1.2 Physical Meaning of Gradient, Divergence, and Curl
3
1.2 Physical Meaning of Gradient, Divergence, and Curl
The aim of this section is not to teach the formulae and calculations of gradient,
divergence, and curl for various electromagnetic systems, as they are already present
in several good quality texts like Feynman’s Lectures Vol. 2 [2], Griffiths [3], Sadiku
[7], and Spiegel [8]. Here we intend to present only the physical meanings of these
popular terms of vector calculus, in order to assist the reader’s imagination to visualize
the behavior of electromagnetic waves in different types of systems that shall be
encountered throughout this book.
1.2.1 Gradient
Whenever one comes across the term gradient, he should immediately think of a
quantity, such as height, increasing or decreasing in a “particular direction.” English
meaning of the term gradient is slope, as in the case of a hill. For example, as shown
in Fig. 1.2a, when an object rolls down a hill its height (h) from the ground level
reduces, whereas during uphill motion (Fig. 1.2b) it increases. In both the cases,
there is a motion and distance traveled along x-direction too. Here, “height” is the
quantity which is varying with respect to x. The variation in case (a) (downhill
motion) is negative while in case (b) is positive, and hence gradient (i.e., slope)
∂h/∂ x is negative in the former while positive in the latter case. By closer inspection,
an additional piece of information is obtained from Fig. 1.2. One should notice that
in the shown physical system, the increase or decrease in the height h is taking place
along a particular direction, “x” in this case. Although height is a scalar quantity,
gradient is a vector! Other scalar physical quantities dependent on height, such as
potential energy (V = mgh), follow the same trend and exhibit similar behavior of
negative gradient in case (a) and positive gradient in case (b). In general, the gradient
of any scalar physical quantity is a vector, since it has a direction and can be shown
to abide by the laws of vector addition.
Fig. 1.2 Physical meaning of gradient
3
1.2 Physical Meaning of Gradient, Divergence, and Curl
The aim of this section is not to teach the formulae and calculations of gradient,
divergence, and curl for various electromagnetic systems, as they are already present
in several good quality texts like Feynman’s Lectures Vol. 2 [2], Griffiths [3], Sadiku
[7], and Spiegel [8]. Here we intend to present only the physical meanings of these
popular terms of vector calculus, in order to assist the reader’s imagination to visualize
the behavior of electromagnetic waves in different types of systems that shall be
encountered throughout this book.
1.2.1 Gradient
Whenever one comes across the term gradient, he should immediately think of a
quantity, such as height, increasing or decreasing in a “particular direction.” English
meaning of the term gradient is slope, as in the case of a hill. For example, as shown
in Fig. 1.2a, when an object rolls down a hill its height (h) from the ground level
reduces, whereas during uphill motion (Fig. 1.2b) it increases. In both the cases,
there is a motion and distance traveled along x-direction too. Here, “height” is the
quantity which is varying with respect to x. The variation in case (a) (downhill
motion) is negative while in case (b) is positive, and hence gradient (i.e., slope)
∂h/∂ x is negative in the former while positive in the latter case. By closer inspection,
an additional piece of information is obtained from Fig. 1.2. One should notice that
in the shown physical system, the increase or decrease in the height h is taking place
along a particular direction, “x” in this case. Although height is a scalar quantity,
gradient is a vector! Other scalar physical quantities dependent on height, such as
potential energy (V = mgh), follow the same trend and exhibit similar behavior of
negative gradient in case (a) and positive gradient in case (b). In general, the gradient
of any scalar physical quantity is a vector, since it has a direction and can be shown
to abide by the laws of vector addition.
Fig. 1.2 Physical meaning of gradient