In calculus III, I recently took an exam. At the end of the test was a True/False question. I never know when to stop analyzing it. Because of my apprehension, I went to her office after the test and she told me that the answer was true; I marked false. I had nothing to go off of at the time, and tried to remember the question and where I could have erred, and I accepted that I probably just messed up somewhere. She also said that the question was simple and not intended to be a trick, and that almost everyone correctly marked true. Well we got the test back today, and I still think I am right, but only if you look at the problem really closely.. http://img535.imageshack.us/img535/7695/calciiitf.png [Broken] They way I see it, one of three possibilities could have gone down: 1.) The student writes the formula down for the directional derivative, barely pays attention to the diagram, sees that the formula is correct, and marks true. 2.) The student writes the formula down for the directional derivative, realizes that theta must be the angle between the gradient and the vector, notes that the vector in the diagram is not defined as the gradient, but then sees the perpendicular contours and realizes that it MUST be the gradient, and marks true. 3.) 1MileCrash realizes that although the contours are perpendicular to a vector, they are perpendicular to the wrong one, the unit vector u. Because of this, u is in the direction of the gradient, and the directional derivative is just ||grad f || since that theta is clearly not 0. 1MileCrash hesitantly marks false, wondering if he's overthought the problem. Did I overthink the problem? What should I do? In my eyes, vector v here means absolutely nothing, so theta means absolutely nothing.