Identify the surface with the given vector equation. r(s, t) = s sin(9t), s², s cos(9t)

Solution:

Given, r(s, t) = s sin(9t), s², s cos(9t)

Here, these are the respective x, y, z axes components.

1) component along x-axis \(r_{i}=s\, sin9t\)

2) component along y-axis \(r_{j}=s^{2}\)

3) component along z-axis \(r_{k}=s\, cos9t\)

From the parameterised equation,

\(r_{i}^{2}+r_{k}^{2}=s^{2}sin^{2}9t+s^{2}cos^{2}9t\)

We know, \(cos^{2}x+sin^{2}x=1\)

So, \(r_{i}^{2}+r_{k}^{2}=s^{2}(sin^{2}9t+cos^{2}9t)\)

\(r_{i}^{2}+r_{k}^{2}=r_{j}\)

This can also be written as x² + z² = y

This is similar to an equation of a parabola in 1 dimension.

By fixing the value of z = 0,

We get y = x² which is the equation of a parabola curving towards the positive infinity of the y-axis and in the x-y plane.

By fixing the value of x = 0,

We get y = z² which is the equation of a parabola curving towards the positive infinity of the y-axis and in the y-z plane.

Thus, by fixing the values of x and z alternatively, we get a circular paraboloid.

Therefore, the given surface is a circular paraboloid.

Identify the surface with the given vector equation. r(s, t) = s sin(9t), s², s cos(9t)

Summary:

The surface with the given vector equation. r(s, t) = s sin(9t), s², s cos(9t) is a circular paraboloid.