Question

(a) What does the equation \(y - {x^2}\) represent in \({\mathbb{R}^2}\) ?

(b) What does it represent in \({\mathbb{R}^3}\) ?

(c) What does the equation \(z - {y^2}\) represent?

Short answer

(a)

The equation \(y = {x^2}\) in \({k^2}\) represents the parabola

(b)

Equation \(y = {x^2}\) in \({A^3}^3\) represents the parabolic cylinder.

(c)

The equation \(z = {y^2}\) in \({\mathbb{R}^3}\) represents the parabolic cylinder

Step-by-step solution

Represent the Equations

(a)

\({\mathbb{R}^2}\)is the two-dimensional coordinate system that contains\(x\)- and\(y\)-coordinates.

The equation \(y = {x^2}\) in \({x^2}\) represents \(\left\{ {(x,y)\mid y = {x^2}} \right\}\), which is the set of all points that intersect and form \(a\) parabola.

The curve of equation is\(y = {x^2}\) shown below.

Thus, the equation \(y = {x^2}\) in \({k^2}\) represents the parabola.

(b)

\({\mathbb{R}^3}\)is the three-dimensional coordinate system that contains\({x_7},{y_y}\)and\(z\)-coordinates.

The equation \(y = {x^2}\) in \({x^3}\) represents \(\left\{ {(x,y,z)\mid y = {x^2}} \right\}\) and does not involve \(z\)-coordinates. Hence, set the \(z\) plane equation as \(z = {k_z}\), where \(k\) is a constant. The \(z\)-plane intersects the curve of equation \(y = {x^2}\).

Therefore, the resultant curve of equation \(y = {x^2}\) is a parabolic cylinder with infinitely many shifted duplicates of the same parabolas. The lines on the cylinder surface are regarded as rulings. The rulings in the representation of \(y = {x^3}\) in \({\mathcal{R}^3}\) are parallel to \(z\)-axis.

Thus, the equation \(y = {x^2}\) in \({A^3}^3\) represents the parabolic cylinder.

(c)

The equation\(z = {y^2}\)in\({{\bf{R}}^3}\)represents\(\left\{ {(x,y,z)\mid z = {y^2}} \right\}\)and does not involve\(x\)-coordinates.Hence, set the \(x\) plane equation as \(x = k\), where \(k\) is a constant. The \(x\)-plane intersects the curve of equation \(z = {y^2}\).

Therefore, the resultant curve of equation \(z = {y^2}\) is parabolic cylinder with infinitely many shifted duplicates of the same parabolas. The rulings in representation of \(y = {x^2}\) in \({R^3}\) are parallel to the \(x\)-axis.

Thus, the equation \(z = {y^2}\) in \({\mathbb{R}^3}\) represents the parabolic cylinder.