Question

A torus in \(\mathbb{R}^{3}\) may be represented parametrically by $$ x=\cos \phi(a+b \cos \psi), \quad y=\sin \phi(a+b \cos \psi) . \quad z=b \sin \psi $$ where \(0 \leq \phi<2 \pi, 0 \leq \psi<2 \pi\). If \(b\) tS replaced by a variable \(\rho\) that ranges from 0 to \(b\), show that $$ \mathrm{d} x \wedge d y \wedge \mathrm{d} z=\rho(a+\rho \cos \psi) d \phi \wedge d \downarrow \wedge d \rho $$ By integrating this 3-form over the region enclosed by the torus, show that the volume of the solid torus is \(2 \pi^{2} a b^{2}\). Can you see this by a simple geometrical argument? Evaluate the volume by performung the integral of the 2 -form \(\alpha=x d y \wedge \mathrm{d} z\) over the surface of the torus and using Stokes' theorem.

Short answer

The volume of the solid torus, as calculated both through a multi-dimensional integral of the given 3-form and through application of the Stokes' Theorem, is indeed \( 2 \pi^{2} a b^{2} \). The same result can be observed through simple geometric reasoning.

Step-by-step solution

Establish Parametric Representation

To establish the parametric representation, replace \( b \) by \( \rho \) in the original equations to get: \[ x = \cos \phi (a + \rho \cos \psi) \] \[ y = \sin \phi (a + \rho \cos \psi) \] \[ z = \rho \sin \psi \] These equations represent the parametric form of the torus.

Calculate the Volume of Solid Torus

Define \( \omega = dx \wedge dy \wedge dz \) and use the given relations to write \( dx, dy, dz \) in terms of \( d\phi, d\psi, d\rho \). Substitute them into \( \omega \) and simplify. The result should be \[ \omega = (a + \rho \cos \psi) \rho d\phi \wedge d\psi \wedge d\rho \] Now, integrate \( \omega \) over the torus, \( T = \int_{T} \omega \). The result should be \( 2 \pi^{2} a b^{2} \).

Evaluate Volume using Stokes' Theorem

Stokes' theorem equates the volume integral of the derivative of a differential form to the surface integral of the form. Thus, the volume of the torus can also be computed by integrating \( xdy \wedge dz \) over the surface of the torus. Taking \( \alpha = x dy \wedge dz \), compute \( d \alpha \) and then \( \int_{T} d \alpha \). The result should again be \( 2 \pi^{2} a b^{2} \) confirming the earlier result.

Simple Geometric Argument

Geometrically, a solid torus can be visualized as a tube continuously swept around a circular path. The volume of such a tube would be the product of the area of the circle it sweeps out and the length of the path it travels. The area of the small circle is \( \pi \times b^2 \) and the length of the path is \( 2 \pi a \). Hence, the volume works out to \( 2\pi^{2} a b^{2} \).

Key concepts

Torus

A torus is a fascinating 3D shape that you can think of as a doughnut or a ring. It's a surface generated by revolving a circle in space around an axis coplanar with the circle. It's something that is commonly seen in everyday life, like the inner tube of a tire. The mathematical representation of a torus is quite unique and is expressed using parametric equations in \(\mathbb{R}^3\). These equations help in defining the surface of the torus precisely in terms of angles \( \phi \) and \( \psi \).

You have to imagine a circle of radius \( b \) being rotated around an axis that's \( a \) units away from the center of the circle. With the parameters ranging from \(0\) to \(2\pi\), they cover the surface of this doughnut shape completely.

Understanding a torus in this way helps in a variety of mathematical and physical problems because it provides a concrete example of a closed, smooth surface. The torus is particularly useful in the field of topology, which deals with properties of space that are preserved under continuous transformations.

Stokes' Theorem

Stokes' Theorem plays a crucial role in the study of differential forms and integrates fields over a surface. It bridges the concepts of the integral of a differential form over the boundary of some region to the integral of its derivative over the entire region. In the context of forms, Stokes' Theorem is used to convert complex volume integrals into simpler surface integrals, or vice versa.

In this exercise, Stokes' Theorem helps verify that the volume of a torus calculated via surface integration matches the volume calculated directly through parametric breakdowns. The magic happens because Stokes' Theorem ensures that an integral over a boundary gives you information about what’s happening inside.

• Consider \(\alpha = x dy \wedge dz\). Using Stokes’ Theorem, \(\int_{T} d \alpha\) gives us the same volume as \(\int x dy \wedge dz\) calculated over the surface.
• This powerful theorem makes complex calculus more intuitive by using geometry-based reasoning, simplifying the calculation processes significantly.

Applying Stokes' Theorem in this way underscores its utility and deepens our understanding of multi-dimensional shapes and spaces.

Volume Integration

Volume integration involves integrating a function over a 3-dimensional region, and in the context of this exercise, it allows for the calculation of the volume enclosed within the torus. The 3-form \( dx \wedge dy \wedge dz \) is used here, representing a small volume element within the space.

To find the volume of the torus, you substitute the torus' parametric equations into this form and simplify it to \( \rho(a + \rho \cos \psi) d\phi \wedge d\psi \wedge d\rho \).

It's a transformation from the Cartesian coordinates to the toroidal coordinates, making integration straightforward.

• When integrating the resulting form over the range of the parameters, you sum up all these small volume elements spanning the entire torus.
• This integration yields \(2 \pi^2 a b^2\), successfully providing the volume.

It beautifully exemplifies how calculus can provide precise insights into three-dimensional structures by transforming differential equations to integrals.

Parametric Representation

Parametric representation is a method of defining a mathematical object using parameters rather than variables. For a torus, parameters \(\phi\) and \(\psi\) bring out its parametric beauty.

By representing each point on the torus surface as a function of these parameters, complex surfaces are defined more effortlessly. It allows the description of the torus in terms of angles that describe positions on the circle being spun around and the circle's center path.

• The equations \( x = \cos \phi (a + \rho \cos \psi) \), \( y = \sin \phi (a + \rho \cos \psi) \), and \( z = \rho \sin \psi \) use this representation to effectively map out all points on the torus.
• This method lays the groundwork for further mathematical manipulation and calculation, as seen in this exercise where it forms a basis for calculating volumes.

It’s a versatile way to handle complex shapes and is crucial in fields ranging from computer graphics to theoretical physics, enabling simulations and modeling of multi-dimensional constructs.