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 \(k=1\). \(T(x, y, z)=100 e^{-x^{2}-y^{2}-z^{2}} ; S\) is the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\).

Short answer

To compute the outward flux, follow these steps: 1. Find the gradient of the temperature function: \(\nabla T = (-200xe^{-x^{2}-y^{2}-z^{2}}, -200ye^{-x^{2}-y^{2}-z^{2}}, -200ze^{-x^{2}-y^{2}-z^{2}})\) 2. Calculate the heat flow vector field: \(\mathbf{F} = (200xe^{-x^{2}-y^{2}-z^{2}}, 200ye^{-x^{2}-y^{2}-z^{2}}, 200ze^{-x^{2}-y^{2}-z^{2}})\) 3. Evaluate the flux integral across the surface \(S\): \(\text{Flux} = \int_{0}^{2\pi} \int_{0}^{\pi} \mathbf{F}(\vec{r}(\theta, \phi)) \cdot \mathbf{n}(\vec{r}(\theta, \phi))\, (a^{2}\sin{\phi}) d\theta d\phi\) The outward flux of \(\mathbf{F}\) across the surface \(S\) can be found by evaluating the integral in Step 3.

Step-by-step solution

Find \(\nabla T\)

The gradient of the temperature function \(T(x, y, z) = 100 e^{-x^{2}-y^{2}-z^{2}}\) is given by \(\nabla T = (\frac{\partial T}{\partial x}, \frac{\partial T}{\partial y}, \frac{\partial T}{\partial z})\). Calculate the partial derivatives of \(T\) with respect to \(x\), \(y\), and \(z\): $$\frac{\partial T}{\partial x} = -200xe^{-x^{2}-y^{2}-z^{2}},$$ $$\frac{\partial T}{\partial y} = -200ye^{-x^{2}-y^{2}-z^{2}},$$ $$\frac{\partial T}{\partial z} = -200ze^{-x^{2}-y^{2}-z^{2}}.$$ So, the gradient is: $$\nabla T = (-200xe^{-x^{2}-y^{2}-z^{2}}, -200ye^{-x^{2}-y^{2}-z^{2}}, -200ze^{-x^{2}-y^{2}-z^{2}}).$$

Find the heat flow vector field \(\mathbf{F}\)

Substitute the gradient \(\nabla T\) into the formula \(\mathbf{F} = -k \nabla T\) with \(k=1\): $$\mathbf{F} = - (-200xe^{-x^{2}-y^{2}-z^{2}}, -200ye^{-x^{2}-y^{2}-z^{2}}, -200ze^{-x^{2}-y^{2}-z^{2}})$$ So, the heat flow vector field \(\mathbf{F}\) is: $$\mathbf{F} = (200xe^{-x^{2}-y^{2}-z^{2}}, 200ye^{-x^{2}-y^{2}-z^{2}}, 200ze^{-x^{2}-y^{2}-z^{2}}).$$

Calculate the outward flux of \(\mathbf{F}\) across the surface \(S\)

