Gradient is a vector operator defined by

∇ =i ∂_x + j ∂_y + ⋯,

where, ∂_𝑥=∂/∂x.

For now consider the 2D case :

∇ A=i ∂_x A + j ∂_y A.

Gradient operating on a scalar "A" and retuns a vector ∇ A. For a given function 𝑓(x,y), ∇ f contains partial differentiate of 𝑓(x,y) with respect to each independent variable and stick the respective unit vector in front of it.

The function f(x,y) has been changed by an amount df and it can be written as

df = (∂_x f(x,y)) dx+ (∂_y f(x,y)) dy = (i ∂_x f(x,y) + j ∂_y f(x,y) ∂y) . (i dx+ j dy)

= ∇ f . dr

where dr = (i dx+ j dy). Therefore,

df = |∇ f | |dr| cos(𝜃)

Physical Inyterpritation : When θ=0° (cos(𝜃)=1), The Gradient vector points in the same direction as the displacement vector. Moreover the change df is maximal in this case (θ=0°), because cos(𝜃) is less than 1 for all other angles.

Hereby we conclude that :

• |∇ f | represents for the maximal rate of change of f.

• The direction ∇ f, gives the direction of maximal change or the "steepest ascent".