Question

Flow Rate In Exercises 39 and \(40,\) use a computer algebra system to find the rate of mass flow of a fluid of density \(\rho\) through the surface \(S\) oriented upward if the velocity field is given by \(\mathbf{F}(x, y, z)=0.5 z \mathbf{k}\) $$ S: z=\sqrt{16-x^{2}-y^{2}} $$

Short answer

The solution involves parametrization of the surface followed by computing the flux integral and then the rate of mass flow. Mathematical computations which involve the usage of a Computer Algebra system are required to solve this problem.

Step-by-step solution

Parametrization of surface S

Here, we can use spherical coordinates for parametrization. In spherical coordinates, \( z=\rho cos\phi \) and \( x^{2}+y^{2}+z^{2}=\rho^{2} \). From the equation of surface S, we can relate \( x^{2}+y^{2} = \rho^{2} sin^{2}\phi = 16 - z^{2} = 16 - \rho^{2} cos^{2}\phi \). After simplifying, \( \rho = 4 \). So, \( z = 4 cos \phi \). This implies the radius of the hemisphere is 4. Hence the parametrization of surface S is axisymmetric disk: \( \mathbf{r}(\phi,z)= 4 sin(\phi) cos(\theta) \mathbf{i} + 4 sin(\phi) sin(\theta) \mathbf{j} + 4 cos(\phi) \mathbf{k}\) with \( 0\leq\phi\leq\frac{\pi}{2} \) and \( 0\leq\theta\leq{2\pi}\)

Compute the Flux Integral

The flux is given by the surface integral of the vector field across surface S. With our parametrization, the flux integral becomes \( \int_S \mathbf{F}(\mathbf{r}(\phi, \theta))) . (\mathbf{r}_{\phi} \times \mathbf{r}_{\theta}) d\phi d\theta , from 0\leq\phi\leq\frac{\pi}{2} \) and \( 0\leq\theta\leq{2\pi} \). Now, substitute \( \mathbf{r}(\phi, \theta) \), \( \mathbf{r}_{\phi} \), and \( \mathbf{r}_{\theta} \) into the above equation and compute the integral. This will give the rate of flow through the surface.

Calculate the rate of Mass flow

The rate of mass flow is given by \( \rho \) times the flux. Simply multiply the result from the above step by \( \rho \) to find the rate of mass flow.

Key concepts

Surface Integral

Understanding surface integrals is crucial for calculating various physical quantities, such as mass flow rate through a surface. In essence, a surface integral allows us to sum up or integrate a function over a surface, much like how a regular integral sums up a function over an interval.

A surface integral can be thought of as finding the total flux of a vector field across a given surface or, in simpler terms, how much of the field penetrates through the surface. Imagine a fishnet spread across a river. The water flowing through the net could be considered analogous to a vector field, and the flow rate through each section of the net would represent the surface integral over that section. Now, if we combine all these small sections, we get the total flow through the net, i.e., the surface integral over the entire surface.

The surface integral of a scalar field represents a somewhat different concept and can be seen as measuring how much of the surface is covered by some property being studied, for example, the accumulated mass or charge over the area.

Parametrization of Surface

Parametrization is a mathematical tool used to describe a surface with an equation or a set of equations. It is a way of defining a surface using two parameters that 'sweep out' the surface in space. Think of a drone mapping a field: it goes back and forth in a pattern, covering the entire area. Parametrization uses mathematical 'coordinates' that mimic this drone's path but on a surface.

In the mass flow problem, spherical coordinates are employed to parametrize the surface of a hemisphere. By using parameters \(\phi\) and \(\theta\), every point on the hemisphere's surface is described as a function of these two variables. Effectively, this reduces a complex 3-dimensional problem to a more manageable 2-dimensional one, making the calculation of the surface integral possible.

Parametrizing surfaces is not just a theoretical exercise; it is crucial for translating real-world problems into solvable mathematical models. Accurate parametrization leads to precise analysis and solutions in fields such as fluid dynamics, electromagnetism, and even in the creation of computer graphics.

Flux of a Vector Field

When considering the flux of a vector field through a surface, we're interested in measuring the 'flow' of the field across that surface. In physical terms, this could represent the amount of fluid, like air or water, passing through a filter or a given section of space.

Mathematically, to calculate this flow, we perform a surface integral over our vector field. But first, we have to ensure that the vector field is projected correctly onto the surface, which is achieved by taking the dot product of the vector field with a normal vector of the surface at each point. The resulting integral, known as the \(flux\) integral, can be used for computing various quantities, such as electric flux in electromagnetism or the rate of fluid flow in fluid dynamics.

Spherical Coordinates

Spherical coordinates are an alternative to Cartesian (x, y, z) coordinates for representing points in three-dimensional space. They are particularly handy when dealing with problems that possess spherical symmetry, like our hemisphere surface.

Here's a simple way to visualize spherical coordinates: Imagine standing at the center of a globe. The value of \(\rho\) represents how far you reach out in any direction, \(\phi\) is the angle you'd turn upward or downward toward the North or South pole, and \(\theta\) is the angle you'd rotate around the world, like spinning a globe on its axis. This coordinate system greatly simplifies the process of computing integrals over curved surfaces, as seen in the hemisphere parametrization for our mass flow rate calculation.