Question

Suppose a solid object in \(\mathbb{R}^{3}\) has a temperature distribution given by \(T(x, y, z) .\) The heat flow vector field in the object is \(\mathbf{F}=-k \nabla T,\) where the conductivity \(k>0\) is a property of the material. Note that the heat flow vector points in the direction opposite that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is \(\nabla \cdot \mathbf{F}=-k \nabla \cdot \nabla T=-k \nabla^{2} T\) (the Laplacian of \(T\)). Compute the heat flow vector field and its divergence for the following temperature distributions. $$T(x, y, z)=100(1+\sqrt{x^{2}+y^{2}+z^{2}})$$

Short answer

Question: Calculate the divergence of the heat flow vector field for the temperature distribution \(T(x, y, z) = 100(1+\sqrt{x^{2}+y^{2}+z^{2}})\), given that the heat flow vector field is \(\mathbf{F} = -k \nabla T\), where k > 0. Answer: The divergence of the heat flow vector field is given by \(\nabla \cdot \mathbf{F} = 100k(x^2+y^2+z^2)^{-1/2}\).

Step-by-step solution

Find the gradient of T(x, y, z)

To find the gradient of the temperature distribution \(T(x, y, z) = 100(1+\sqrt{x^{2}+y^{2}+z^{2}})\), we need to find the partial derivatives with respect to each variable and arrange them as a vector: $$\nabla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right)$$ Compute the partial derivatives: $$\frac{\partial T}{\partial x} = 100 \cdot \frac{x}{\sqrt{x^2+y^2+z^2}}$$ $$\frac{\partial T}{\partial y} = 100 \cdot \frac{y}{\sqrt{x^2+y^2+z^2}}$$ $$\frac{\partial T}{\partial z} = 100 \cdot \frac{z}{\sqrt{x^2+y^2+z^2}}$$ Combining these partial derivatives, we have the gradient of \(T\): $$\nabla T = \left( 100 \cdot \frac{x}{\sqrt{x^2+y^2+z^2}}, 100 \cdot \frac{y}{\sqrt{x^2+y^2+z^2}}, 100 \cdot \frac{z}{\sqrt{x^2+y^2+z^2}} \right)$$

Compute the heat flow vector field F(x, y, z)

We are given that \(\mathbf{F} = -k \nabla T\), where \(k > 0\) is the conductivity. Substitute the gradient of \(T\) found in Step 1: $$\mathbf{F} = -k \cdot \left( 100 \cdot \frac{x}{\sqrt{x^2+y^2+z^2}}, 100 \cdot \frac{y}{\sqrt{x^2+y^2+z^2}}, 100 \cdot \frac{z}{\sqrt{x^2+y^2+z^2}} \right)$$

Compute the divergence of the heat flow vector field F(x, y, z)

From \(\nabla \cdot \mathbf{F} = -k \nabla^{2} T\), we want to compute \(\nabla^{2} T\). The Laplacian of \(T\) is given by: $$\nabla^{2} T = \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2}$$ Compute the second partial derivatives and sum them up: $$\frac{\partial^2 T}{\partial x^2} = -100 \cdot \frac{x^2}{(x^2+y^2+z^2)^{3/2}}$$ $$\frac{\partial^2 T}{\partial y^2} = -100 \cdot \frac{y^2}{(x^2+y^2+z^2)^{3/2}}$$ $$\frac{\partial^2 T}{\partial z^2} = -100 \cdot \frac{z^2}{(x^2+y^2+z^2)^{3/2}}$$ Summing them up, we have: $$\nabla^{2} T = -100 \cdot \frac{x^2+y^2+z^2}{(x^2+y^2+z^2)^{3/2}} = -100(x^2+y^2+z^2)^{-1/2}$$ Finally, calculate the divergence of the heat flow vector field as \(\nabla \cdot \mathbf{F} = -k \nabla^{2} T\): $$\nabla \cdot \mathbf{F} = -k(-100(x^2+y^2+z^2)^{-1/2}) = 100k(x^2+y^2+z^2)^{-1/2}$$

Key concepts

Temperature Distribution

In the study of heat flow, understanding the temperature distribution within a solid object is crucial. This distribution reflects how temperature varies at different points in the object. For example, consider the function \(T(x, y, z) = 100(1+\sqrt{x^2+y^2+z^2})\). Here, \(T\) denotes the temperature at any given point \((x, y, z)\) in the three-dimensional space \(\mathbb{R}^3\).

