91Ó°ÊÓ

A vector function is simply a function where each input value, typically represented as a parameter like \(t\), is associated with a vector as the output. In the context of the exercise, the vector function is \(\mathbf{r}(t) = \langle \sqrt{t} \cos t, \sqrt{t} \sin t, t \rangle\). This function consists of three components:

• \(x = \sqrt{t} \cos t\)
• \(y = \sqrt{t} \sin t\)
• \(z = t\)

These components describe how each coordinate \(x, y,\) and \(z\) changes with respect to the parameter \(t\). Vector functions are useful in many areas of mathematics and physics to describe curves or paths in space. They allow us to move away from static coordinate forms and describe dynamic movement along a path.

To determine if a curve, defined by a vector function, lies entirely on a surface, we compare the mathematical descriptions of both the curve and the surface. In the exercise, the surface is defined by the equation \(z = x^2 + y^2\), which represents a paraboloid. To confirm that the curve \(\mathbf{r}(t)\) lies on this surface, we verify whether its components satisfy the surface equation.
Start by substituting the vector function components into the surface equation. From the vector function, \(x = \sqrt{t} \cos t\) and \(y = \sqrt{t} \sin t\). By squaring these, we derive:

• \(x^2 = t \cos^2 t\)
• \(y^2 = t \sin^2 t\)

Adding these gives \(x^2 + y^2 = t\), which matches \(z = t\) in the vector function, proving the curve lies on the surface. This method of matching components is a common technique to analyze such problems.

Rectangular coordinates, often referred to as Cartesian coordinates, are a system used to denote a point in space by assigning it a coordinate tuple like \((x, y, z)\). In our task, these coordinates help us express the spatial position of points on our vector function \(\mathbf{r}(t)\) and on the surface \(z = x^2 + y^2\).While dealing with problems, transforming between vector function formats and rectangular coordinates gives us the flexibility to analyze points readily using geometry and algebra. For example, by knowing \(x = \sqrt{t} \cos t\) and \(y = \sqrt{t} \sin t\), we can easily check if they meet the condition of the surface equation \(z = x^2 + y^2\) by working in a system where operations like squaring and addition have clear geometric meanings.

Parametric equations express a curve using one or more parameters rather than directly relating variables like \(x\), \(y\), and \(z\). In the vector function for \(\mathbf{r}(t)\), each coordinate \(x, y,\) and \(z\) is expressed in terms of a single parameter \(t\). These equations are valuable because they allow the precise tracing of a path in space as \(t\) varies across any range of interest. Parametric equations can easily deal with complex curves and movements, offering a broad and flexible framework for descriptions. By setting \(t\), you effectively "animate" the path the curve will trace through space, showcasing its position at every instance of \(t\) and ensuring comprehension of its interaction with any given surface, like the paraboloid in our example. Mastering parametric equations is crucial for studying curves and surfaces because they simplify complex interactions and enhance geometric interpretation.