Main Question or Discussion Point
In the context of height fields, the geometric meaning of partial derivatives and gradients is more visible than usual. Suppose that near the point (a, b), f(x, y) is a plane (the above figure). There is a specific uphill and downhill direction. At right angles to this direction is a direction that is level with respect to the plane. Any intersection between the plane and the f(x, y) = 0 plane will be in the direction that is level. Thus the uphill/downhill directions will be perpendicular to the line of intersection f(x, y) = 0.
I read a book about gradient vector but I don't understand what the above underlined sentences meant.
Can you tell me what they mean?