Question

If \(\alpha=x d y \wedge d z+y d z \wedge d x+z d x \wedge d y\) compute \(\int_{\text {a\Omega }} \alpha\) where \(\Omega\) is (i) the unit cube, (ii) the unit ball in \(\mathbb{R}^{3}\). In each case verify Stokes' theorem, $$ \int_{\text {an }} \alpha=\int_{\Omega} d \alpha $$

Short answer

After performing the above steps, the values of \(\int_{a\Omega} \alpha\) for the unit cube and ball are obtained. Further, the values of \(\int_{\Omega} d \alpha\) for both cases are computed. The verification of Stokes' theorem is done by equating \(\int_{a\Omega} \alpha\) and \(\int_{\Omega} d \alpha\) for both geometries.

Step-by-step solution

Compute dα

First, compute the exterior derivative of \(\alpha\). In this case, \(d \alpha=2(x d y \wedge d x \wedge d z+y d z \wedge d x \wedge d y+z d x \wedge d y \wedge d z)\).

Setup Integration for Unit Cube

For \(\Omega\) as the unit cube [0,1]x[0,1]x[0,1], the boundary has six faces. For each face, find the outward unit normal vector and then calculate the integral of \(\alpha\) on each face. Add these six integrals to find \(\int_{a\Omega} \alpha\). For example, for the face x=0, the outward unit normal vector is (-1,0,0) and the integral of \(\alpha\) on this face is \(-\int_{0}^{1} \int_{0}^{1} z d y \wedge d z\). Compute similar integrals over the rest five faces and add them to obtain the total integral over the boundary of the cube.

Verify Stokes' Theorem for Unit Cube

Verify Stokes' theorem for the cube. Evaluate \(\int_{\Omega} d \alpha\) and check if it equals to \(\int_{a\Omega} \alpha\) obtained in Step 2. For the cube, \(\int_{\Omega} d \alpha=2 \int_{0}^{1} \int_{0}^{1} \int_{0}^{1}(x+y+z) d x d y d z\). Calculate this triple integral and compare with the result of Step 2 to verify Stokes' theorem.

Setup Integration for Unit Ball

For \(\Omega\) as the unit ball in \(\mathbb{R}^{3}\), the boundary is a unit sphere. Convert \(\alpha\) into spherical coordinates and compute the integral of \(\alpha\) over the surface of the sphere to find \(\int_{a\Omega} \alpha\). For example, in spherical coordinates, \(\alpha=sin\theta cos\phi dr \wedge d\theta \wedge d\phi + sin^2\theta sin\phi d\theta \wedge d\phi \wedge dr + cos\theta d\phi \wedge dr \wedge d\theta\). Compute the integral of this form over the sphere.

Verify Stokes' Theorem for Unit Ball

Verify Stokes' theorem for the unit ball. Evaluate \(\int_{\Omega} d \alpha\) and confirm that it equals to \(\int_{a\Omega} \alpha\) obtained in Step 4. The computation is similar to Step 3 but with limits of integral suitable for unit ball. Compare the results to verify Stokes' theorem.

Key concepts

exterior derivative

The exterior derivative is a crucial concept in differential geometry and is denoted by the symbol \(d\). It helps in generalizing the notion of differentiation to higher dimensional spaces and forms.
When you have a differential form \(\alpha\), taking its exterior derivative \(d\alpha\) results in a higher order form. This new form encapsulates the changes of \(\alpha\) over its domain.
For example, if \(\alpha = x dy \wedge dz + y dz \wedge dx + z dx \wedge dy\), then its exterior derivative \(d\alpha\) becomes a form that includes components \(2(x dy \wedge dx \wedge dz + y dz \wedge dx \wedge dy + z dx \wedge dy \wedge dz)\).
This result comes from applying the rules of exterior differentiation which include linearity, the product rule, and \(d^2 = 0\), which means that applying the exterior derivative twice results in zero for any form.

boundary integrals

Boundary integrals are integral computations performed over the boundary of a geometric region. They are essential in the verification of Stokes' Theorem, which relates surface or line integrals to integrals over volumes or higher-dimensional regions.
When applying Stokes' Theorem, you often need to evaluate \(\int_{\partial \Omega} \alpha\), where \(\partial \Omega\) is the boundary of a region \(\Omega\).

• The boundary of a unit cube, for instance, consists of six flat rectangular faces. You compute the integral of the form \(\alpha\) over each face and sum them up.
• In contrast, the boundary of a unit ball is a spherical surface, which requires integration using spherical coordinates.

This process is not only important for validating theoretical results but also for practical computational problems in physics and engineering.

unit cube

The unit cube is a simple yet fundamental geometric shape in mathematics. It is defined as a cube with sides of length 1, typically oriented along the coordinate axes.
In this context, the cube has vertices such as \((0,0,0)\), \((1,0,0)\), \((0,1,0)\), and so on, leading to a total volume of 1 cubic unit.
The boundary of the unit cube consists of six square faces, each with an area of 1. These faces are aligned with the Cartesian planes, making computations straightforward.

• For integration over this boundary, you compute the integral over each face separately, typically using the respective normal vectors.
• Once all face integrals are calculated, they are summed up to provide the boundary integral over the cube.

Understanding the unit cube is crucial when applying Stokes' Theorem because it serves as a simple example to illustrate how volume integrals inside \(\Omega\) relate to boundary integrals on \(\partial \Omega\).

unit ball

The unit ball in \(\mathbb{R}^{3}\) represents all points in 3D space that are within a distance of 1 from a central point, typically the origin. This creates a sphere, which serves as the boundary of the unit ball.
Unlike the unit cube, the ball's boundary is continuous and not flat, thus requiring spherical coordinates for efficient integration.
The transition to spherical coordinates involves variables \(r\), \(\theta\) (polar angle), and \(\phi\) (azimuthal angle). These are more suited to the symmetry of a sphere than Cartesian coordinates.

• The integral over the unit sphere is performed using these coordinates, simplifying the computation of boundary integrals for forms specified in spherical terms.
• Stokes' Theorem applies here by translating the problem into areas and integrating over the continuous boundary surface.

Grasping the concept of the unit ball and its implications in integration is vital for solving problems in vector calculus and physics, especially when dealing with spherical symmetrical fields.