Sora began at the base. To find the fastest way up, she used her . "The gradient vector
Finally, Sora saw the peak, but there was a catch. A sacred boundary line—a circular fence defined by
She invoked the . She looked for the spot where the gradient of the mountain was perfectly parallel to the gradient of the fence (
She planted the flag, knowing that in Cartesia, every curve had a story, and every surface had a slope.
Near the summit, Sora reached a strange clearing. To her left and right, the ground rose like high walls. In front and behind, the ground dropped off into deep canyons."A ," she whispered. Her compass spun wildly; the slope was zero, but she wasn't at the top. She used the Second Derivative Test . By calculating the discriminant (
), she realized she was at a critical point that was neither a peak nor a valley. She had to push past the equilibrium to find the true summit. The Lagrange Constraint
always points toward the steepest ascent," she reminded herself. Every step she took was in the direction of the greatest change. If she turned 90 degrees, she’d be walking along a , staying at the exact same altitude—safe, but getting nowhere. The Fog of Partial Derivatives