Page-by-Page Format: Each handwritten page is shown followed by its accessible content
The handwritten notes indicate there is "No context in my question" - the answer depends on what space we're considering. Unless specified, assume \(\mathbb{R}^3\).
Possible answers listed:
Possible answers listed:
In \(\mathbb{R}^1\): The equation \(x = 2\) represents a single point at \(x = 2\) on the real line.
In \(\mathbb{R}^2\) (xy plane): The equation \(x = 2\) represents a vertical line where the x-coordinate is fixed at 2, but y can be anything. Since y is "Free", we stack all the points in the y-direction and get a line.
Diagram Description: The page shows a coordinate system with the x-axis horizontal and y-axis vertical. A vertical line is drawn at \(x = 2\). Several points are marked on this line: \((2, 2)\), \((2, 1)\), \((2, 0)\), \((2, -1)\), and \((2, -2)\). The annotation states "y can be Anything" and identifies y as a "Free" variable. The note explains that by stacking all the points in the y-direction, we get a line.
In \(\mathbb{R}^3\): The equation \(x = 2\) represents a plane perpendicular to the x-axis. Both y and z are free variables that can take any values.
Key Concept: When both y and z are free variables, we stack all the lines (from the \(\mathbb{R}^2\) case) in the z-direction to get a plane.
Diagram Description: A three-dimensional coordinate system shows the x, y, and z axes. A plane is drawn perpendicular to the x-axis at \(x = 2\). The plane extends infinitely in the y and z directions. Multiple points are marked including \((2, 0, 25)\), \((2, 0, 11)\), \((2, 0, 0)\), \((2, 4, 0)\), \((2, 4, 11)\), \((2, 4, -1)\), and \((2, 0, -25)\). The annotations indicate that "y is Free" and "z is Free", and that we "stack all the lines in z-direction" to "Get a Plane."
The equation \(x^2 + y^2 = 1\) has different interpretations depending on dimension. In \(\mathbb{R}^3\), z is a Free Variable.
In the xy plane (\(z = 0\)): This is a circle of radius 1 centered at the origin.
In \(\mathbb{R}^3\): By stacking circles of radius 1 in the z-direction, we obtain a circular cylinder parallel to the z-axis.
Definition: A circular cylinder parallel to the z-axis is formed when z is a free variable and the equation in x and y describes a circle.
Diagram Description: On the left, a circle of radius 1 is shown in the xy-plane, labeled "Circle of Radius 1 in xy Plane (z=0)". On the right, a three-dimensional view shows this circle extended vertically along the z-axis to form a cylinder. Multiple circular cross-sections are shown at different heights (some solid, some dashed to indicate depth). The cylinder extends infinitely in both positive and negative z-directions. The annotation reads "Stack Circles of Radius 1 in z-direction" resulting in a "Circular Cylinder Parallel to z-axis".
Definition: Traces (also called slices, curves, or cross-sections) are the curves we obtain by intersecting a three-dimensional surface with a plane. We stack these traces to get a complete 3D picture.
Types of traces:
\(z = 0\) trace (xy trace): The intersection of the surface with the xy plane.
\(z = k\) trace: The intersection of surface with the \(z = k\) plane, which is a plane parallel to the xy plane.
\(x = k\) trace: The intersection of surface with the \(x = k\) plane, which is a plane parallel to the yz plane.
\(y = k\) trace: The intersection of surface with the \(y = k\) plane, which is a plane parallel to the xz plane.
Important: By examining traces in different directions, we can understand the complete three-dimensional structure of a surface.
In this equation, y is Free. We stack traces (circles) in the y-direction to form a circular cylinder parallel to the y-axis.
Diagram Description: On the left, a circle is shown in the xz-plane. On the right, a three-dimensional cylinder is shown oriented horizontally along the y-axis, with circular cross-sections visible at various y-values.
In this equation, x is Free. We stack traces (circles) in the x-direction to form a circular cylinder parallel to the x-axis.
Diagram Description: A three-dimensional cylinder is shown oriented along the x-axis. The cylinder has circular cross-sections in planes perpendicular to the x-axis, drawn at various x-values with some in solid blue and others dashed to show depth.
In this equation, y is the free variable. The base curve is a parabola in the xz-plane.
For any fixed value of y, the equation \(x = 7 - z^2\) describes a parabola. When we stack these parabolic traces in the y-direction, we obtain a parabolic cylinder parallel to the y-axis.
