Question

In Exercises \(1-8,\) describe and sketch the surface. $$ y^{2}+z=4 $$

Short answer

The surface \(y^{2}+z=4\) represents a parabolic cylinder that opens downwards and extends infinitely in both the positive and negative x-direction.

Step-by-step solution

Identify kind of surface

Given the equation \(y^{2}+z =4\), it doesn't involve the variable \(x\). Hence, there are no restrictions on the \(x\)-values. That means the surface must extend infinitely in both the positive and negative \(x\)-direction and will be a cylindrical surface.

Describe the surface

Now rearrange the equation to \(z = 4 - y^{2}\) to get a clear idea about the projection in the z-axis. This is an equation of a downward parabola in the yz-plane. However because of the absence of the inclusion of \(x\) variable, this extends to all \(x\)-values and hence forms a parabolic cylinder.

Sketch the Surface

Sketching this surface needs to consider the shape of the function in the yz-plane and then recognize that the shape is consistent at all points along the x-axis. So, one may start with sketching a downward opening parabola in the yz-plane and then introduce arrows along the x-axis to indicate the surface extends infinitely in the x-direction.

Key concepts

Parabolic Cylinder

Understanding the parabolic cylinder is crucial in visualizing complex surfaces within the realm of calculus. A parabolic cylinder is formed when a parabola in a two-dimensional plane is translated along an axis perpendicular to the plane of the parabola, creating a three-dimensional surface. For example, consider the equation \(y^{2}+z=4\). This represents a parabola that opens downwards on the yz-plane.

In the case of our exercise, since the equation does not involve the variable \(x\), the resulting figure is a parabolic cylinder extending infinitely along the x-axis. The absence of \(x\) in the equation indicates that for every point \(y,z\) lying on the parabola \(y^{2}+z=4\), there is a line parallel to the x-axis. These lines are copies of the parabola at different \(x\)-coordinates, forming the cylindrical surface.

It's helpful to improve your grasp on the concept by imagining how a shape that's typically 2D, like a parabola, can 'sweep' or slide along an axis to cover an infinite three-dimensional space. This is precisely what happens with the parabolic cylinder—it is the sweeping of the 2D parabolic shape along an axis.

3D Surface Sketching

The skill of sketching three-dimensional surfaces is quite beneficial in the study of multivariable calculus. In our example with the equation \(y^{2}+z=4\), 3D sketching requires visualizing a parabola in the yz-plane and then extending this shape along the x-axis.

To sketch this, one typically starts by drawing the parabola \(z=4-y^{2}\) as it appears when \(x=0\). This would look like a downward opening parabola. The next step is to illustrate that this parabola is constant along the x-axis. By adding a set of parallel lines to this parabola and extending these lines infinitely in both positive and negative x-directions, the sketch starts to represent a cylindrical surface—the parabolic cylinder. Be mindful to include arrows to demonstrate the infinite nature of this surface along the x-direction.

By practicing the technique of extending a 2D curve into 3D space, students can better understand the concepts behind the shapes they encounter in multivariable calculus. Visual aids such as drawing software or modeling clay can also be useful in enhancing one's ability to visualize and sketch 3D surfaces.

Cylindrical Surface Equation

A cylindrical surface equation in calculus is a way to describe a surface that extends infinitely in one direction. Such surfaces are called cylindrical due to their similarity to the shape of a cylinder. The key to understanding these equations is recognizing that one variable is conspicuously missing, indicating the direction in which the surface extends infinitely.

In the exercise equation \(y^{2}+z=4\), \(x\) is not present, which implies that in the direction of the x-axis, the surface goes on forever. We can say that the given equation is a specific example of a cylindrical surface. The general form of a cylindrical surface equation is one that relates two coordinates, say \(y\) and \(z\), while the third coordinate, \(x\), does not appear in the equation at all.

To work with these equations, it's important to assign the proper shape in the plane of the variables present and then extend that shape in the direction of the missing variable. Doing this will allow you to harness the full potential of understanding and working with 3D surfaces in calculus. Remember, cylindrical surfaces, while infinite, are not necessarily circular—they take the shape of the equation in the plane perpendicular to their axis of extension, as shown in our parabolic cylinder example.