2.11:
Gradient and Del Operator
The gradient of a scalar field represents the direction and magnitude of the maximum spatial rate of change of the field. The gradient is always normal to a surface of constant value.
To understand this, consider the scalar field of the partial pressure of carbon dioxide emitted in smoke.
At any point in space, the partial pressure can be represented as a function of three coordinate axes.
The partial derivative of this pressure along the axes, when added together, produces a vector quantity called the gradient of partial pressure of carbon dioxide.
Its magnitude indicates the maximum rate at which the pressure changes, whereas its direction shows the direction in which the pressure changes most.
Mathematically, the gradient of a scalar field can be written as a vector operating over a scalar. This vector is called the del operator.
In cylindrical and spherical coordinates, the gradient of a scalar field is expressed by the transformation relation, where the first term denotes the del operator in these coordinate systems.
2.11:
Gradient and Del Operator
In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The gradient is a mathematical operator that gives both the magnitude and direction of the maximum spatial rate of change. Consider a person standing on a mountain. The slope of the mountain at any given point is not defined unless it is quantified in a particular direction. For this reason, a "directional derivative" is defined, which is a vector that gives the slope and direction. The gradient of the scalar field satisfies both these conditions.
The gradient has the following general properties: (1) It operates on a scalar function and results in a vector function. (2) It is normal to a constant value surface. This property is used extensively to identify the direction of vector fields. (3) The gradient always points toward the maximum change in the scalar function.
Mathematically, the gradient of a scalar function is expressed as
Here, 'p' is the scalar function. The term in the parenthesis is called the del operator. The del operator is a vector operator that acts on vector and scalar fields. It is a mathematical operator that, by itself, has no geometrical meaning. It is the interaction of the del operator with other quantities that gives it geometric significance.
Suggested Reading
- Griffiths, D.J. (2013). Fourth Edition. Introduction to ELectrodynamics. San Francisco, CA: Pearson. pp.13.
- IDA, N. (2015). Fourth Edition. Engineering Electromagnetics. Switzerland: Springer International Publishing. pp.57.