Diagram Description: On the left, a parabola opening to the left is shown in the xz-plane with vertex near \((7, 0)\). The equation \(x = 7 - z^2\) is labeled, with y noted as Free. On the right, the three-dimensional parabolic cylinder is shown extending along the y-axis. Multiple parabolic cross-sections are drawn at different y-values (some solid, some dashed), illustrating how the parabola is "stacked" in the y-direction to form the cylinder.
Break activities:
Next topics: Sphere, Cone, and Paraboloid
The equation \(x^2 + y^2 + z^2 = 25\) represents a sphere of radius 5 centered at the origin.
We can rewrite the equation as \(x^2 + y^2 = 25 - z^2\) to see how horizontal traces change with height:
No traces above 5 or below -5 because \(25 - z^2\) would be negative.
Diagram Description: A sphere of radius 5 is shown centered at the origin. Horizontal circular cross-sections are drawn at various z-values, shown as solid and dashed purple ellipses. The x, y, and z axes pass through the center. Three great circles (in purple, blue, and green) are drawn on the surface to help visualize the spherical shape.
Note: This page shows different trace calculations, including:
Diagram Description: A three-dimensional view shows horizontal circular cross-sections at various z-heights. The circles are drawn in purple/magenta, with some solid and some dashed to indicate which are in front and which are behind. The cross-sections grow larger as we move away from \(z = 0\), illustrating how the surface expands.
To distinguish a cone from other surfaces, we need to find traces in other directions beyond just horizontal slices.
\(x = 0\) (yz trace): Setting \(x = 0\) gives \(y^2 = z^2\), which simplifies to \(y = z\) or \(y = -z\). These are two lines passing through the origin.
\(y = 0\) (xz trace): Setting \(y = 0\) gives \(x^2 = z^2\), which gives \(x = z\) or \(x = -z\). Again, two lines through the origin.
Diagram Description: At the top, two red sketches show rough outlines of the double cone from different viewing angles. Below, two coordinate plane diagrams show the linear traces. The left diagram shows the yz-plane with two lines \(y = z\) and \(y = -z\) intersecting at the origin. The right diagram shows similar lines in the xz-plane: \(x = z\) and \(x = -z\). Both sets of lines pass through the origin at 45-degree angles to the vertical axis.
Key Characteristic of Cones: Horizontal traces are circles that grow linearly with height, while traces in coordinate planes are intersecting lines.
Diagram Description: A complete three-dimensional double cone is shown with its vertex at the origin and axis along the z-axis. The cone consists of two nappes (upper and lower portions) extending infinitely in both the positive and negative z-directions. Circular cross-sections are drawn at various heights, showing how the radius increases linearly with \(|z|\). Blue lines on the surface indicate the linear traces from the coordinate planes (the lines \(x = z\), \(x = -z\), \(y = z\), and \(y = -z\) that pass through the vertex).
This surface opens upward along the positive z-axis, forming a bowl or cup shape.
No traces for \(z < 0\) because \(x^2 + y^2 \geq 0\) always.
The circles' radii increase as \(\sqrt{z}\) as we go up in the z-direction.
\(x = 0\) (yz trace): \(z = y^2\) - an upward-opening parabola
\(y = 0\) (xz trace): \(z = x^2\) - an upward-opening parabola
Diagram Description: On the left, two parabolas are shown: one in the yz-plane (\(z = y^2\)) and one in the xz-plane (\(z = x^2\)), both opening upward with vertices at the origin. On the right, the three-dimensional paraboloid is displayed showing the bowl shape opening upward. Horizontal circular cross-sections are visible at various heights (\(z = 0\) as a point, and larger circles at \(z = 1, 2, 4\)).
Diagram Description: A detailed three-dimensional illustration shows the paraboloid \(z = x^2 + y^2\) as a smooth bowl-shaped surface opening upward. The surface extends from the vertex at the origin upward and outward in all directions. Horizontal circular cross-sections are visible at various heights, and the parabolic curves forming the surface extend from the vertex along different radial directions. The smooth curvature is characteristic of a paraboloid. Vertical parabolic curves (shown in blue and green) represent the traces in the xz and yz planes.
Summary: The paraboloid \(z = x^2 + y^2\) is characterized by circular horizontal traces whose radii grow as \(\sqrt{z}\), and parabolic traces in vertical planes through the z-axis.
Summary of Surface Types:
Strategy: To identify and sketch surfaces, examine traces in multiple directions. The combination of trace shapes (circles, lines, parabolas, etc.) and how they change reveals the surface type.