Question

Verifiziere den Stokesschen Integralsatz für \(\underline{V}(\underline{x})=\left[x+y,-x^{2} z y, z+x y\right]^{\top}\) und die „obere Hemisphäre“ \(\vec{H}=\left\\{\underline{x} \in \mathbb{R}^{3} \| \underline{x} \mid=1, z \geq 0\right\\}\)

Short answer

To verify Stokes' theorem, calculate the curl of the vector field \(V\), the line integral of \(V\) around the boundary of the hemisphere, and the surface integral of the curl of \(V\) over the hemisphere. Compare the results of the line integral and the surface integral to confirm that they are equal, thus verifying Stokes' theorem.

Step-by-step solution

Calculating the Curl of \(V\)

First, calculate the curl of the vector field \(V\). Using the formula for the curl in Cartesian coordinates, we find that \(\nabla \times V = (z, -2xzy, 1+y).\)

Evaluating the Line Integral

Next, calculate the line integral of the vector field \(V\) around the boundary curve of the hemisphere \(H\). This is a unit circle in the xy-plane, so we can parametrize it with \(\underline{x}(t) = (\cos(t), \sin(t), 0)\) for \(0 \leq t < 2\pi\). The line integral then becomes an integral from 0 to \(2\pi\) of \(V(\underline{x}(t)) \cdot \underline{x}'(t) dt = \int_0^{2\pi} (\cos(t) + \sin(t), 0, 0) \cdot (-\sin(t), \cos(t), 0) dt = -\int_0^{2\pi} (\cos(t)\sin(t) + \sin^2(t)) dt.\)

Evaluating the Surface Integral

Lastly, calculate the surface integral of the curl of \(V\) over \(H\). Using spherical coordinates, we parametrize the hemisphere with \(\underline{x}(\theta, \phi) = (\sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi))\) for \(0 \leq \theta < 2\pi\) and \(0 \leq \phi \leq \pi/2\). The surface integral is then an integral from 0 to \(2\pi\) and from 0 to \(\pi/2\) of \(\nabla \times V(\underline{x}(\theta, \phi)) \cdot \underline{x}_\theta \times \underline{x}_\phi d\theta d\phi = \int_0^{2\pi}\int_0^{\pi/2} (\cos(\phi), -2\sin^2(\phi)\cos(\theta)\cos(\phi), 1 + \sin(\phi)\sin(\theta)) \cdot (\sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi)) d\phi d\theta\).

Conclusion: Verifying Stokes' Theorem

Finally, compare the results of the line integral and the surface integral. Stokes' theorem is verified if they are equal.

Key concepts

Vector Calculus

Vector calculus is a branch of mathematics that focuses on vector fields, often in three-dimensional space. It extends the concepts of single variable calculus to higher dimensions. In vector calculus, we explore different types of integrals.
The main ones are surface integrals and line integrals. Such integrals help us calculate quantities like fluid flow, or to understand the behavior of fields influenced by various forces.

Stokes' Theorem is a fundamental result in vector calculus. It connects the surface integrals of a vector field with the line integrals over the boundary of the surface. Understanding vector calculus is essential to deal with problems that involve spatial quantities and their interactions in physical space.

Surface Integrals

Surface integrals allow us to extend the idea of line integrals to surfaces. We calculate the "flow" across a surface in a vector field using these integrals. Surfaces can be any 2D space in a 3D region, like a curve or a part of a sphere.
To perform a surface integral, you need to parametrize the surface first. Parametrization simplifies the integration process. For example, a hemisphere can be parametrized using spherical coordinates, giving us a systematic way to perform calculations across the surface.

In the original exercise, the hemisphere was parametrized as \(\underline{x}(\theta, \phi) = (\sin(\phi)\cos(\theta), \sin(\phi)\sin(\theta), \cos(\phi))\). This way, the surface integral of the curl of the vector field over the hemispherical surface can be computed.

Line Integrals

Line integrals focus on integrating a vector field along a curve or path. Imagine evaluating the effect of a field along a wire or boundary of an object. This is done with line integrals.
To calculate a line integral, parametrize the curve or boundary. This gives the means to integrate the field along the chosen path. The circle boundary of the hemisphere is a classic example where we can easily use line integrals.

• Parametrize the boundary: For the hemisphere, the boundary is a unit circle on the xy-plane. Using \(\underline{x}(t) = (\cos(t), \sin(t), 0)\) from \(0\) to \(2\pi\).
• Evaluate: Compute \(V(\underline{x}(t)) \cdot \underline{x}'(t) dt\) over the parametrized curve.

Line integrals are crucial for applications involving fields across curves or boundaries.

Curl of a Vector Field

The curl is a measure of the rotational or swirling effect in a vector field. It's like understanding which way and how strongly a tiny paddle wheel would spin when placed in the field.
Mathematically, it is a vector derived from an existing vector field using the cross product of vector calculus operators. It is calculated using the nabla operator (∇) and is crucial for understanding field behaviors such as rotation and circulation.

In this exercise, the curl of \(\underline{V} = \left[x+y,-x^{2} z y, z+x y\right]^{\top}\) was computed: \(abla \times V = (z, -2xzy, 1+y)\). This curl shows the potential rotation or twisting at every point in the vector field. Calculating the curl is a key step in verifying Stokes' Theorem, as we relate it to the surface integral over the hemispherical surface.