To compute the outward flux, we will use the flux integral formula: $$\text{Flux} = \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS,$$ where \(\mathbf{n}\) is the outward unit normal vector to the surface \(S\). For a sphere with equation \(x^{2}+y^{2}+z^{2}=a^{2}\), the outward unit normal vector is given by: $$\mathbf{n} = \frac{1}{a}(x, y, z).$$ Next, we parametrize the surface \(S\) in spherical coordinates as: $$\vec{r}(\theta, \phi) = (a\sin{\phi}\cos{\theta}, a\sin{\phi}\sin{\theta}, a\cos{\phi}),$$ where \(\theta \in [0, 2\pi]\) and \(\phi \in [0, \pi].\) Calculate the cross product of partial derivatives of \(\vec{r}\) with respect to \(\theta\) and \(\phi\): $$\frac{\partial \vec{r}}{\partial \theta} = (-a\sin{\phi}\sin{\theta}, a\sin{\phi}\cos{\theta}, 0),$$ $$\frac{\partial \vec{r}}{\partial \phi} = (a\cos{\phi}\cos{\theta}, a\cos{\phi}\sin{\theta}, -a\sin{\phi}).$$ Now find the surface element \(dS = ||\frac{\partial \vec{r}}{\partial \theta} \times \frac{\partial \vec{r}}{\partial \phi}||d\theta d\phi\): $$\text{dS} = ||(a^{2}\sin^{2}{\phi}\cos{\theta}, a^{2}\sin^{2}{\phi}\sin{\theta}, a^{2}\sin{\phi}\cos{\phi})||d\theta d\phi = a^{2}\sin{\phi}d\theta d\phi.$$ Now we can set up the flux integral: $$\text{Flux} = \int_{0}^{2\pi} \int_{0}^{\pi} \mathbf{F}(\vec{r}(\theta, \phi)) \cdot \mathbf{n}(\vec{r}(\theta, \phi))\, (a^{2}\sin{\phi}) d\theta d\phi$$ Evaluate the integral, and we will get the outward flux of \(\mathbf{F}\) across the surface \(S\).

Key concepts

Heat Flow Vector Field

In the realm of thermodynamics and heat transfer, the concept of a heat flow vector field, often denoted as \(\mathbf{F}\), is key to understanding how heat energy moves through a material. This vector field represents the rate and direction at which heat travels. The negative sign indicates that heat naturally flows from higher to lower temperatures, as dictated by the second law of thermodynamics.

To exemplify, consider an iron rod heated at one end; the generated heat flow vector field would point from the hot end to the cooler areas, illustrating the path and intensity of heat transfer within the rod. In mathematical terms, this vector field is expressed as \(\mathbf{F} = -k abla T\), where \(k\) is the material's thermal conductivity and \(abla T\), the gradient of temperature, signifies the rate of change and direction of the temperature field within the object.

Gradient of Temperature

The gradient of temperature, written as \(abla T\), is a critical player in the equation of the heat flow vector field. Essentially, it is a vector that points in the direction of the greatest increase of the temperature field, and its magnitude tells us how rapidly the temperature changes in that direction.

Consider being on a hillside where the gradient tells you the steepest path upwards. In the context of temperature, the gradient propels you in the direction where the temperature climbs fastest. We calculate it by finding the partial derivatives of the temperature function \(T(x, y, z)\) with respect to each spatial dimension—\(x\), \(y\), and \(z\)—and combining them to form a vector. This vector then provides a complete picture of how temperature varies throughout the space, pivotal for computing heat flow.

Surface Integral

The surface integral plays a pivotal role when determining the total quantity that flows across a given surface, in this case, the surface of a sphere. The surface integral of a vector field, like our heat flow field \(\mathbf{F}\), over a surface \(S\), essentially tallies up the field's normal component (perpendicular to the surface) over every point on \(S\).

The mathematical representation of the outward flux through surface \(S\) is given by \(\iint_S \mathbf{F} \cdot \mathbf{n} \, dS\), where \(\mathbf{n}\) is the outward unit normal vector to the surface and \(dS\) represents an infinitesimally small area on the surface. For the sphere in our example, you can picture a myriad of tiny flat patches, each contributing to the total heat flow exiting the sphere.

Spherical Coordinates

The spherical coordinate system is an alternative to the Cartesian coordinate system, primarily used when the geometry of the problem exhibits spherical symmetry—a perfect fit for problems involving spheres, such as the one we're discussing. It describes a point in space with three coordinates: \(\rho\), the radial distance from the origin; \(\theta\), the azimuthal angle in the \(xy\)-plane from the \(x\)-axis; and \(\phi\), the polar angle from the positive \(z\)-axis.

In the context of calculating an outward flux across the surface of a sphere, it is often more convenient to parametrize the surface using \(\theta\) and \(\phi\) and express the position vector as \(\vec{r}(\theta, \phi)\). This system simplifies the calculations significantly because it aligns with the inherent symmetry of the sphere, making the complex three-dimensional problem more tractable.