91Ó°ÊÓ

Vector functions are a powerful tool in mathematics that allow us to describe geometric shapes in a multi-dimensional space using parameters. In this particular exercise, the vector function given is \( \mathbf{r}(u, v)=\sin v \mathbf{i}+\cos u \sin 2 v \mathbf{j}+\sin u \sin 2 v \mathbf{k} \). Here, \( \mathbf{r}(u, v) \) is a parametric surface depending on two parameters, \(u\) and \(v\). A vector function like this consists of:

• Parameters \(u\) and \(v\), which affect the shape and position of the surface.
• The unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\), which indicate directions along the x, y, and z axes respectively.
• Sinusoidal components, such as \(\sin v\), \(\cos u\), and \(\sin 2v\), that introduce wave-like characteristics to the shapes.

By varying \(u\) and \(v\), we explore different configurations of the surface, explaining its huge versatility. This specific function maps a point on a surface in 3D space for given values of \(u\) and \(v\), enabling visualization of complex geometric forms.

Grid curves are an essential concept when exploring parametric surfaces, like the one we analyzed in this exercise. They help us understand how the surface is generated by the vector function when either \(u\) or \(v\) is held constant. Let's delve into the details:

• u-constant Grid Curves: When \(u\) is constant, the vector function simplifies to \( \mathbf{r}(v) = \sin v \mathbf{i} + (\cos u \sin 2v) \mathbf{j} + (\sin u \sin 2v) \mathbf{k} \). Here, we observe that the curve is influenced by changes in \(v\), creating wave-like or sinusoidal patterns. This means these grid curves reflect how the surface behaves with varying \(v\), like horizontal stripes over the surface.


• v-constant Grid Curves: Conversely, keeping \(v\) constant, the vector function reduces to \( \mathbf{r}(u) = \sin v \mathbf{i} + (\cos u C) \mathbf{j} + (\sin u C) \mathbf{k} \), with \( C = \sin 2v \). This results in circular paths, as \(u\) varies, effectively showing vertical "slices" of the surface.

Understanding grid curves assists in visualizing complex surfaces by focusing on the impact of one variable while keeping the other fixed, and is crucial in interpreting parametric graph shapes.

Unit vectors are fundamental in the world of vector math, acting as the building blocks for expressing direction. In our given vector function \(\mathbf{r}(u, v) = \sin v \mathbf{i} + \cos u \sin 2 v \mathbf{j} + \sin u \sin 2 v \mathbf{k}\), they have specific roles:

• Direction Indicators: The unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) denote directions along the x, y, and z axes, respectively. Each has a magnitude of 1, ensuring that they only indicate direction and not distance.


• Permitting Combination: By combining them with coefficients derived from the sinusoidal functions, we define positions and shapes in space for every point described by the function \(\mathbf{r}(u, v)\). For instance, \(\sin v \mathbf{i}\) focuses on the x-direction behavior dictated by \(v\).


• Maintaining Orthogonality: The orthogonal nature of \(\mathbf{i}, \mathbf{j},\) and \(\mathbf{k}\) means they stay perpendicular to each other within the 3D coordinate system, simplifying calculations and understanding of the spatial geometry involved.

Understanding unit vectors helps in grasping how directions and magnitudes determine the geometry and orientation of surfaces described by vector functions. They are a key aspect whenever interpreting such parametric functions.