Question

Explain the meaning of the integral \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) in Stokes' Theorem.

Short answer

In the context of Stokes' theorem, the integral term \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) refers to the total circulation of the vector field within the region bounded by the curve C. This surface integral measures the flux of the curl of the vector field through the surface S, summing up the local contributions to the circulation. Stokes' theorem states that this total circulation is equal to the circulation around the boundary of the region, represented by the line integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\). Thus, the integral term in Stokes' theorem highlights the relationship between the curl of a vector field and its circulation along a closed curve.

Step-by-step solution

Interpret the Surface Integral

The term \(\iint_{S} (\nabla \times \mathbf{F}) \cdot \mathbf{n} dS\) is a surface integral which measures the total "flux" of the curl of the vector field \(\mathbf{F}\) through the surface \(S\). The dot product \((\nabla \times \mathbf{F}) \cdot \mathbf{n}\) is the component of the curl of the vector field normal to the surface. Integrating this product (flux) over the entire surface sums up the contributions of all infinitesimal areas, giving us the total flux through the surface.

Relate Curl to Circulation

The curl of a vector field, \((\nabla \times \mathbf{F})\), is a measure of the "rotation" or "circulation" of the field around a point. If the curl is nonzero at a point, it means that the vectors of the field "rotate" around that point. Integrating the curl over the entire surface enclosed by the curve results in the sum of all these local contributions to the circulation along the surface. Therefore, the surface integral of the curl is a measure of the overall circulation of the vector field within the closed region bounded by the curve.

Meaning of Stokes' Theorem

The meaning of the integral \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S\) in Stokes' theorem is that it gives the total circulation of the vector field within the region bounded by the curve C. According to Stokes' theorem, this total circulation is equal to the circulation around the boundary of the region, which is represented by the line integral \(\oint_{C} \mathbf{F} \cdot d\mathbf{r}\). This implies that the circulation of the vector field around a closed curve can be calculated by integrating the curl of the vector field over a surface spanning the curve (and vice versa). Understanding the meaning of the integral term in Stokes' theorem allows us to see the relationship between the curl of a vector field and its circulation along a closed curve.