Question

Stokes' Theorem on closed surfaces Prove that if \(\mathbf{F}\) satisfies the conditions of Stokes' Theorem, then \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S=0\) where \(S\) is a smooth surface that encloses a region.

Short answer

Question: Prove that the surface integral of the curl of a vector field dot product with the normal vector over an enclosed surface is zero, according to Stokes' Theorem. Answer: For a smooth vector field and an enclosed surface with no boundary curve, Stokes' Theorem states that the surface integral of the curl of the vector field dot product with the normal vector over the enclosed surface will be zero. This is because the line integral of the vector field around the degenerate boundary curve is zero, which leads to: $$\iint_{S}(\nabla\times\mathbf{F})\cdot\mathbf{n}\:dS = 0$$

Step-by-step solution

State Stokes' Theorem

Stokes' Theorem states that for a given smooth vector field \(\mathbf{F}\) and a smooth surface \(S\) with boundary curve \(C\), oriented consistently, we have: $$\oint_{C}\mathbf{F}\cdot d\mathbf{r} = \iint_{S}(\nabla\times\mathbf{F})\cdot\mathbf{n}\:dS$$ where \(\nabla\times\mathbf{F}\) is the curl of the vector field \(\mathbf{F}\), and \(\mathbf{n}\) is the unit normal vector to the surface S.

Define an enclosed surface

An enclosed surface S is a smooth, closed surface that encloses a volume. This means that the boundary curve \(C\) around S is the degenerate curve with no actual length. In other words, the curve C does not exist, or we can think of it as a single point.

Apply Stokes' Theorem to enclosed surfaces

Since the boundary curve C does not exist or is a point, we can say that the line integral of the vector field around it must be zero: $$\oint_{C}\mathbf{F}\cdot d\mathbf{r} = 0$$ Now, applying Stokes' Theorem, we can write: $$0 = \iint_{S}(\nabla\times\mathbf{F})\cdot\mathbf{n}\:dS$$

Conclusion

We have proven that if \(\mathbf{F}\) satisfies the conditions of Stokes' Theorem, then the surface integral of the curl of the vector field dot product with the normal vector over an enclosed surface S will be zero: $$\iint_{S}(\nabla\times\mathbf{F})\cdot\mathbf{n}\:dS = 0$$

Key concepts

Surface Integrals

Surface integrals are a major concept in vector calculus, used to evaluate multidimensional integrals over a surface within three-dimensional space. Essentially, a surface integral quantifies the flux, or, loosely speaking, the amount of a vector field that flows through a given surface.

To understand this better, imagine a surface like a piece of cloth fluttering in the wind—the surface integral would calculate how much wind—or any other vector quantity like an electric field—is passing through that cloth. Mathematically, the surface integral of a vector field \textbf{F} over a surface S is denoted as \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS \), where \mathbf{n}\ is the unit normal vector to the surface at each point, and \(dS\) represents a differential element of the surface area.

In the context of Stokes' Theorem, the surface integral is related to the curl of the vector field, offering a bridge between the behavior of the field around a closed loop and the cumulative behavior across a surface.

Vector Field

A vector field is a mathematical entity that assigns a vector to every point in a space. It's a way to represent physical quantities that have both magnitude and direction, like velocity, force, or magnetic fields.

For example, to visualize a vector field imagine placing a tiny wind vane at every point in a space; each wind vane shows the wind's direction and speed at that exact location. When we talk about a 'smooth' vector field in the context of Stokes' Theorem, we mean that the vectors change continuously as you move through the field—there are no abrupt changes or breaks.

Line Integral

A line integral is another fundamental concept in calculus, pivotal to multivariable calculus and physics, specifically when working with vector fields. Unlike normal integrals that sum up over an interval on the number line, a line integral sums up a function along a curve.

The line integral of a vector field \mathbf{F}\ along a curve C is written as \( \oint_{C} \mathbf{F} \cdot d\mathbf{r} \), where \(d\mathbf{r}\) denotes the infinitesimal tangent vector along the curve. Essentially, it measures the work done by the field in moving a particle along the curve. In simpler terms, it's like adding up the force felt by a hiker (from a vector field such as gravity) along a path in the hills, taking into account both the force's strength and the direction in which the hiker is moving.

Curl of a Vector Field

The curl is a vector operation that describes the rotation or swirling strength at each point within a vector field. For a given vector field \mathbf{F}\, the curl is denoted by \(abla \times \mathbf{F}\).

Imagine a paddle wheel placed in flowing water, where the water represents the vector field. The way the water makes the wheel spin—that's what the curl measures. In three dimensions, the concept of curl is fundamental in connecting line integrals and surface integrals via Stokes' Theorem. The theorem essentially equates the 'circulation' around the boundary of a surface (the line integral of our field) to the 'sum of the curl' inside that surface area (the surface integral of the curl of the field).