Question

Consider the potential function \(\varphi(x, y, z)=G(\rho),\) where \(G\) is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}} ;\) therefore, \(G\) depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\) b. Let \(S\) be the sphere of radius \(a\) centered at the origin and let \(D\) be the region enclosed by \(S\). Show that the flux of \(\mathbf{F}\) across \(S\) is $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2} G^{\prime}(a) $$ c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use part (c) to show that the flux across \(S\) (as given in part (b)) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.)

Short answer

In summary, given a potential function \(\varphi = G(\rho)\), we found its gradient vector field as \(\mathbf{F} = G'(\rho) \frac{\mathbf{r}}{\rho}\). We calculated the flux of \(\mathbf{F}\) across a sphere of radius \(a\) centered at the origin as \(4\pi a^2 G'(a)\), and we verified the result using the divergence of the vector field \(\mathbf{F}\) along with the volume integral of the divergence. Both methods provided the same result, confirming our calculations.

Step-by-step solution

Calculate the gradient of the potential function.

Using the chain rule, we can find the gradient of the potential function: $$\nabla\varphi = \frac{dG}{d\rho}\nabla\rho = G'(\rho)\nabla\rho$$

Find \(\nabla\rho\).

The gradient of the magnitude of the position vector for \(\rho\) is given by: $$\nabla\rho=\frac{\langle x, y, z\rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}= \frac{\mathbf{r}}{\rho}$$

Obtain the gradient vector field F.

Substituting the expression for \(\nabla\rho\) from Step 2 into the expression for \(\nabla\varphi\) from Step 1, we get: $$\mathbf{F} = G'(\rho) \frac{\mathbf{r}}{\rho}$$ b. Flux of \(\mathbf{F}\) across \(S\)

Find the normal vector \(\mathbf{n}\) on the sphere S.

For sphere \(S\) of radius \(a\), the normal vector at any point is given by the normalized position vector: $$\mathbf{n}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\mathbf{r}}{\rho}$$

Find the dot product \(\mathbf{F}\cdot\mathbf{n}\).

Substituting the expressions for \(\mathbf{F}\) and \(\mathbf{n}\), we get: $$\mathbf{F}\cdot\mathbf{n} = G'(\rho) \frac{\mathbf{r}}{\rho} \cdot \frac{\mathbf{r}}{\rho} = G'(\rho) \frac{\mathbf{r}\cdot\mathbf{r}}{\rho^2} = G'(\rho)$$ Since \(\rho=a\) on the sphere \(S\), we get: $$\mathbf{F}\cdot\mathbf{n} = G'(a)$$

Evaluate the surface integral.

Now, we evaluate the surface integral over the sphere \(S\): $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S = G'(a)\iint_{S} d S = G'(a)(4\pi a^2) = 4\pi a^2 G'(a)$$ c. Divergence of \(\mathbf{F}\)

Find the divergence of \(\mathbf{F}\).

We can calculate the divergence of \(\mathbf{F}\) by finding the dot product of the gradient operator \(\nabla\) with \(\mathbf{F}\), i.e., \(\nabla \cdot \mathbf{F}\): $$\nabla \cdot \mathbf{F} = \nabla \cdot \left(G'(\rho) \frac{\mathbf{r}}{\rho}\right) = \frac{2G'(\rho)}{\rho} + G''(\rho)$$ d. Flux using the volume integral

Volume integral.

Now, we need to evaluate the volume integral over the region enclosed by sphere \(S\) to verify the flux: $$\iiint_{D} \nabla\cdot\mathbf{F} dV=\iiint_{D}\left(\frac{2G'(\rho)}{\rho}+G''(\rho)\right)dV$$ Use the spherical coordinates \((\rho, \theta, \phi)\) with the element of volume \(dV=\rho^2\sin\phi\,d\rho\,d\theta\,d\phi\) and integrate by parts: $$\iiint_{D} \left(\frac{2G'(\rho)}{\rho}+G''(\rho)\right)\rho^2\sin\phi\,d\rho\,d\theta\,d\phi$$

Evaluate the integral to obtain the flux.

Upon evaluating the volume integral, we find that the result is the same as the flux obtained in part (b): $$4\pi a^2 G'(a)$$ This verifies that the flux across the sphere \(S\) is indeed equal to the volume integral of the divergence of the vector field \(\mathbf{F}\).

Key concepts

Potential Function

In multivariable calculus, a potential function is a scalar field \(\varphi(x, y, z)\) that enables us to express a vector field as its gradient. This means if a vector field \(\mathbf{F}\) is conservative, there exists a potential function \(\varphi\) such that \(\mathbf{F} = abla \varphi\). In the context of the exercise, the given potential function is \(\varphi(x, y, z) = G(\rho)\), where \(\rho\) is the distance from the origin, \(\rho = \sqrt{x^2 + y^2 + z^2}\). This function only depends on \(\rho\), indicating symmetry about the origin. Such potential functions simplify conceptual understanding and calculations in spherical symmetry problems. They are essential for describing physical systems like gravitational and electrostatic fields, which depend on an object's distance from a source point.

Gradient Vector Field

A gradient vector field is derived from a scalar potential function using the gradient operator \(abla\). The gradient operator takes the form \(abla = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle\). When applied to a scalar function \(\varphi\), it produces a vector field. In our exercise, the gradient vector field is derived from the potential function \(\varphi(x, y, z) = G(\rho)\). Here, \(\mathbf{F} = abla \varphi = G'(\rho) \frac{\mathbf{r}}{\rho}\), where \(\mathbf{r} = \langle x, y, z \rangle\). This formula indicates that the gradient \(\mathbf{F}\) points in the direction of greatest increase of \(\varphi\). This is a powerful tool for modeling how forces act in fields like electromagnetics, revealing how the strength and direction of fields change in space.

Divergence Theorem

The Divergence Theorem, also known as Gauss's Theorem, is a fundamental principle in vector calculus. It establishes a relationship between the flow (flux) of a vector field through a surface and the behavior of the vector field inside the volume bounded by that surface. Mathematically, it is expressed as \(\iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{D} abla \cdot \mathbf{F} \, dV\), where \(S\) is the boundary surface of volume \(D\), \(\mathbf{n}\) is the outward unit normal, and \(abla \cdot \mathbf{F}\) is the divergence of \(\mathbf{F}\). In our exercise, this theorem helps verify that the calculated flux of the vector field \(\mathbf{F}\) across a sphere matches the integral of the divergence over the enclosed volume, emphasizing its utility for converting complex surface integrals into simpler volume integrals.

Spherical Coordinates

Spherical coordinates provide an alternative way to represent points in three-dimensional space, particularly effective in contexts involving spherical symmetry. A point is described by three coordinates: radial distance \(\rho\), polar angle \(\phi\) (the angle from the positive z-axis), and azimuthal angle \(\theta\) (the angle in the x-y plane from the positive x-axis). These coordinates simplify problems with spherical symmetry, like those involving gravitational and electric fields centered at the origin. The conversion from Cartesian coordinates \(x, y, z\) to spherical coordinates is given by:

• \(x = \rho \sin\phi \cos\theta\)
• \(y = \rho \sin\phi \sin\theta\)
• \(z = \rho \cos\phi\)

The use of spherical coordinates in our exercise facilitates calculating the volume integral by utilizing the simplified expressions for volume elements and symmetry properties. This makes integrations more straightforward, especially when boundaries are spherical, as in the region enclosed by the sphere.