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3.21 Contours

A suitable function $f:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}$ of the form $f(x,y)$ can give rise to a curve in the plane.

Example (Contour).

Let $f(x,y)$ be the height of land at point $(x,y)$ . A walker decides to walk at constant height $h$ , so his path is the level curve

C_{h}=\{(x,y):f(x,y)=h\}

which follows the contours of his Ordnance Survey map. His direction of travel is a tangent to $C_{h}$ , and we regard this as parallel to $(dx,dy)$ .