Question

The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T,\) where \(T(x, y, z)\) is the temperature in the object and \(k>0\) is a constant that depends on the material. Compute the outward flux of \(\mathbf{F}\) across the following surfaces S for the given temperature distributions. Assume \(k=1\). \(T(x, y, z)=-\ln \left(x^{2}+y^{2}+z^{2}\right) ; S\) is the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\).

Short answer

Question: Calculate the outward flux of the heat flow vector field through the surface S, given that \(T(x, y, z) = -\ln \left(x^2 + y^2 + z^2 \right)\), \(k=1\), and surface S is the sphere \(x^2+y^2+z^2=a^2\). Answer: The outward flux of the heat flow vector field through the surface S is \(8\pi a^4\).

Step-by-step solution

1. Calculate the gradient of the temperature function T(x, y, z)

We are given the function T(x, y, z) as \(T(x, y, z) = -\ln \left(x^2 + y^2 + z^2 \right)\). To find \(\nabla T\), we need to find the partial derivatives of T with respect to \(x\), \(y\), and \(z\). The gradient is given by: \(\nabla T = \left( \frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z} \right)\) Now, let's differentiate T(x, y, z) with respect to \(x\), \(y\), and \(z\). We get: \(\frac{\partial T}{\partial x} = \frac{-2x}{x^2+y^2+z^2}\) \(\frac{\partial T}{\partial y} = \frac{-2y}{x^2+y^2+z^2}\) \(\frac{\partial T}{\partial z} = \frac{-2z}{x^2+y^2+z^2}\) So, the gradient \(\nabla T = \left( \frac{-2x}{x^2+y^2+z^2}, \frac{-2y}{x^2+y^2+z^2}, \frac{-2z}{x^2+y^2+z^2} \right)\).

2. Calculate the heat flow vector field F(x, y, z)

The heat flow vector field \(\mathbf{F}\) is given by \(\mathbf{F} = -k \nabla T\). Since \(k=1\), we can simply invert the gradient vector to get: \(\mathbf{F} = \left( \frac{2x}{x^2+y^2+z^2}, \frac{2y}{x^2+y^2+z^2}, \frac{2z}{x^2+y^2+z^2}\right)\).

3. Compute the outward flux of F across the surface S using the divergence theorem

The surface S is the sphere \(x^2+y^2+z^2=a^2\). We can write this surface S in the parametric form: \(\mathbf{r}(\theta, \phi) = a\left(\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta\right)\), where \(0\le\theta\le\pi\) and \(0\le\phi\le2\pi\). Let's calculate the normal vector \(\mathbf{n}\) by finding the cross product of \(\frac{\partial\mathbf{r}}{\partial\theta}\) and \(\frac{\partial\mathbf{r}}{\partial\phi}\): \(\mathbf{n} = \frac{\partial\mathbf{r}}{\partial\theta}\times\frac{\partial\mathbf{r}}{\partial\phi} = a^2\left(\sin^2\theta\cos\phi, \sin^2\theta\sin\phi, \sin\theta\cos\theta\right)\) Now, let's compute the outward flux of \(\mathbf{F}\) across the surface S. The flux integral is given by: \(\iint_{S} \mathbf{F}\cdot\mathbf{n} \, dS = \iint_{S} \left( \frac{2x}{x^2+y^2+z^2}, \frac{2y}{x^2+y^2+z^2}, \frac{2z}{x^2+y^2+z^2}\right)\cdot\left(\sin^2\theta\cos\phi, \sin^2\theta\sin\phi, \sin\theta\cos\theta\right) \, dS\) First, let's compute the dot product: \(\mathbf{F}\cdot\mathbf{n} = \frac{4a^4\sin^3\theta\cos^2\phi}{a^2}\ + \frac{4a^4\sin^3\theta\sin^2\phi}{a^2}\ + \frac{4a^4\sin\theta\cos\theta}{a^2} = 4a^2\sin^3\theta(\cos^2\phi+\sin^2\phi) + 4a^2\sin\theta\cos\theta\) Next, let's set up the flux integral: \(\iint_{S} \mathbf{F}\cdot\mathbf{n} \, dS = \int_{0}^{\pi}\int_{0}^{2\pi} [4a^2\sin^3\theta(\cos^2\phi+\sin^2\phi) + 4a^2\sin\theta\cos\theta] \, a^2\sin\theta\,d\phi d\theta\) Evaluating this integral, we get: \(\iint_{S} \mathbf{F}\cdot\mathbf{n} \, dS = 8\pi a^4\) This is the outward flux of the heat flow vector field \(\mathbf{F}\) through the surface S.

Key concepts

Vector Field

A vector field is a mathematical construct where each point in space is associated with a vector. In the context of heat flow in conducting objects, the vector field \(abla T\) represents the direction and magnitude of the temperature change at every point.

• The vector field assigns a vector to every point in the field.
• The direction of these vectors shows the direction of maximum rate of increase of the scalar field, in this case, temperature.
• The length of the vector indicates how rapidly the temperature is changing at that point.

In our example, with temperature \(T(x, y, z) = -\ln(x^2 + y^2 + z^2)\), the gradient \(abla T\) is computed by differentiating \(T\) with respect to each variable to find the partial derivatives. The gradient becomes a vector field that tells us how temperature is changing in space and helps determine the flow direction of heat.

Flux Calculation

Flux is a measure of how much of something (like heat or gas) passes through a surface. To calculate it, we need the vector field (in this case, heat flow \(\mathbf{F}\)) and the surface through which the flow is passing. Flux across a surface can be described by the integral of the dot product of the vector field and the surface's normal vector over the surface.

• The dot product of the vector field \(\mathbf{F}\) and the normal vector \(\mathbf{n}\) at each point on the surface indicates how much of the field passes through perpendicularly at that point.
• Integration over the surface adds up contributions across the entire surface area.

For a sphere \(S\), the calculation involves parametric representation of the surface and carrying out a surface integral to find total flux. In our example, computing the flux of the heat vector field through the sphere resulted in \(8\pi a^4\), giving us the measure of the outward heat movement.

Divergence Theorem

The divergence theorem is a powerful mathematical tool that links the flow of a vector field through a surface to the behavior of the field inside the volume bounded by the surface. It simplifies complex flux calculations by relating a surface integral to a volume integral.

• The theorem states that the flux across a closed surface is equal to the integral of the divergence of the vector field over the volume enclosed by the surface.
• Useful for converting calculations from a potentially difficult surface to a simpler volume calculation.
• In practical problems, this can mean shifting from a two-dimensional integration problem to a simpler three-dimensional one.

Applying the divergence theorem in our heat flow problem provides an elegant way to understand the movement of heat through the sphere surface by examining the volume within it, potentially simplifying the computational workload.