Let's define a function f(x,y,z) ,where (x,y,z) are points in 3-d space.Let's take value of our function at some point in space (x,y,z) . Now on moving from this point to SOME point in near by space, gives us maximum rate of change to our function .WHICH point????????
Answer to this question is given by Gradient of that function. Mathematically,Gradient of a function is a vector that represents both the magnitude and direction of maximum space rate of increase of that function.
gradient of function ∇ f= ∂ f/ ∂ x Ax + ∂ f/∂ y Ay +∂ f/∂ z Az.
- Now,rate of change of our function "f" in 3-D space along any direction can be found by multiplying magnitude of gradient vector with cosine of angle b/w given direction and gradient vector.!!!!.This is called DIRECTIONAL DERIVATIVE .
-Now,consider a region in 3d space where magnitude of our function remains same.this region or surface is called equipotential surface.Here rate of change of function is zero. Applying above result , |∇f|COSΘ=0 . So,Gradient of a function is perpendicular to equipotential surface !!!!.