The given vector equation can be analyzed to identify the surface it represents. The equation is:
r(s,t) = (s sin(9t), s^2, s cos(9t))
To understand the surface, we start by examining the components of the vector:
- The first component is x = s sin(9t).
- The second component is y = s^2.
- The third component is z = s cos(9t).
From the second component, we see that y = s^2, which suggests a parabolic relationship because it squares the variable s. This indicates that the surface opens up in the y-direction as s varies.
Next, if we eliminate s from the first and third components, we can express x and z in terms of y:
- From y = s^2, we can express s = √y.
- Substituting this into the equations for x and z:
x = √y sin(9t) and z = √y cos(9t).
Now we can rewrite this system by using the identities of sine and cosine:
Using the fact that sin²(θ) + cos²(θ) = 1, we can form:
sin²(9t) + cos²(9t) = 1.
Thus, if we consider x and z:
x² + z² = y
This equation describes a circular cylinder along the y-axis. The angle 9t indicates that the circle formed by the projections of points in the xz-plane completes a full revolution as t varies, creating a helical structure along the y-axis.
In conclusion, the surface represented by the vector equation is a parabolic cylinder that is also helical in the xz-plane, indicated by the trigonometric functions of 9t.