Answer
See proof
Work Step by Step
Step 1: We are given a figure of a surface and we want to determine if the given surface can be oriented. To explain what is meant by an orientable surface, we can imagine a vector \(\mathbf{n}(x, y, z)\) that is normal to the tangent of the surface at some point. If we can 'drive' this vector over the surface for all the points on one side of the surface, and this vector never crosses to the other side (it doesn't take both the values \(\mathbf{n}(x, y, z)\) and \(-\mathbf{n}(x, y, z)\)), then the surface is orientable. Step 2: The given figure can confuse us into thinking that this is a Möbius strip, which is a non-orientable surface obtained by taking one side of a rectangle and twisting it once by \(180^\circ\) and then connecting it to the other side. If you now 'drive' the vector \(\mathbf{n}\) over a Möbius strip, you will reach all the points and 'both' sides. So this is a non-orientable surface. The given surface is not the Möbius strip, since it has two \(180^\circ\) twists, so topologically thinking, it has come back to the initial state of reaching both sides of the former rectangle. Also, note that if we twist for an odd number of times, we get a non-orientable surface, and for an even number of times, it is orientable. So, the given surface is orientable.
