Suppose that a scalar quantity is represented by 'A'. Value of 'A' depends upon its position in space. In other words we can say that there is a Scalar Field ‘A’ and this scalar field 'A' is function of p, q and r. Equation of this scalar field can be given as:
A = A (p, q, r)
It is always found that for a given equi - scalar surface
A (p, q, r) = constant,
Above equation defines the surface for which function A (p, q, r) is found. Equi - scalar surface is the surface for which 'A' is defined.
For different values of 'A' different surfaces can be defined in a systematic Set.
Ascendant of 'A' can be expressed as:
(∂A / ∂p)i^ + (∂A / ∂q)j^ + (∂A / ∂r)k^,
Where i^, j^, and k^ are unit vectors.
It may also be written as:
Ascendant of A ≡ ▼A, where ▼ is 'del' operator and is given by
▼= (∂/ ∂p) i^ + (∂ / ∂q) j^ + (∂ / ∂r) k^,
When we use a negative (-) sign in front of ▼A, then it will be called as gradient of 'A' (or grad A);
That is grad A ≡ - ▼A.
Grad A indicates vector has been directed towards lower values of A.
It can be seen that ▼A is normal to equi scalar surface at the Point p, q, and r.
The gradient is also called gradient vector field which is normally defined for a Scalar Function.
If we take a function f (y1, y2, y3, …., yn), then scalar function is denoted by ▼f
where ▼ is differential or del operator.