A particle exists in three dimensions and is trapped inside a solid S. The cross section of the cylinder C on the xy plane is the region bounded between r = cos (0) and r = sin (0) in the first quadrant. All the points in the solid S exists inside the cylinder C bounded between the planes z = -y and z = y. Before we can get into the particle dynamics, we need to introduce the notion of "inner product." You are already familiar with the dot product of two vectors. Dot products give a sense of how much the two vectors are aligned with each other. In layman's terms, a function can be thought of as a vector with infinite components. Lets put this in perspective, a 3d vector a=a+a+ak has three components ar, ay, and az. Now, lets consider a function f defined by the map f: x 7x, where x = [-3,2]; then the "th" component of the "vector" f is just f (x), or equivalently 7x. Since x can take the value of any real number between -3 and 2 and that there are infinite real numbers between any two numbers, then it follows that the "vector" f has infinite components. In the case of the dot product, we have to take a summation, and in the case of an inner product, we have to take an integral. The definition of the inner product of two functions g and h (denoted by (g|h)) over a volume S is defined as g(x, y, z)* h (x, y, z) dV,