Intuition behind stationary points in a multivariable function Why should the local optima for a differentiable function $f(x,y)$ be at the stationary point(s)? What is the intuition behind setting the first partials equal to zero? Is it because if one of the partials is positive (negative), then we can move in (away from) that variable's direction and increase the function value?
From geometry, $f(x,y)$ is a 2D surface, $df$ is the vector field on this 2D surface, while its dual $\star df$ tangent to the contours on the surface.
From the aspect of vector field $df$. The stationary points of $f(x,y)$ are those points, the vectors at which are of zero length. In other words, all the vectors either converge to those point, or emit from them.
From the aspect of dual vector field $\star df$. The stationary points of $f(x,y)$ are those points, around which all the dual vectors are running.
From geometry, $f(x,y)$ is a 2D surface, $df$ is the vector field on this 2D surface, while its dual $\star df$ tangent to the contours on the surface.
From the aspect of vector field $df$. The stationary points of $f(x,y)$ are those points, the vectors at which are of zero length. In other words, all the vectors either converge to those point, or emit from them.
From the aspect of dual vector field $\star df$. The stationary points of $f(x,y)$ are those points, around which all the dual vectors are running.