What is meant by the "contour of a function?"

Question

Suppose that we have $f(x,y)=(x+y)^2.$ What is meant by the "contour of a function," and what is an analytic expression for it? All software, such as Matlab, Mathematica,.. gives just a function like ContourPlot[] without details of the analytic expression. Further, I read in a book that the direction of the gradient gives the normal to the contour. Thanks for suggesting any details.

Answer 1

A contour line of a function, $f$, is the set of solutions to $f(x)=c$ for some fixed $c$. Usually this is used in the context of multivariate calculus to help visualize functions and the connections between the values of the variables.

A contour plot of a function is the graph of a series of contour lines, usually with some fixed incremental value for the $c_i$. Contour plots are generally in the number of dimensions as the function, and don't actually display the values of the function (though sometimes the contour lines are labeled). For example, one might use the set $\{c_i=i\}$ and graph the contour lines $$f(x)=0,f(x)=1,f(x)=2,\ldots$$This is a great way to display a function when the goal is to understand how it grows, as the density of contour lines shows how steep the function is.

The gradient is, as you note, always normal to the contour lines, and so contour plots can be used to visualize how that turnstile changes, much like using slope fields to represent differential equations.

In real life applications, contour plots allow us to easily represent maps that are useful for exploration that might feature important changes in altitude in two dimensions.