Problem 14 - Thomas Calculus

Question

In Exercises \(1-16,\) give a geometric description of the set of points in space whose coordinates satisfy the given pairs of equations. $$x^{2}+y^{2}+z^{2}=4, \quad y=x$$

Step-by-Step Solution

Answer

The intersection is an elliptical cylinder centered on the line where \(x = y\), within the sphere of radius 2.

Step 1: Identify the Shape Defined by the First Equation

The equation \(x^2 + y^2 + z^2 = 4\) represents a sphere centered at the origin \((0, 0, 0)\) with a radius of 2. This is because the general equation for a sphere is \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\), where \((h, k, l)\) is the center and \(r\) is the radius.

Step 2: Understand the Condition Given by the Second Equation

The equation \(y = x\) describes a plane in the three-dimensional space. In this plane, every point has its \(y\) coordinate equal to its \(x\) coordinate, creating a diagonal plane that passes through the origin and makes an angle of 45 degrees with both the \(x\)-axis and \(y\)-axis.

Step 3: Combine the Equations to Find the Intersection

To find the points satisfying both conditions, we look for the intersection of the sphere and the plane. Substitute \(y = x\) into the sphere's equation: \[(x^2 + x^2 + z^2 = 4)\]which simplifies to \[(2x^2 + z^2 = 4)\].

Step 4: Simplify and Describe the Intersection

The equation \(2x^2 + z^2 = 4\) represents an elliptical cylinder. This cylinder is centered on the \(z\)-axis with its axis along the line \(x = y\), as the section \(z = 0\) would give the ellipse \(2x^2 = 4\), or \(x^2 = 2\), showing the semi-major axis of length \(\sqrt{2}\) along the plane defined by \(x = y\).

Key Concepts

Sphere

A sphere is one of the simplest three-dimensional shapes you can think of. It is perfectly round, like a basketball or a bubble.
In mathematical terms, a sphere is defined by the equation \((x - h)^2 + (y - k)^2 + (z - l)^2 = r^2\). This tells us everything about where the sphere is and how big it is:

• (h, k, l) is the center of the sphere. It's like the point right in the middle of the ball, or bubble.
• r is the radius, which is the distance from the center to any point on the sphere.

In our case, the given equation \(x^2 + y^2 + z^2 = 4\) means:

• The center of the sphere is at \((0, 0, 0)\), which is the origin of the coordinate system.
• The radius of the sphere is \(2\).

Visualize the sphere as a big round shape in space, perfectly centered around the origin, reaching out to a radius of 2 units in all directions.

Plane

A plane in geometry is like a vast, flat sheet of paper that stretches endlessly in two directions. In three-dimensional space, a plane slices through the space like a knife through butter.
The equation \(y = x\) is a specific type of plane. Here's what it represents:

• Every point on this plane has its y value equal to its x value. This means if you pick any point with a certain x, it will have the same y coordinate.
• This plane forms a 45-degree angle with the x and y axes.
• It runs diagonally through the three-dimensional space.

Imagine a slanted ramp that perfectly lines up with the origin and extends up and to the right, continuing forever without edges.

Elliptical Cylinder

An elliptical cylinder is a type of cylinder where the base is not a circle, as in a regular cylinder, but an ellipse. You can think of it like squeezing a basketball into an oval shape and stretching it up into a cylindrical form.
In our exercise, we found an elliptical cylinder as the intersection of the sphere and plane. Here's the breakdown:

• The cylindrical surface is defined by the equation\(2x^2 + z^2 = 4\).The meaning of this is:
• If you set \(z = 0\), then the elliptical base is structured as \(2x^2 = 4\). This simplifies to \(x^2 = 2\), giving the semi-major axis as \(\sqrt{2}\).
• The intersection creates a continuous loop, aligned with the line \(x = y\).
• This elliptical cylinder is perfectly aligned with the space where \(x = y\).

Picture it as an infinitely tall oval tube standing upright, with the oval base lying flat in the plane where \(x = y\)describes a diagonal path in space.