Question

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 if a vector field F satisfies the conditions of Stokes' Theorem and the surface S is a smooth surface that encloses a region, then the surface integral of the curl of the vector field F will be zero. Answer: Given that the vector field F satisfies the conditions of Stokes' Theorem and the surface S is smooth and encloses a region, we have applied Stokes' Theorem to this setup and found that the closed curve C is not present (S has no boundary). Then, we can say that the line integral over the closed curve C is zero, leading to: \(0 = \iint_S (\nabla \times \mathbf{F})\cdot\mathbf{n}dS\) Hence, the surface integral of the curl of the vector field F will be zero.

Step-by-step solution

Assume the conditions of Stokes' Theorem are satisfied.

In order to use Stokes' Theorem, the vector field F must satisfy its conditions, which are continuity and differentiability throughout the region that the smooth surface S encloses.

State that the surface has no boundary, and hence there's no closed curve.

Since the surface S encloses a region and has no boundary, this implies that there's no closed curve C. By definition, the curve C is the boundary of the surface S, but in this case, we don't have any boundary for the surface.

Apply Stokes' Theorem to the given vector field and surface.

We're now going to apply Stokes' Theorem to this setup. As per Stokes' Theorem, we have: \(\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_S (\nabla \times \mathbf{F})\cdot\mathbf{n}dS\) However, since there is no closed curve C, we can say that the left-hand side of the equation is zero, as there is no curve to integrate over. Therefore, we get: \(0 = \iint_S (\nabla \times \mathbf{F})\cdot\mathbf{n}dS\)

Conclude the proof.

According to the equation derived in the previous step, we have shown that if the vector field F satisfies the conditions of Stokes' Theorem and the surface S is a smooth surface that encloses a region, then the surface integral of the curl of the vector field F will be zero, i.e., \(\iint_S (\nabla \times \mathbf{F})\cdot\mathbf{n}dS = 0\) This completes the proof.

Key concepts

Vector Field

A vector field is a function that assigns a vector to each point in a space. Imagine blowing wind across a field. At each point in the field, the wind might have a different speed and direction. This is akin to what a vector field represents, with every vector indicating a direction and magnitude at its corresponding point in space.

Key properties of vector fields include:

• Components: Vector fields can be described using components like \( \mathbf{F}(x, y, z) = ai + bj + ck \), where \(a, b, \) and \(c\) are functions of \(x, y, z\).
• Dimensions: Vector fields exist in multiple dimensions, such as 2D or 3D, depending on the context.
• Differentiability and Continuity: For applications like in Stokes' Theorem, differentiability and continuity are necessary to ensure meaningful integrations.

Surface Integral

Surface integrals extend the concept of integration to summing over curved surfaces. When you perform a surface integral, you sum up values over a surface's extent rather than along a line or within a volume.

When dealing with a surface integral of a vector field, the formula used is \( \iint_{S} \mathbf{F} \cdot \mathbf{n} \, dS \), where:

• \(\mathbf{F}\): The vector field being integrated over the surface.
• \(\mathbf{n}\): The unit normal vector to the surface \(S\), indicating direction perpendicular to \(S\).
• \(dS\): A differential area element on the surface \(S\).

Surface integrals help in finding quantities like the flux of a vector field through a surface, providing insight into how the field behaves and interacts with the surface in question.

Curl of a Vector Field

The curl of a vector field measures the tendency of the field to rotate around a point. It is a vector pointing in the direction of the axis of rotation, with magnitude representing the strength of rotation.

For a vector field \( \mathbf{F}(x, y, z) = ai + bj + ck \), the curl \( abla \times \mathbf{F} \) is computed using:

• \( abla \times \mathbf{F} = \left( \frac{\partial c}{\partial y} - \frac{\partial b}{\partial z}\right)i + \left( \frac{\partial a}{\partial z} - \frac{\partial c}{\partial x}\right)j + \left( \frac{\partial b}{\partial x} - \frac{\partial a}{\partial y}\right)k \)

This can be remembered using a determinant format involving the basis vectors \(i, j, k\) and partial derivatives.

• Applications: Curl is crucial in fluid dynamics, electromagnetism, and studying rotational motion.
• Visual Representation: If you visualize a field where tiny "paddles" or "wheels" are placed everywhere, the curl tells you how those paddles would rotate.