Question

Let \( \vec{F} \) be a vector point function defined inside and on the surface (S) of the sphere \( x^2+y^2+z^2=1 \). Then \( \oint_S (\text{Curl} \vec{F}) \cdot \hat{n} ds = \)

• A. \( \iint_E \vec{F} \cdot d\vec{R} \), where E is the region of the sphere
• B. 0
• C. \( \frac{4}{3}\pi \)
• D. \( \oint_S \vec{F} \cdot d\vec{R} \)

Hint:

The Divergence Theorem: \( \oint_S \vec{A} \cdot d\vec{S} = \iiint_V (\nabla \cdot \vec{A}) dV \).
Vector Identity: \(\text{div}(\text{curl } \vec{F}) = \nabla \cdot (\nabla \times \vec{F}) = 0\).
The flux of the curl of a vector field through any closed surface is always zero.

Answer 1

The Correct Option is B

We need to evaluate the surface integral \( \oint_S (\text{Curl} \vec{F}) \cdot \hat{n} ds \), where S is a closed surface (a sphere). Let \(\vec{G} = \text{Curl} \vec{F} = \nabla \times \vec{F}\). The integral is \( \oint_S \vec{G} \cdot \hat{n} ds \). By the Divergence Theorem (Gauss's Theorem), for a closed surface S enclosing a volume V: \[ \oint_S \vec{G} \cdot \hat{n} ds = \iiint_V (\nabla \cdot \vec{G}) dV \] Substitute \(\vec{G} = \nabla \times \vec{F}\): \[ \oint_S (\nabla \times \vec{F}) \cdot \hat{n} ds = \iiint_V \nabla \cdot (\nabla \times \vec{F}) dV \] A standard vector identity states that the divergence of the curl of any sufficiently differentiable vector field \(\vec{F}\) is identically zero: \[ \nabla \cdot (\nabla \times \vec{F}) = 0 \] Therefore, the integral becomes: \[ \iiint_V (0) dV = 0 \] This result holds for any closed surface S, including the given sphere. \[ \boxed{0} \]