The gradient of a two dimensional function is effectively the same as solving for a normal vector at (x,y) on a curve in the x y plane. Setting f(x,y) to a constant can be thought of visually as isobars on a weather map. The three variables of pressure variation by x and y are represented two dimensionally by drawing lines through points of constant value f. The normal to one of these isobars points toward the center of a high pressure or low pressure system. The gradient therefore is pointing in the direction of greatest incline. So in both uses of gradient above, that is, f(x,y) = constant and F(x,y,z) = constant, the gradient is used the same way to find the normal vector. How can this be, if both represent 3 D surfaces? f(x,y) represents a 3 D surface, but f(x,y) = constant does not.
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