2 Notes Vector Calculus 1
2 Notes Vector Calculus 1
2 Notes Vector Calculus 1
1. Vector Calculus
Because the electric and magnetic fields are vector quantities, because they are all around us (in three-
dimensional space), and because they are governed by differential equations, we have to introduce
some new mathematical concepts in order to describe them. While you will be used to differentiation
and integration, we need to be able to think about differentiating and integrating vector quantities -
the ideas we will use are called, for this reason, Vector Calculus. Vector calculus can look a bit
daunting, however we will avoid anything too unpleasant.
1.1 Scalars
Some of the basic quantities used in electromagnetics either scalars (quantities which have only a
magnitude). Scalars can be real or complex and they can be functions of space and/or time.
An example of a scalar is the number 6 - this is a constant (not a function).
An example of a scalar function is f(x)=x2 (this is a function of one variable, x). The ideas of
differentiation and integration of such a function are straightforward.
If we wanted to describe the temperature (a scalar quantity) at all points in a room, at one point
in time, we would require a scalar function of 3 variables, say g(x,y,z). Differentiation and
integration of such functions are more involved (see below).
If this temperature varied with time we would need 4 variables, say g(x,y,z,t).
1.2 Vectors
Many of the quantities in electromagnetics have direction as well as magnitude. We call these
“vectors” and print them in bold (or underline them if you are writing by hand).
Vectors can also be functions of space and/or time:
If we wanted to describe the movement of air (a vector quantity) at all points in a room, at one
point in time, we would require a vector function of 3 variables, say G(x,y,z). In other words, at
any point in the room, the air can be moving in any direction.
If the air movement varied with time we would need 4 variables, say G(x,y,z,t).
Lines and Waves Notes 2020
where Ax, Ay and Az are the components of A in the direction of each of the three Cartesian unit
vectors.
Vector manipulation (addition, subtraction, multiplication) is necessary because many of the
electromagnetic quantities are vectors. Furthermore, the use of certain differential operators allows
the fundamental postulates and formulas to be expressed in a concise and general manner.
B B B
C C
A A A
(a) Two vectors, A and B (b) Parallelogram rule (c) Head-to-tail rule
2
Lines and Waves Notes 2020
If for example the first vector represents force and the second direction then the dot product is a
measure of work. The dot product is widely used in electromagnetic theory, for example when
calculating the work done in moving through a magnetic or electric field, and in calculating the flux
of a vector field.
Note: the order of the dot product is not important, i.e. A·B=B·A. Furthermore, the dot product of a
vector taken with itself is equal to its magnitude squared, i.e. A·A = A2
A B
ABsin AB
n AB A
Figure 2: Vector multiplication - Cross Product
The vector cross product is an important mathematical operation in electromagnetic theory; for
example, the power density of an electromagnetic wave is defined in terms of the vector
multiplication of the electric and magnetic fields, i.e. EH.
When talking about the rate of change of T(x, y, z), we therefore have to remember that it will depend
on the direction in which we move. This is the idea of partial differentiation:
𝜕𝐶 𝜕𝐶
gives the change in temperature as we move in the x direction, 𝜕𝑧 gives the change in temperature
𝜕𝑥
as we move in the z direction. If we collect together all the possible partial derivatives we get the
gradient of our scalar field T (usually written grad T or T):
𝜕T 𝜕T 𝜕T
In Cartesian co-ordinates: ̂ +𝒚
∇𝑇 = 𝒙 ̂ + 𝒛̂
𝜕𝑥 𝜕𝑦 𝜕𝑧
At a given point in space it is the vector which represents the direction and magnitude of the
maximum spatial rate of increase of that scalar. To find the rate of change in any particular direction,
given by a unit vector 𝒂 ̂, we form the dot product 𝒂
̂ ⋅ ∇T. The most obvious example of this is if
𝜕T
aˆ = xˆ , in which case 𝒂
̂ ⋅ ∇𝑇 = 𝒙
̂ ⋅ ∇𝑇 = - which is indeed the rate of change in the x direction.
𝜕𝑥
For example, the static electric field intensity E is derivable as the negative gradient of a scalar
electric potential V; that is E = - V. Imagine a landscape where the height above ground represents
the electric potential V - this concept is illustrated in Figure 3; At any particular point the gradient
of the potential is the direction and magnitude of the steepest slope.