- #1
kidsasd987
- 143
- 4
This is a bit counterintuitive to me that the gradient vector is always normal to the level curve
and the level surface.
lets say we have a function f(x,y)=z
then the gradient is,
f(x,y) partial derivative with respect to x*i +f(x,y) partial derivative with respect to y*j
what we actually get is,
dz/dx*i+dz/dy*j=grad_f(x,y)
then,
[(dz/dx*i)+(dz/dy)*j]/sqrt((dz/dx*i)^2+(dz/dy*j)^2)
this is always perpendicular to the level curve. But why does that direction always maximize the function?
and the level surface.
lets say we have a function f(x,y)=z
then the gradient is,
f(x,y) partial derivative with respect to x*i +f(x,y) partial derivative with respect to y*j
what we actually get is,
dz/dx*i+dz/dy*j=grad_f(x,y)
then,
[(dz/dx*i)+(dz/dy)*j]/sqrt((dz/dx*i)^2+(dz/dy*j)^2)
this is always perpendicular to the level curve. But why does that direction always maximize the function?