91Ó°ÊÓ

A parametrized surface is a surface in three-dimensional space described using two parameters. Essentially, you can think of it as a map which takes two variables, say \( u \)and \( v \), and assigns a point in space to each pair of values. In this problem, the parametrized surface is given by:
\[\textbf{x}(u, v) = (f(u) \, \text{cos} \, v, f(u) \, \text{sin} \, v, g(u)) \]
Here, both \( f(u) \) and \( g(u) \) are functions of \( u \). When you change the values of \( u \) and \( v \), you get different points on the surface. This surface wraps around the \( z \)-axis, and its exact shape depends on the functions \( f \) and \( g \).

For example, if \( f(u) = u \) and \( g(u) = u \), this describes a helical surface. The surface spirals up around the \( z \)-axis. By varying the parameterizations, you can create a wide variety of surfaces.

A normal vector is a vector that is perpendicular (orthogonal) to a surface at a given point. In the context of our parametrized surface, we can find this vector by using the cross product of the partial derivatives of the surface equation with respect to the parameters \( u \) and \( v \).

The partial derivatives of \( \textbf{x}(u, v) \) with respect to \( u \) and \( v \) are: \[\textbf{x}_u = \frac{\text{d} \textbf{x}}{\text{d} u} = (f' \text{cos} \ v, f' \text{sin} \ v, g'(u)) \] \[\textbf{x}_v = \frac{\text{d} \textbf{x}}{\text{d} v} = (-f \text{sin} \ v, f \text{cos} \ v, 0) \]
The cross product of these two vectors gives us the normal vector, \( \textbf{N} \): \[\textbf{N} = \textbf{x}_u \times \textbf{x}_v \]
After computing the cross product, we get the normal vector: \[\textbf{N} = (-g'(u)f \ \text{cos} \ v, -g'(u)f \ \text{sin} \ v, ff') \]
This vector is orthogonal to the surface at every point \( (u,v) \) on the surface.

The cross product is a binary operation on two vectors in three-dimensional space. It results in a third vector which is orthogonal to the plane containing the original vectors. This is particularly useful in finding normal vectors to surfaces.

If you have two vectors, \( \textbf{A} = (A_x, A_y, A_z) \) and \( \textbf{B} = (B_x, B_y, B_z) \), their cross product \( \textbf{C} = \textbf{A} \times \textbf{B} \) is defined as: \[\textbf{C} = (A_yB_z - A_zB_y, A_zB_x - A_xB_z, A_xB_y - A_yB_x) \]
This cross product is particularly useful in differential geometry for determining the normal vector of a surface at a specific point. Let's apply it in our problem:
Given our partial derivatives: \[\textbf{x}_u = (f' \cos v, f' \sin v, g'(u)) \] \[\textbf{x}_v = (-f \sin v, f \cos v, 0) \]
The cross product \( \textbf{x}_u \times \textbf{x}_v \) is computed as:
\[\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ f' \cos v & f' \sin v & g'(u) \ -f \sin v & f \cos v & 0 \ \end{vmatrix} \]
Carrying out the determinant, we find: \[\textbf{N} = (-g'(u)f \cos v, -g'(u)f \sin v, ff') \]
This final vector represents the normal to the surface at any point specified by \( (u,v) \).