Hello. I can't seem to wrap my head around the geometry of the gradient vector in ℝ3(adsbygoogle = window.adsbygoogle || []).push({});

So for F=f(x(t),y(t)), [tex]\frac{dF}{dt}=\frac{dF}{dx}\frac{dx}{dt}+\frac{dF}{dy}\frac{dy}{dt}[/tex]

This just boils down to

[tex]\frac{dF}{dt}=∇F \cdot v[/tex]

Along a level set, the dot product of the gradient vector and velocity vector equals zero. But going uphill/downhill, the dot product no longer equals zero.

To my understanding, the velocity vector lies on the tangent plane, which is perpendicular to the gradient vector. By this logic, the dot product should always equal zero.

What am I missing here? What is the geometry of the gradient vector and velocity vector in relation to one another?

Edit: Is it possible that I'm just confusing the gradient vector with the normal vector? In that case, the dot product of the normal vector and velocity vector will always equal zero, right?

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# Vector Calculus - gradient geometry

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