In this equation, the temperature depends on the distance from the origin, represented by \(\sqrt{x^2 + y^2 + z^2}\). This means that points further from the origin tend to have different temperatures.

Understanding temperature distribution helps us determine how heat will flow within the object. Areas with higher temperature will typically transfer heat to areas with lower temperature, influenced by the material's conductivity that dictates how well the heat is conducted.

Gradient of a Scalar Field

The gradient of a scalar field, like our temperature function \(T\), is a vector field that points in the direction of the greatest increase of the function. Mathematically, it is denoted by \(abla T\).

For our temperature function \(T(x, y, z) = 100(1+\sqrt{x^2+y^2+z^2})\), the gradient is calculated by finding the partial derivatives with respect to \(x\), \(y\), and \(z\):

• \(\frac{\partial T}{\partial x} = 100 \cdot \frac{x}{\sqrt{x^2+y^2+z^2}}\)
• \(\frac{\partial T}{\partial y} = 100 \cdot \frac{y}{\sqrt{x^2+y^2+z^2}}\)
• \(\frac{\partial T}{\partial z} = 100 \cdot \frac{z}{\sqrt{x^2+y^2+z^2}}\)

The gradient vector then becomes \(abla T = \left( 100 \cdot \frac{x}{\sqrt{x^2+y^2+z^2}}, 100 \cdot \frac{y}{\sqrt{x^2+y^2+z^2}}, 100 \cdot \frac{z}{\sqrt{x^2+y^2+z^2}} \right)\).

This vector indicates the direction and rate of the greatest temperature increase at each point.

Divergence and Laplacian

The divergence of a vector field measures how much the field is spreading out from a point. For the heat flow vector field \(\mathbf{F} = -k abla T\), the divergence is \(abla \cdot \mathbf{F}\).

The Laplacian \(abla^2 T\) of the scalar field \(T\) is a measure of the field's curvature. It's critical in understanding how temperature changes across space.

To compute the Laplacian for \(T(x, y, z)\), we sum the second partial derivatives:

• \(\frac{\partial^2 T}{\partial x^2} = -100 \cdot \frac{x^2}{(x^2+y^2+z^2)^{3/2}}\)
• \(\frac{\partial^2 T}{\partial y^2} = -100 \cdot \frac{y^2}{(x^2+y^2+z^2)^{3/2}}\)
• \(\frac{\partial^2 T}{\partial z^2} = -100 \cdot \frac{z^2}{(x^2+y^2+z^2)^{3/2}}\)

Adding these, the Laplacian is \(abla^2 T = -100(x^2+y^2+z^2)^{-1/2}\).

Finally, the divergence of \(\mathbf{F}\) is \(abla \cdot \mathbf{F} = 100k(x^2+y^2+z^2)^{-1/2}\), showing how heat flows away from or towards a point in the material.

Partial Derivatives

Partial derivatives are essential tools for analyzing functions of multiple variables, like our temperature function \(T(x, y, z)\).

They represent the rate of change of the function concerning one variable while keeping the others constant. In our example, to find \(\frac{\partial T}{\partial x}\), you treat \(y\) and \(z\) as constants and find how \(T\) changes as \(x\) changes.

Computing partial derivatives:

• \(\frac{\partial T}{\partial x} = 100 \cdot \frac{x}{\sqrt{x^2+y^2+z^2}}\)
• \(\frac{\partial T}{\partial y} = 100 \cdot \frac{y}{\sqrt{x^2+y^2+z^2}}\)
• \(\frac{\partial T}{\partial z} = 100 \cdot \frac{z}{\sqrt{x^2+y^2+z^2}}\)

These derivatives combine to form the gradient \(abla T\), illustrating how the temperature changes in each direction.

Such calculations refine our understanding of the temperature distribution and the resulting heat flow within an object.