Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(S\) be a solid sphere (or any solid enclosed by a "nice" surface \(\partial S)\). Show that $$ \iint_{\partial S}(\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} d S=0 $$ (a) By using Stokes's Theorem. (b) By using Gauss's Theorem. Hint: Show \(\operatorname{div}(\operatorname{curl} \mathbf{F})=0\).

Step-by-Step Solution

Answer

Both Stokes's and Gauss's Theorems show the integral is zero over the closed surface.

Step 1: Understand Stokes's Theorem

Stokes's Theorem relates a surface integral over a surface \( \partial S \) to a line integral over its boundary curve. It states: \( \iint_{\partial S} (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r} \), where \( C \) is the boundary of \( \partial S \). For a closed surface (like the boundary of a solid sphere), the boundary curve \( C \) does not exist, hence the line integral is zero.

Step 2: Apply Stokes's Theorem

Since the boundary \( C \) of the closed surface \( \partial S \) does not exist, the right side of Stokes's Theorem becomes zero: \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} = 0 \). Therefore, \( \iint_{\partial S} (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \). This shows that the integral of curl over a closed surface is zero using Stokes's Theorem.

Step 3: Understand Gauss's Theorem

Gauss's Theorem, also known as the Divergence Theorem, relates the flux of a vector field \( \mathbf{F} \) through a closed surface \( \partial S \) to the divergence of \( \mathbf{F} \) over the volume \( S \) enclosed by the surface: \( \iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{S} \operatorname{div} \mathbf{F} \, dV \).

Step 4: Show Divergence of Curl is Zero

The divergence of the curl of any vector field \( \mathbf{F} \) is zero: \( \operatorname{div}(\operatorname{curl} \mathbf{F}) = 0 \). This is a well-known vector calculus identity.

Step 5: Apply Gauss's Theorem

Using Gauss's Theorem, \( \iint_{\partial S} (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \iiint_{S} \operatorname{div}(\operatorname{curl} \mathbf{F}) \, dV \). Since \( \operatorname{div}(\operatorname{curl} \mathbf{F}) = 0 \), the volume integral becomes zero: \( \iiint_{S} 0 \, dV = 0 \). Thus, \( \iint_{\partial S} (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = 0 \).

Key Concepts

Stokes's Theorem

Stokes's Theorem is a powerful tool in vector calculus that connects the surface integral of the curl of a vector field over a surface to a line integral around its boundary. This theorem states:\[\iint_{\partial S} (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} \, dS = \oint_{C} \mathbf{F} \cdot d\mathbf{r}\]Here, \(\partial S\) represents the surface, \(C\) is the boundary curve of this surface, \(\mathbf{n}\) is the unit normal to the surface, and \(\mathbf{F}\) is a vector field. When dealing with a closed surface such as a sphere, there is no boundary curve \(C\). Thus, the right-hand side of the theorem, which includes the line integral, is zero. This results in the left-hand side, i.e., the integral of the curl over the surface, also being zero. It simplifies conceptualizing how circulation around closed paths (boundaries non-existent in a closed surface) influences the curl's integral.

Gauss's Theorem

Gauss's Theorem, often referred to as the Divergence Theorem, creates a link between the flux of a vector field across a closed surface and the divergence of that field over the volume enclosed by the surface. It is formulated as:\[\iint_{\partial S} \mathbf{F} \cdot \mathbf{n} \, dS = \iiint_{S} \operatorname{div} \mathbf{F} \, dV\]In this formula, \(\mathbf{F}\) is a vector field, \(\partial S\) signifies the closed surface, \(\mathbf{n}\) is the outward normal to that surface, and \(S\) is the volume enclosed. Gauss's Theorem effectively helps to transform a surface integral into a volume integral, making it particularly useful for computing flux when dealing with closed surfaces. For a curl field \(\operatorname{curl} \mathbf{F}\), the divergence is zero as shown using vector identities, thereby simplifying many problems in vector calculus.

Divergence and Curl

Understanding divergence and curl is crucial in vector calculus, as these are foundational operations applied to vector fields. The divergence of a vector field, \(\operatorname{div} \mathbf{F}\), measures how much a vector field is spreading out or converging at any given point:\[\operatorname{div} \mathbf{F} = abla \cdot \mathbf{F}\]It gives insight into the rate of change of density under the influence of the vector field. On the other hand, the curl of a vector field, \(\operatorname{curl} \mathbf{F}\), describes the rotation or the circulation tendency at a point:\[\operatorname{curl} \mathbf{F} = abla \times \mathbf{F}\]The divergence of the curl is always zero:\[abla \cdot (abla \times \mathbf{F}) = 0\]This key identity simplifies many formulas and proofs in vector calculus, including those demonstrating that the surface integral of the curl over a closed surface is zero.

Surface Integrals

Surface integrals allow us to integrate over a curved surface in space, playing a critical role in physics and geometry. When computing a surface integral of a vector field \(\mathbf{F}\) over a surface \(S\), the integral captures the cumulative contributions of \(\mathbf{F}\) through infinitesimal patches \(\mathbf{n} \, dS\) on \(S\):\[\iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS\]The vector \(\mathbf{n}\) denotes the unit normal vector to the surface at each point, and \(dS\) represents an infinitesimally small piece of surface area. Surface integrals are used to determine quantities like flux across a surface, providing valuable insights in electromagnetism, fluid flow, and other fields. For closed surfaces, which fully enclose a volume, these integrals are linked to the volume's divergence via Gauss's Theorem.