise known as the Gauss-
Ostrogradsky theorem.
• Mathematically, we can write as
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Practice Problem#1
• Determine the divergence of these vector field:
• Solution:
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Practice Problem#2
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Cont…
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CURL OF A VECTOR AND STOKES'S THEOREM
• The curl of A is an axial (or rotational) vector whose magnitude is the
maximum circulation of A per unit area as the area lends to zero and
whose direction is the normal direction of the area when the area is
oriented so as to make the circulation maximum.
Figure: Illustration of a curl: (a) curl at P points out of the page;
(b) curl at P is zero EE3005 by Dr. Arshad Hassan
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Cont…
• The curl of A in cylindrical coordinates
• The curl of A in cylindrical coordinates
• The curl of A in spherical coordinates
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Properties of the Curl
• The following properties of the curl:
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Stokes’ Theorem
• This is called Stokes's theorem.
• Stokes'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. (see Figure) provided that A and
∇ × 𝐀 are continuous on S.
• The flux lines cancel within the surface and only contributions
from the outer closed curve remain!
• Use: The link between differential and integral calculus
• Instead of integrating over a complete region, just
integrate over the boundary
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Figure: Illustration of Stokes's theorem
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Home Assignment
• [Griffiths – Intro. to Electrodynamics]
• Problems 1.46 up to 1.48
• Problems 1.53 up to 1.62
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Practice Problem#1
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Cont…
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LAPLACIAN OF A SCALAR
• The Laplacian of a scalar field V, written as ∇2 𝑉 is the divergence of the gradient
of V.
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Cont…
• In cylindrical coordinates,
• in spherical coordinates,
• A scalar field V is said to be harmonic in a given region if its Laplacian
vanishes in that region.
• This is called Laplace's equation.
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Laplacian of a Vector
• Laplacian of a Vector is defined as the gradient of the divergence of A
minus the curl of the curl of A. That is,
• Laplacian of a vector is the sum of the Laplacian of each of the scalar
components
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Practice Problem#1
• Find the Laplacian of the scalar field
• Solution:
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CLASSIFICATION OF VECTOR FIELDS
• A vector field is uniquely characterized by its divergence and curl.
Neither the divergence nor curl of a vector field is sufficient to
completely describe the field. All vector fields can be classified in
terms of their vanishing or nonvanishing divergence or curl as follows:
Figure: illustrates typical fields
in these four categories.
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Divergence-less or Solenoidal Fields
• A vector field A is said to be solenoidal (or divergence less) if ∇. 𝐀 = 0.
• Solenoidal means that the field resembles the magnetic force field
produced by a solenoid.
• Such a field has neither source nor sink of flux. From the divergence
theorem
• Flux lines of A entering any closed surface must also leave it.
• Examples of solenoidal fields are incompressible fluids, magnetic fields, and
conduction current density under steady state conditions.
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Cont…
• To generate a solenoidal field, start with any vector field F and take its
curl. The resulting vector field (𝐀) will be solenoidal because ∇. (∇ ×
𝐅) = 0
• Conversely, if a vector field (𝐀) is solenoidal, it must be the curl of
another vector field (𝐅) ; ∇. 𝐀 = 0 where 𝐀 = ∇ × 𝐅
• Here F is called the vector potential
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Irrotational or Potential or conservative Field
• A vector field A is said to be irrotational (or potential) if ∇ × 𝐀 = 𝟎
• Irrotational means that the field does not have the ability to do work
around any point (or closed path)
• Potential means that the vector field would necessarily be the
gradient of some scalar field (called the scalar potential)
• Conservative means that the line integral of the vector field between
two points of space is independent of the path taken to reach from
one point to the other
• Examples of irrotational fields include the electrostatic field and the
gravitational field.
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Cont…
• In general, the field of gradient V (for any scalar V) is purely irrotational
since
• For this reason, A may be called a potential field and V the scalar potential
of A. The negative sign in above eq. has been inserted for physical reasons
that will be covered in this course.
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Helmholtz 's theorem
• A vector A is uniquely prescribed within a region by its divergence and its
curl. If we let,
and
• 𝜌𝑣 can be regarded as the source density of A and 𝜌𝑠 its circulation density.
• Any vector A satisfying above equations with both 𝜌𝑣 and 𝜌𝑠 vanishing at infinity
can be written as the sum of two vectors: one irrotational (zero curl), the other
solenoidal (zero divergence). This is called Helmholtz 's theorem.
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