Question

Let \(D\) be a solid heat-conducting cube formed by the planes \(x=0, x=1, y=0, y=1, z=0,\) and \(z=1 .\) The heat flow at every point of \(D\) is given by the constant vector \(\mathbf{Q}=\langle 0,2,1\rangle\) a. Through which faces of \(D\) does \(Q\) point into \(D ?\) b. Through which faces of \(D\) does \(\mathbf{Q}\) point out of \(D ?\) c. On which faces of \(D\) is \(Q\) tangential to \(D\) (pointing neither in nor out of \(D\) )? d. Find the scalar component of \(\mathbf{Q}\) normal to the face \(x=0\). e. Find the scalar component of \(\mathbf{Q}\) normal to the face \(z=1\). f. Find the scalar component of \(\mathbf{Q}\) normal to the face \(y=0\).

Short answer

- The faces where y=1 and z=1. 2. Which faces have heat flow pointing out of D? - The faces where y=0 and z=0. 3. On which faces is Q tangential to the cube? - Faces x=0 and x=1. 4. What is the scalar component of Q normal to the face x=0? - 0 5. What is the scalar component of Q normal to the face z=1? - 1 6. What is the scalar component of Q normal to the face y=0? - 2

Step-by-step solution

Identify the faces of the cube with heat flow pointing into D

Notice that - for the face x=0, since Q_x = 0, there is no heat flow pointing in or out of this face. - for the face x=1, since Q_x = 0, there is no heat flow pointing in or out of this face. - for the face y=0, since Q_y = 2, the heat flow is pointing outward. - for the face y=1, since Q_y = 2, the heat flow is pointing inward. - for the face z=0, since Q_z = 1, the heat flow is pointing outward. - for the face z=1, since Q_z = 1, the heat flow is pointing inward. The faces where Q points into D are the faces where y=1 and z=1. b. Determine the faces where Q points out of the cube

Identify the faces of the cube with heat flow pointing out of D

Based on the conclusions above, the faces where Q points out of D are the faces where y=0 and z=0. c. Determine the faces where Q is tangential to the cube

Check for tangential heat flow

As mentioned earlier, for faces x=0 and x=1, the heat flow does not point into or out of the cube, which means the heat flow is tangential to these faces. Hence, faces x=0 and x=1 have tangential heat flow. d. Scalar component normal to the face x=0

Scalar component for x=0 face

Since Q_x = 0, the scalar component of Q normal to x=0 face is 0. e. Scalar component normal to the face z=1

Scalar component for z=1 face

Since Q_z = 1, the scalar component of Q normal to z=1 face is 1. f. Scalar component normal to the face y=0

Scalar component for y=0 face

Since Q_y = 2, the scalar component of Q normal to y=0 face is 2.

Key concepts

Scalar Components

Scalar components are values that represent the magnitude of a vector in a specific direction. In the context of heat flow, these components show how much heat moves perpendicularly through a surface.
To find the scalar component of a vector \(\mathbf{Q} = \langle 0, 2, 1 \rangle\) normal to a face, you look at the component of \(\mathbf{Q}\) that is perpendicular to that surface:

• For the face \(x=0\), the normal vector is \(\langle 1, 0, 0 \rangle\) and \(Q_x = 0\), so the scalar component is 0.
• For the face \(z=1\), the normal vector is \(\langle 0, 0, 1 \rangle\) and \(Q_z = 1\), so the scalar component is 1.
• For the face \(y=0\), the normal vector is \(\langle 0, 1, 0 \rangle\) and \(Q_y = 2\), so the scalar component is 2.

These components help understand the distribution of heat flow through each face.

Heat Conduction

Heat conduction is the process by which heat energy is transmitted through collisions between neighboring molecules in a material. In solids, heat typically flows from a region of higher temperature to a region of lower temperature.
The rate of heat conduction can be represented by a vector, in this case, \(\mathbf{Q} = \langle 0, 2, 1 \rangle\), which tells us the direction and magnitude of the heat flow.

• The vector \(\mathbf{Q}\) provides a clear understanding of how heat is distributed across different parts of a solid.
• In a cube, analyzing each face separately gives insights into the patterns of heat flow and conduction through the material.

By understanding these principles, we can make predictions about how the material behaves in different thermal environments.

Vector Fields

Vector fields represent the distribution of vectors over a specific region in space. Here, the vector field is \(\mathbf{Q} = \langle 0, 2, 1 \rangle\), showing uniform heat flow across the solid.
This vector field provides critical information about the heat flow:

• The constant vector \(\mathbf{Q}\) means that the heat flow is consistent everywhere inside the cube.
• By examining the components of \(\mathbf{Q}\), we can determine directionality and behavior on each face of the cube.
• The heat flow's direction can help find which faces of the cube experience inward, outward, or tangential flows.

Understanding vector fields is vital for modeling heat distribution and evaluating thermal performance in various engineering applications.

Solid Geometry

Solid geometry involves the study of three-dimensional shapes like cubes. Here, the cube is defined by its boundaries: planes at \(x = 0, x = 1, y = 0, y = 1, z = 0, z = 1\).
The study of solid geometry allows us to explore how vectors interact with the surfaces of these shapes:

• Faces of the cube provide a structured environment to measure heat flow precisely.
• Understanding the interactions on each face helps predict how the structure may respond to thermal changes.
• Analyzing geometric properties ensures accurate calculations of quantities like heat flux and tangential flow.

By using these principles of geometry, we can apply theoretical concepts to solve practical problems related to heat distribution.