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http://math.stackexchange.com/quest...always-be-directed-in-an-increasing-direction

I understand the proof here. But.. the idea here is del(f)*dl = df is maximized when del(f) and dl point to the same direction, and that maximizes df. Then we have to first consider the direction of dl to verify where del(f) points to.

If we assume that there is a multivariable function f(x1, x2, x3, . . . xn)

and lets say that the derivative with respect to xj is a negative value at p0.

(also derivatives with respect to other variable x1, x2, x3 . . . xn are positive)

which indicates that the peak is at the left of the graph (at the negative direction with respect to point p0)

then, del(f)*dl = df will be maximized when df/dxj*(-dxj) because it would give positive incremental df, since the derivative is negative at p0.

also, we can do this because dl is a vector quantity so we can define its direction as we want it to be.

But, total differential doesnt take this into account. It just multiplies a small increment of each variable, dxi. and they all have the same sign.

I think this is a contradiction.