A point function is a physical quantity, which may be expressed as a continuous function of the position of a point in a region of space.
The region in which the physical quantity is specified is known as field.
Scalar field: When a scalar physical quantity is expressed by a continuous scalar point function θ (x, y , z), then the region (x, y, z) specified by θ is the scalar field and the function θ is a scalar point function or scalar field function. Now if the temperature of a room varies from one position to another inside the room, then it may be expressed as a function of space coordinates (x, y, z). This temperature is the scalar field and may be expressed as
T = T (x,y,z) =T (r)
where r is the position vector of a point (x, y, z).
Vector field. When a vector physical quantity is expressed by a continuous vector point function A (x, y , z) then the region specified by this function is a vector field. For example, the velocity v of the wind if varying from place to place, then
v = v (x, y , z) = v (r)
grad ϕ = ∂ ϕ / + ∂ϕ/∂y +k ∂ϕ/∂z
=( I ∂ /∂x +j ∂/∂y +k ∂/∂z) ϕ
is known as a vector field.