Problem 414 - Calculus Volume 3

Question

. & \text { Consider } & \text { radial } & \text { vector }\end{array}\( field \)\mathbf{F}=\frac{\mathbf{r}}{|\mathbf{r}|}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{1 / 2}}, \quad\( Compute \)\quad\( the \)\quad\( surface integral, where \)S\( is the surface of a sphere of radius \)a$ centered at the origin.

Step-by-Step Solution

Answer

The surface integral is zero.

Step 1: Understand the Problem

We need to compute the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} \), where \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|} = \frac{\langle x, y, z \rangle}{(x^2 + y^2 + z^2)^{1/2}} \) and \( S \) is the surface of a sphere of radius \( a \) centered at the origin.

Step 2: Recognize the Symmetry of the Problem

Since \( \mathbf{F} \) is a radial vector field and \( S \) is a sphere centered at the origin, the problem has spherical symmetry. This means the divergence theorem or Gauss's theorem may simplify the computation.

Step 3: Apply Divergence Theorem

According to the divergence theorem: \( \iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V(abla \cdot \mathbf{F}) \; dV \), where \( V \) is the volume enclosed by \( S \). Here, find \( abla \cdot \mathbf{F} \).

Step 4: Compute Divergence of \( \mathbf{F} \)

Calculate \( abla \cdot \mathbf{F} = abla \cdot \left( \frac{\langle x, y, z \rangle}{(x^2 + y^2 + z^2)^{1/2}} \right) \). Use the formula for the divergence of \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|} \), which simplifies as \( abla \cdot \mathbf{F} = \frac{2}{(x^2 + y^2 + z^2)^{3/2}} \cdot 0 \) because it reduces to zero.

Step 5: Evaluate the Volume Integral

Since \( abla \cdot \mathbf{F} = 0 \), the volume integral \( \iiint_V(abla \cdot \mathbf{F}) \; dV = 0 \) because the divergence is zero everywhere in the volume \( V \).

Step 6: Conclude Result

By applying the divergence theorem and calculating that the divergence of the vector field is zero, the surface integral \( \iint_S \mathbf{F} \cdot d\mathbf{S} = 0 \).

Key Concepts

Radial Vector Field

A radial vector field is a type of vector field that radiates outward from or inward toward a central point, often referred to as the origin. In these fields, the vector directions are always aligned along the radial lines extending from a central point. This characteristic makes them particularly useful in problems involving spherical coordinates. In mathematical terms, a radial vector field can be expressed as \( \mathbf{F} = \frac{\mathbf{r}}{|\mathbf{r}|} \), where \( \mathbf{r} = \langle x, y, z \rangle \) is the position vector, and \( |\mathbf{r}| \) is its magnitude. This expression means the vector field points in the same direction as \( \mathbf{r} \) and has a constant magnitude of 1. Because of their directional nature, radial vector fields play a crucial role in calculating surface integrals across spheres or similar symmetrical objects.

Spherical Symmetry

Spherical symmetry implies that a system or field looks the same from any direction around a given point. This type of symmetry is very helpful for simplifying calculations in vector calculus and physics, as it ensures uniformity and often simplifies integrals and other computations. In terms of mathematical problems, when a vector field and objects involved (such as a sphere) exhibit spherical symmetry, it means their configurations or mathematical expressions do not change if the object is rotated around its center point. This consistency can greatly simplify the use of techniques like surface integrals, where the geometric or physical properties being studied are dependable in every direction. The sphere in our problem has spherical symmetry because it is centered at the origin, and the given radial vector field is also symmetric with respect to the origin.

Divergence Theorem

The divergence theorem, also known as Gauss's divergence theorem, is a significant result in vector calculus. It creates a link between the flux of a vector field through a closed surface and the divergence of the field inside the surface. Mathematically, it is expressed as\[\iint_S \mathbf{F} \cdot d\mathbf{S} = \iiint_V (abla \cdot \mathbf{F}) \, dV\]where \( S \) is a closed surface, \( V \) is the volume enclosed by \( S \), and \( \mathbf{F} \) is the vector field. The theorem essentially turns a difficult surface integral calculation into a potentially simpler volume integral calculation.In this specific exercise, the divergence theorem simplifies the task by confirming that the surface integral of the radial vector field over the sphere is zero because the divergence within every point of the sphere's volume is zero.

Gauss's Theorem

Gauss's theorem, another name for the divergence theorem, is a powerful tool used to convert a problem from a difficult surface-level task to a more straightforward volume-level computation. It allows engineers and physicists to translate physical accumulation or flux across a boundary into something calculable within the boundary's enclosed space. This theorem aids in bridging the differential properties at a point and the global properties over a closed surface. For example, Maxwell's equations in electromagnetism commonly use Gauss's theorem to solve for electric fields given charge distributions. In the context of this exercise, Gauss's theorem helps to confirm that because the divergence of our radial vector field is zero throughout the enclosed volume, the net flux across the spherical boundary is zero. This was used effectively to simplify the calculation of the surface integral over the sphere.