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A parametrized surface is a way of describing a surface in space where two parameters, often labeled as \(u\) and \(v\), each take on values that define coordinates in three-dimensional space. You can think of these parameters as coordinates in a new system that uniquely determines points on a surface. For instance, in the case of the cone given in the problem, the parameters \(r\) and \(\theta\) define every possible point on the surface of the cone through the use of trigonometric functions.
The parametrized vector function \(\mathbf{r}(u, v) = f(u, v) \mathbf{i} + g(u, v) \mathbf{j} + h(u, v) \mathbf{k}\) helps in expressing surfaces as sets of equations.
This allows us to describe complex surfaces in a simpler way, where \(f\), \(g\), and \(h\) dictate the shape and structure of the surface, and \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) are the standard unit vectors in 3D Cartesian space. It is often used in conjunction with calculus to analyze the properties of surfaces, such as curvature and tangent planes.

The cross product is a vector operation fundamental in finding a vector perpendicular to two given vectors in three-dimensional space. Given two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) results in a third vector that is orthogonal to both \(\mathbf{a}\) and \(\mathbf{b}\). This property is extremely useful in finding normal vectors to surfaces.
For the exercise at hand, the tangent plane to a surface at a given point is specified using the normal vector derived from the cross product of the partial derivative vectors \( \mathbf{r}_u \) and \( \mathbf{r}_v \). In forming the determinant, the i, j, and k unit vectors help structure the cross product, resulting in a mathematically concise vector orthogonal to the surface at the point of interest.
This step is crucial for establishing the orientation of the tangent plane, as the plane's equation uses the normal vector to describe all points lying on it that are also tangent to the surface.

Partial derivatives extend the concept of a derivative to functions with multiple variables. Instead of considering the rate of change with respect to one variable, partial derivatives look at how a function changes as just one of its multiple inputs varies.
In this context, dealing with the parametrized surface \(\mathbf{r}(r, \theta)\), we calculate partial derivatives \(\mathbf{r}_r\) and \(\mathbf{r}_\theta\), which represent the rate of change of the vector function in the directions of \(r\) and \(\theta\), respectively.

• \(\mathbf{r}_r\) gives the change as \(r\) changes
• \(\mathbf{r}_\theta\) gives the change as \(\theta\) changes

Derivative calculation is fundamental in assessing the directional growth of the surface and in this exercise, it serves as a preliminary step toward establishing the tangent plane by determining the vectors required for the cross product.

A Cartesian equation is a mathematical representation of surfaces or curves in space using x, y, and z coordinates. For surfaces derived from parametric equations, it's often useful to express the relationship between coordinates directly.
In this exercise, the parametric cone can ultimately be described in Cartesian form with the equation \(z^2 = x^2 + y^2\). The transformation from parametric to Cartesian form provides a significant simplification in understanding the surface shape.
Such an equation is powerful for sketching surfaces and solving geometrical problems, as it directly relates the spaces in x, y, and z without additional parameters. Converting to a Cartesian form allows one to consider the symmetries and specific geometric features of the surface, making it more intuitive for spatial reasoning and further mathematical manipulation.