Relationship between partial derivatives

[Niishi]

Hello,
Can anyone please tell me how to get the relationship between partial derivatives at a point, that is, dy/dx|x = - df/dx|y / df/dy|x ?

 

[epenguin]

You maybe meant dy/dx|f = - df/dx|y / df/dy|x ?

 

[mathwonk]

f(x,y) = c, implies df/dx dx + f/dy dy = 0, now solve for dy/dx.

 

[mathwonk]

the geometry is that a curve x(t),y(t) moving in the level set where f= c, has velocity vector (dx/dt, dy/dt) which is perpendicular to the gradient vector of f: (df/dx, df/dy).

 

[Niishi]

Thanks mathwonk i was able to get it. I did not understand the explanation in the second the post particularly why t has been introduced etc.

 

[mathwonk]

well saying df/dx dx/dt + df/dy dy/dt = 0, is a dot product statement involving grad f and the velocity vector of (x(t),y(t)).

it is a little harder to make geometric sense out of the similar equation df/dx dx + df/dy dy = 0, in terms of differentials.