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?