Question

2. There is a field on a hill whose surface can be closely appoximated by the raph of f(x, y) 1800 (2y2). A farmer wants to plow the field in a manner that will minimize the amount erosion during a rain storm. Our farmer was once a mathematics major but has forgotten much of his Calculus III. Use methods from Calculus III to explain to the farmer why the field should be plowed so that the furrow created by the plow runs along a contour of f in order to minimize the amount of erosion during a rain storm.

Answer 1

Functions of two variables behave similarly as a single variable function; their maxima and minima are often found at points where the tangent plane to the graph of the function is horizontal, i.e. where the gradient is zero. for example, Imagine that you are hiking along the path to the top of a hill. At the maximum of your elevation, the steepest direction of the hillside would be to your sides, perpendicular to your path. That is, your path is perpendicular to the gradient of the function that describes the elevation, and tangent to the level sets (or contours) of the function.