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Chapter 12: Problem 7

Describe in words the level curves of the paraboloid \(z=x^{2}+y^{2}.\)

Short Answer

Expert verified
The level curves of the paraboloid \(z=x^{2}+y^{2}\) are circles centered at the origin with radius \(\sqrt{c}\), where \(c\) represents the height of the paraboloid. As the value of \(c\) increases, the radius of the circle increases, which means the level curves will become larger.

Step by step solution

01

Set up the level curve equation for arbitrary constant height (c)

Level curves are formed by intersecting the given surface (paraboloid) with a set of horizontal planes at different heights. Mathematically, this is done by setting \(z=c\), where \(c\) is a constant. In our case, we have: $$z = x^{2}+y^{2}$$ and we set \(z=c\) to get the level curve equation: $$c = x^{2}+y^{2}$$
02

Recognize the equation of a circle

The equation we derived in step 1, \(c = x^{2}+y^{2}\), is the equation of a circle with radius \(\sqrt{c}\) and center at the origin (0, 0).
03

Describe the level curves

From step 2, we know that the level curves are circles with radius \(\sqrt{c}\) and center at the origin (0, 0). The level curves of the paraboloid \(z=x^{2}+y^{2}\) are circles centered at the origin with radius \(\sqrt{c}\), where \(c\) represents the height of the paraboloid. As we increase the value of \(c\), the radius of the circle increases, which means the level curves will become larger.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Paraboloid
A paraboloid is a three-dimensional shape resembling an elongated bowl, which opens upwards or downwards. Mathematically, it can be expressed with the equation \( z = x^2 + y^2 \). The shape is defined such that each cross-section parallel to the base is a circle, forming the surface of a paraboloid.

When we cut through the paraboloid horizontally at different heights, the slices reveal level curves. These level curves tell us the shape of horizontal sections through the paraboloid. Visualize a stack of concentric circles rising up or receding down the paraboloid's surface. This geometric structure is seen in applications ranging from optics to designing satellite dishes.
Equation of a Circle
The equation of a circle is a fundamental concept in geometry, defining a set of points in a plane that are equidistant from a fixed point known as the center. A standard circle equation is given by \( x^2 + y^2 = r^2 \), where \(r\) is the radius.

In the context of level curves on a paraboloid, when a horizontal plane cuts through, the resulting intersection is a circle. This emerges from setting \( z = x^2 + y^2 = c \), leading to \( x^2 + y^2 = c \). Here, \(c\) plays the role of \( r^2 \), and describes a circle's radius squared centered at the origin (0, 0) in the xy-plane.
Radius of a Circle
The radius of a circle is the distance from the center of the circle to any point on its boundary. It serves as a crucial measure when describing circles.

In the exercise, the equation \( x^2 + y^2 = c \) gives the radius as \( \sqrt{c} \). This results because the square of the radius \( r^2 \) equals the constant house height \( c \) from our set level surface. As \( c \) increases, the corresponding radius \( \sqrt{c} \) also increases, depicting the expansion of level curves on a paraboloid.
Center of Circle
The center of a circle is a pivotal point from which every point on the circle is equidistant, serving as a symmetrical focal spot for geometric shapes.

For the equation \( x^2 + y^2 = c \), the center of each corresponding circle is at the origin, (0, 0). This is because the paraboloid is centered at its vertex, where x and y are zero, hence making each cross-sectional level curve concentric to the center of the coordinate plane. Being repeatedly centered at the origin simplifies many mathematical models and is instrumental in discussions involving symmetry in circles and paraboloids.

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Most popular questions from this chapter

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