Question

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and parametric description of the sphere.

Short answer

In this problem, we are given a radial field \(\mathbf{F}\) and asked to find the outward flux across the surface of a sphere of radius \(a\). Using the explicit method, we begin by finding the normal vector to the surface. We then compute the dot product of the field with the normal vector, which gives us an expression to integrate over the sphere's surface. After converting to polar coordinates and evaluating the double integral, we find that the outward flux is \(4 \pi / a^{p-3}\).

Step-by-step solution

Calculate the gradient of the surface's function

We need to find the gradient of \(f(x, y) = \sqrt{a^2 - x^2 - y^2}\) to find the normal vector to the surface of the sphere. To compute the gradient, we'll find the partial derivatives of \(f\) with respect to \(x\) and \(y\): \(\frac{\partial f}{\partial x} = \frac{-x}{\sqrt{a^2 - x^2 - y^2}}\) \(\frac{\partial f}{\partial y} = \frac{-y}{\sqrt{a^2 - x^2 - y^2}}\) The gradient of the surface function is thus: \(\nabla f(x, y) = \left\langle \frac{-x}{\sqrt{a^2 - x^2 - y^2}}, \frac{-y}{\sqrt{a^2 - x^2 - y^2}}\right\rangle\)

Calculate the normal vector of the surface

To get the normal vector to the surface, we take the cross product of the gradient of \(f\) and the vector \(\left\langle 1, 1, 0 \right\rangle\): \(\mathbf{n} = \left\langle 0, \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right\rangle = \left\langle 0, \frac{-x}{\sqrt{a^2 - x^2 - y^2}}, \frac{-y}{\sqrt{a^2 - x^2 - y^2}}\right\rangle\)

Calculate the outward flux of the field

The outward flux of the radial field across the sphere is the surface integral of the dot product of the field with the normal vector over the sphere's surface. In other words, we need to find: \(\iint_S \mathbf{F} \cdot \mathbf{n}\,\mathrm{d}S\) Using our given field \(\mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|^p}\) and computed normal vector \(\mathbf{n}\), the dot product \(\mathbf{F} \cdot \mathbf{n}\) becomes: \(\frac{-x^2 - y^2}{(x^2 + y^2 + z^2)^{\frac{p+1}{2}}}\) Next, we need to compute the surface integral of this dot product over the sphere's surface. This involves converting to polar coordinates and evaluating the double integral: \(\iint_S \mathbf{F} \cdot \mathbf{n}\,\mathrm{d}S = \int_{0}^{2\pi} \int_{0}^{\pi} \frac{-a^2 \sin^2\theta}{(a^2)^{\frac{p+1}{2}}} a^2 \sin\theta \,\mathrm{d}\theta\,\mathrm{d}\phi\) This simplifies to: \(\int_{0}^{2\pi} \int_{0}^{\pi} - \frac{a^3 \sin^3\theta}{a^{p+1}} \,\mathrm{d}\theta\,\mathrm{d}\phi\) After evaluating this integral, we find the outward flux to be: \(4 \pi / a^{p-3}\)

Key concepts

Surface Integrals

Surface integrals are a critical tool in vector calculus for analyzing how vector fields interact with surfaces. They allow us to quantify the total influence of a field over an entire surface, such as the rate at which fluid flows across a boundary. Think of a surface integral as a way to 'sum up' a field's effect, just like you'd add up individual items to find a total cost.

To calculate a surface integral, we consider a vector field and a surface over which we want to measure the field's effect. The surface integral measures the component of the vector field that is normal, or perpendicular, to the surface at each point. Mathematically, we express this as the dot product of the field with a normal vector to the surface, integrated over the entire surface. In simpler terms, it's analogous to adding up the effect of the field at each infinitesimal patch of the surface to get the total effect.

Vector Calculus

Vector calculus is the branch of mathematics that deals with vectors and the various operations that can be performed on them in space. It's the key to solving problems in physics and engineering that involve fields like velocity, force, and electric and magnetic fields. Vector calculus can seem complex, but it's based on a few fundamental concepts: scalar and vector fields, differentiation and integration of vectors, and theorems like the gradient, divergence, and curl. Each of these tools helps us describe how vector fields behave and change over space and provide a way to calculate physical quantities over different geometries, such as curves and surfaces. Put simply, vector calculus is the language we use to describe and analyze how things flow and move in space.

Parametric Surfaces

Parametric surfaces are a way to describe surfaces using parameters, rather than the standard x, y, and z coordinates. This is extremely handy in complex scenarios, where describing a surface in the usual way is difficult or impossible. To create a parametric surface, we use two parameters (usually denoted as u and v) and connect them to the x, y, and z coordinates through a set of equations.

This approach is flexible and powerful because it allows us to describe a wide variety of shapes and surfaces that would be otherwise cumbersome to handle. For a sphere, as in our original exercise, the parameters could be angles that define position in spherical coordinates. This makes calculating properties and integrals over the surface much easier because we can adapt our coordinate system to the problem's geometry, streamlining the math and often leading to simpler calculations.

Calculus of Vector Fields

The calculus of vector fields extends the ideas of calculus to fields described by vectors. It involves operations such as taking the gradient, which measures how a scalar field changes in space, or the divergence and curl, which tell us how a vector field diverges or curls around a point.

For example, the outward flux across a sphere involves calculating the surface integral of a radial vector field. This requires us to understand how the vector field behaves in three-dimensional space and how it interacts with the sphere's surface. By calculating the outward flux, we're essentially measuring how much of the field 'flows' out of the sphere. The rigorous application of the calculus of vector fields allows us to understand physical phenomena like electromagnetism, fluid dynamics, and more. It's a way of taking the intuitive concept of flow and making it concrete and quantitative.