Differential Geometry

When studying curves, we studied how the curve twisted and turned in space. We now turn to surfaces, two-dimensional objects in three-dimensional space and examine how the concept of curvature translates to surfaces. In Calculus 3, you have encounter surfaces defined as graphs of real valued functions of two variables z = f(x, y). This function also can take the form x = f(y, z) or y = f(x, z). In some cases, on the other hand, this function is given implicitly as F (x, y, z) = 0. For example, a sphere of radius a is given by x + y + z = a and it is impossible to get a single two variable function that would describe the whole sphere. Cylinder x + y = a is another such example. Let us review some examples from Calculus 3.