91Ó°ÊÓ

Cylindrical coordinates provide a fascinating method to describe points in a three-dimensional space. They are like an extension of polar coordinates where you add the height or depth to the mix. In cylindrical coordinates, any point is described using three values:

• \(r\) - the radial distance from the origin to the point projected onto the \(xy\)-plane.
• \(\theta\) - the angle that the radial line makes with the positive \(x\)-axis.
• \(z\) - the height of the point from the \(xy\)-plane.

This form is particularly useful for problems involving rotational symmetry around an axis, as it aligns well with cylindrical shapes like tubes or washers. By expressing Cartesian equations through these coordinates, you often simplify complex geometry, making analyses more manageable.

A surface equation defines the z-value of a surface based on x and y coordinates or, in a broader sense, based on some parameters like in our example with cylindrical coordinates. The given equation for the surface is a straightforward algebraic expression:\[ z = \frac{1}{1 + x^2 + y^2} \]This equation sets up a surface where the z-coordinate depends inversely on the square sum of the x and y coordinates. This sort of equation is typical for creating surfaces like paraboloids or spheres in a 3D space.
In cylindrical coordinates, substituting \(x = r\cos\theta\) and \(y = r\sin\theta\) into this equation transforms it with polar elements, yielding \[ z = \frac{1}{1+r^2} \]. This highlights how the surface shrinks towards the origin as either \( x \) or \( y \) increases, creating a bowl-like shape.

Algebraic expressions serve as the building blocks of mathematical surfaces. In the context of surfaces in 3D, they functionally relate coordinates of a point on a surface. Algebraic expressions use operations like addition, multiplication, and exponentiation on numbers and variables to shape an equation, just like the one seen in this exercise.The key algebraic step in this problem is simplifying the expression when changing variables. The formula's denominator \(1 + x^2 + y^2\) in Cartesian coordinates transforms via substitutions based on the radial and angular description in cylindrical format:

• Express \(x\) and \(y\) using \(r\) and \(\theta\).
• Simplify \(r^2(\cos^2\theta + \sin^2\theta)\) knowing \(\cos^2\theta + \sin^2\theta = 1\).
• Finalize it as \(1 + r^2\).

Such simplification helps in achieving a more manageable expression to describe surfaces parametrically, enhancing both comprehension and use in practical applications like engineering and physics.