How can we identify the surface represented by the vector equation r(s, t) = (s, sin(4t), s^2, cos(4t))?

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The given vector equation can be expressed in the parametric form as:

  • x = s
  • y = sin(4t)
  • z = s^2
  • w = cos(4t)

To identify the surface defined by this vector equation, we can analyze the relationship between the variables x, y, z, and w.

Firstly, observe that:

  • From the first equation x = s, we can substitute s in the other equations.
  • Substituting s = x into the third equation gives us z = x^2 (which represents a parabolic relationship in the x-z plane).
  • The second equation shows that y is periodic with respect to t through the sine function.
  • Likewise, w = cos(4t) shows that w is also periodic and oscillates between -1 and 1.

Next, consider the relationship between y and w. Since both are functions of t, they can be represented as:

  • y^2 + w^2 = sin^2(4t) + cos^2(4t) = 1

This means that as t varies, the points (y, w) move along a circle of radius 1 in the yw-plane. Therefore, we can derive that:

  • The surface is generated by varying x (which insists s can take any value) along a parabolic column represented by z = x^2.
  • Simultaneously, for each value of x, the coordinates (y, w) lie on a unit circle.

In summary, the surface represented by the given vector equation is a parabolic cylindrical surface (due to z = x^2) extended along curves that form a circle in the yw-plane. This means for every point on the parabola defined in the xz-plane, there exists a circle in the yw-plane, describing a 3-dimensional surface that is connected by these circles at every level of the parabola.

This is a unique and visually interesting surface, as it showcases both a parabolic structure along the xz dimensions while varying in a circular motion on the yw attributes.

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