Chapter 3:
Vector Calculus
In Electromagnetics, the following special set of
vector operators are extensively used :
3-1 Gradient of a Scalar Field
3-2 Divergence of a Vector Field
3-3 Curl of a Vector Field
3-4 Laplacian Operator
3-1 Gradient of a Scalar Field
The derivative of a scalar function , in regard
to a given direction x, that is , describes
the rate of change of with x.
The gradient provides the means to describe
the space rate of change of the scalar when
this is also a function of y and z in a Cartesian
coordinate system.
(cont) Example of a Gradient of a Scalar Field
T1(x, y, z) temperature at P1(x, y, z) ,
T2(x+dx, y+dy, z+dz) at nearby point P2 .
The differential temperature:
Can be rewritten as
Where the vector , or grad T, is the gradient of T, that is
(cont) Gradient of a Scalar Field
The symbol is read del (or also nabla) and it denotes a (gradient)
differential operator (also called the Hamilton operator).
The operator can be defined in Cartesian coordinates as
The result of the operation is a vector whose magnitude is equal
to the maximum rate of change of the scalar T per unit distance, and
whose direction is along the direction of maximum increase.
Note: The gradient of a vector is meaningless under the rules of
vector calculus.
(cont) Gradient of a Scalar Field
With , the directional derivative
of T is given by
If is a known function, the difference in temperature at points P2
and P1 is calculated by:
(read Example 3-1)
Gradient Operator in
Cylindrical and Spherical Coordinates
Using the coordinate transformation relations, the gradient
operator can be defined as
In cylindrical coordinates:
In spherical coordinates:
3-2 Divergence of a Vector Field
A vector field is usually represented by field lines.
Lets consider, for example, the flux lines of the electric field E
induced by a positive charge q in the space around it (from
Coulombs law)
The presence of the electric field is
regarded as a flux and we refer to its
fields lines as flux lines.
(cont) Divergence of a Vector Field
The outward flux crossing a unit surface ds
is defined as flux density, and for the
electric field example is
The total flux crossing a closed surface
This surface integral can be transformed into a volume integral,
and the result is known as the divergence theorem (or also the
Gauss-Ostrogradski theorem):
(cont) Divergence Theorem
The divergence theorem, , can be
enounced as follows:
The flux of a vector through any closed surface is equal to the
volume integral of the divergence of the same vector over the
volume bounded by this closed surface.
The divergence of the vector E in this example, is defined as:
It is common in practice to denote div E as .
That is, in Cartesian coordinates:
(cont) Divergence of a Vector Field
A vector field has positive divergence if the net flux out of
surface S is positive the volume contains a source of
flux.
If the vector field has negative divergence, the net flux is inward
into the volume is viewed as a flux sink.
In the case of a uniform field E, due to the fact that the same
amount of flux enters the volume as leaves from it, the
divergence is zero.
If , the vector field E is called solenoidal, or
divergenceless.
3-3 Curl of a Vector Field
Besides the gradient of a scalar and the divergence of a vector,
a third fundamental operator used in vector analysis is the curl
operator. This is used to describe the rotational property, or the
circulation of a vector field.
The circulation of a vector B is defined as the
line integral of B around a closed contour C
situated in the field of vectors B:
Circulation =
In Fig. (a), the circulation of the uniform field is zero. For the field
B induced by the infinite wire carrying a dc current I in Fig. (b),
the circulation result is
(cont) Curl of a Vector Field
The line integral (vector circulation) can be transformed into a
surface integral, and the result represents the Stokess theorem:
The Stokess theorem, can be enounced as follows:
The circulation of an arbitrary vector B along a closed curve C
is equal to the flux of the curl of this vector through a surface
S bounded by the curve C.
If curl B = 0, the field B is said to be irrotational or conservative,
because its circulation is zero.
(cont)
The curl of a vector field B, denoted curl B
or , is defined as the circulation of B
per unit area:
-The direction of curl B is , the unit normal of , defined in
accord to the right-hand rule.
-The area of the contour C is oriented such that the
circulation is maximum.
-In the case in which S is a closed surface, ,
because the contour C reduces to a point
and the circulation becomes zero.
(cont)
For a vector B expressed in Cartesian coordinates as
It can be shown that the curl definition leads to
3-4 Laplacian Operator
In Electromagnetics a frequently encountered combination is
divergence of a gradient of a scalar, i.e., div grad V, or .
For convenience, the divergence of the gradient of a scalar V, is called
the Laplacian of V, and is denoted by , (pronounced "del square").
That is,
Note that the Laplacian of a scalar function is a scalar.
(find the expressions for the Laplacian of a scalar in cylindrical and spherical
coordinates on the inside back cover of the textbook)
