Question

Show that if \(\iint_{S} u_{n} d \sigma=0\) for every oriented spherical surface \(S\) in a domain \(D\) and the components of \(\mathbf{u}\) have continuous derivatives in \(D\), then \(\mathbf{u}\) is solenoidal in \(D\). Does the converse hold?

Short answer

Answer: We used the divergence theorem and showed that the divergence of \(\mathbf{u}\) is zero. This implies that \(\mathbf{u}\) is solenoidal in the domain \(D\). We then considered the converse and found it to be true: If \(\mathbf{u}\) is solenoidal in the domain \(D\), then \(\iint_{S} \mathbf{u}_{n} d\sigma = 0\) for every spherical surface \(S\) in \(D\).

Step-by-step solution

Use the divergence theorem

According to the given condition, we have \(\iint_{S} \mathbf{u}_{n} d\sigma = 0\). Now, we will apply the divergence theorem, which states that: $$\iint_{S} \mathbf{u}_{n} d\sigma = \iiint_{V} (\nabla \cdot \mathbf{u}) dV$$ Here, \(V\) is the volume inside the surface \(S\). So, we have: $$\iiint_{V} (\nabla \cdot \mathbf{u}) dV = 0$$

Show that the divergence of \(\mathbf{u}\) is zero

Since the components of \(\mathbf{u}\) have continuous derivatives in \(D\), we can say that \(\mathbf{u}\) is smooth. We know that the integral \(\iiint_{V} (\nabla \cdot \mathbf{u}) dV = 0\). This integral is zero for every sphere in the domain \(D\), which means the integrand \((\nabla \cdot \mathbf{u})\) must be zero in \(D\). Thus, we can conclude that: $$\nabla \cdot \mathbf{u} = 0$$

Conclude that \(\mathbf{u}\) is solenoidal

The condition \(\nabla \cdot \mathbf{u} = 0\) implies that \(\mathbf{u}\) is solenoidal in the domain \(D\). Therefore, we have proven the first part of the problem.

Consider whether the converse is true

Now, let's consider if the converse is true. If \(\mathbf{u}\) is solenoidal in \(D\), is it true that \(\iint_{S} \mathbf{u}_{n} d\sigma = 0\) for every spherical surface \(S\) in \(D\)? We can revisit the divergence theorem: $$\iint_{S} \mathbf{u}_{n} d\sigma = \iiint_{V} (\nabla \cdot \mathbf{u}) dV$$ Since we have assumed that \(\mathbf{u}\) is solenoidal, this means \(\nabla \cdot \mathbf{u} = 0\). Therefore, the right-hand side of the divergence theorem equation becomes: $$\iiint_{V} (\nabla \cdot \mathbf{u}) dV = \iiint_{V} 0 dV = 0$$ Thus, the left-hand side also becomes zero: $$\iint_{S} \mathbf{u}_{n} d\sigma = 0$$ So, the converse is also true: If \(\mathbf{u}\) is solenoidal in the domain \(D\), then \(\iint_{S} \mathbf{u}_{n} d\sigma = 0\) for every spherical surface \(S\) in \(D\).

Key concepts

Divergence Theorem

The Divergence Theorem connects a surface integral with a volume integral, relating the flow of a vector field through a surface to the behavior of the vector field inside the surface. This theorem states that the surface integral of a vector field over a closed surface is equal to the volume integral of the divergence of the vector field inside the volume bounded by the surface. Formally, for a vector field \( \mathbf{F} \) with continuous derivatives, if \( V \) is the volume enclosed by the closed surface \( S \), the theorem can be expressed as: \[ \iint_{S} \mathbf{F} \cdot d\mathbf{S} = \iiint_{V} (abla \cdot \mathbf{F}) \, dV \]

• The left-hand side represents how much the vector field is "flowing out" through the surface \( S \).
• The right-hand side gives the total divergence within the volume \( V \), indicating how sources and sinks affect the flow.

By applying this theorem, one can conclude that if the divergence over a volume is zero, the net flux through any enclosing surface is also zero.

Oriented Spherical Surface

An oriented spherical surface is a fundamental concept in vector calculus. It refers to a sphere with a specified direction, typically indicated by the normal vectors of the surface pointing outward from the center. Orientation describes the direction in which the surface is considered, influencing calculations like flux and surface integrals.
In many mathematical and physical applications, spheres are preferred over other shapes due to their symmetry and simplicity.

• Choosing a consistent orientation ensures calculations of flow across the surface are consistent.
• Orientation affects integrals since it dictates the sign based on the direction of the normal vector relative to the field.

These oriented surfaces provide a symmetrical and straightforward scenario to analyze properties like divergence or flux, making them ideal for testing vector field properties.

Continuous Derivatives

In vector calculus, a fundamental requirement for applying several theorems, such as the Divergence Theorem, is that the vector field should have continuous derivatives. This means each component of the vector field vector function is continuously differentiable, making the field "smooth" over its domain.
Continuous derivatives ensure that subtle behaviors of the vector field can be reliably interpreted and calculated, as no abrupt changes or discontinuities occur in the field.

• Ensures stability in numerical computations, avoiding erratic changes in the vector field.
• Allows the derivation of predictable trends across the domain, which is essential when integrating over surfaces and volumes.

Knowing that derivatives are continuous assures that the vector field obeys the necessary conditions to apply the Divergence Theorem precisely.

Spherical Surface Integral

A spherical surface integral is a type of surface integral that specifically uses a spherical surface to evaluate the flow of a vector field across that surface. These integrals are critical in determining aspects like flux, representing the total quantity passing through the surface.
When applying a spherical surface integral, the idea is to sum up contributions of the field across small patches of the sphere, each patch weighed by its area and the field's magnitude at that location.

• Integral expression: \( \iint_{S} \mathbf{u}_{n} d\sigma \), where \( \mathbf{u}_{n} \) is the vector field component normal to the surface, and \( d\sigma \) is the differential area element.
• The result provides insight into whether the vector field "enters" or "exits" the boundary.

For a solenoidal field, the spherical surface integral over any closed spherical surface is zero, indicating that the flow out is balanced by the flow in, confirming no net change in the field's quantity within the volume enclosed.