Contour Maps, Gradient Vectors, and Directions Derivatives

For our final project in Multivariable Calculus, we chose to focus on contour maps, gradient vectors, and directional derivatives. To help visualize what a contour map is, one can compare a contour map to a collapsible camping bowl. The bowl can represent the graph of f(x,y)=x​2​+y​2​. When the bowl is collapsed the x and y coordinates are preserved, but the once three-dimensional object is now two dimensional. Each circle on this new flat surface corresponds to a certain z value or height that the bowl had before it was collapsed. Each of these circles is known as a contour line. Another way to think about what a contour map is is to picture the three-dimensional graph with a horizontal plane intersecting it. As you move the plane the places where the two graphs intersect are the contour lines for that specific z value. This idea can then be applied to more much complex functions as well. From these contour maps, one can easily visualize a special vector known as the gradient. This gradient vector points in the direction of greatest change at a specific point, also known as perpendicular to the contour curve. In order to find the gradient vector, one must calculate ∇f=(∂f/∂x, ∂f/∂y) where f is a scalar function. The gradient vector is then also used the calculate the directional derivative of a function using the equation f​v​=∇f・v. The directional derivative is the derivative, or rate of change, of a function as we move in a specific direction defined by the unit vector (a vector of the length one) v.

From a variety of contour plots found, we chose simple maps that could be easily understood and came from several different fields where one can utilize contour maps. From there we decided to create our own functions to further demonstrate contour maps and to introduce gradient vectors. We used Mathematica in order to create the contour maps for these functions. We then calculated the general formula for a gradient vector at any point for those functions and graphed that as a vector field. When graphed together, you can see the greatest rate of change, the gradient vectors, from any point on the contour map are perpendicular to the contour lines. Since many of the gradient vectors didn’t line up exactly with the curves we decided to make a simpler example using the function f(x,y)=-x​2​-y​2​ to better show that relationship. From researching different contour maps we were able to find a topographical map of the entire world, so we decided to use Bryn Mawr College and Haverford College to calculate estimations for the directional derivatives between buildings we frequently visit. First, we simply calculated the direct distances using the approximate coordinates at different places. We also determined the total change in function value, i.e. height as we go from one place to another. Using these values, we calculated an estimate of the rate of change in height per unit distance traveled as a student/staff goes from one building inside the premises to another, which is the directional derivative along the path taken. In addition, we also calculated an estimation of the average elevation of Bryn Mawr’s campus.

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