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Explanation of the Animation There's only one function being considered in the animation above. It doesn't matter what it is (roughly speaking it's f(x,y) = cos2(x)sin2(y)). It is however differentiable everywhere, which means anywhere we choose to infinitely magnify the surface representing the function, the result will look like a plane. We can do this to the surface that appears in the animation by imagining that the portion of it we see lies above a rectangle on the x,y-plane. This rectangle is represented by the red rectangle in the upper left corner of the animation. Click anywhere on it after the plot is complete and a new plot will appear: a magnification of the portion of the undulating surface above the point on the x,y-plane we chose. This can be done repeatedly. To reset back to the initial undulating surface, click on the reset button. It's a simple animation: just two things to do. Extrema. Multivariable functions arise in many contexts. For example, suppose you're a bug on the salt flats out in Utah, and suppose you don't like salt. You can't fly, so feel all around you with your antennae until you determine the direction in which it is least salty. Let c(x,y) be the salt concentration function for the salt flats, and let cx(x,y) and cy(x,y) be the partial derivatives of c(x,y). Ok, once you determine the the direction of least saltiness you step in that direction. Well, the gradient [cx, cy] is the direction of maximum increase in concentration, so [-cx, -cy] determines the direction in which you stepped. (This would be really easy to simulate on a computer.) Ok, so you take a step, feel around, take a step, feel around, and with each step the concentration of salt is less than it was before, until (and if) you get to a point where in every direction the saltiness is greater than where you are (actually, all sorts of things could happen on the way that complicate the picture, but we don't have the time to discuss every conceivable scenario (do we?)). At this point [cx, cy] must be equal to [0,0] (so the surface infinitely magnified at that point would look like a plane parallel to the x,y-plane), and the slope of c(x,y) must be positive just a short distance away from where we are. That means we're at a local minimum. It may not be the least salty point on the plane, but it's less salty than all the points in a local neighborhood around where we are. Even our path to this point was probably not the shortest; it probably wound around a bit, but we're simple little bugs, and for us the best we could do was to follow a path always opposite the direction of the gradient. As is true of 1-dimensional functions, having both partial derivatives equal to zero is not enough to ensure we're at a local minimum (or maximum). For a point (x,y) to be a local minimum, for every unit vector [u,v], the directional derivative of c in that direction (ucx + vcy) must be positive a short distance away from (x,y) in the direction [u,v]. There are other ways of determining if a point is a local extrema using second partial derivatives, but we'll give it a rest for now. |