This problem concerns a surface of revolution in $\mathbb{R}^3$ that results from rotating a Jordan curve $\Gamma$ around the $zz$-axis. Let me break down each part:

Part (a)

The goal is to show that the surface area $A(S)$ of the surface $S$ is given by the integral: $A(S) = 2\pi \int_{\Gamma} \rho \, ds,$ where $\rho$ is the radial coordinate, and $ds$ is the differential arc length element along the curve $\Gamma$.

Explanation:

For a surface of revolution, the formula for the surface area is obtained by rotating a curve around an axis (here, the $z$-axis). The general formula for the surface area of revolution is: $A = 2\pi \int \rho(s) \, ds,$ where $\rho(s)$ is the distance from the curve to the axis of rotation (in this case, the $z$-axis). The curve $\Gamma$ is parameterized in terms of arc length $s$, and $ds$ represents an infinitesimal element of this curve.

Thus, the formula $A(S) = 2\pi \int_{\Gamma} \rho \, ds$ is a general result for the surface area of revolution, obtained by integrating over the curve $\Gamma$.

Part (b)

Now, we are tasked with using the result from part (a) to show that the surface area of a torus with radii $0 < r < R$ is $4\pi^2 Rr$.

Explanation:

A torus can be described as the surface obtained by rotating a circle of radius $r$ (lying in the plane perpendicular to the $z$-axis) around a circle of radius $R$ (centered on the $z$-axis). Using cylindrical coordinates, the circle that is rotated has a radial distance $\rho = R + r\cos(\theta)$, where $\theta$ is the angle parameter of the small circle of radius $r$.

From part (a), the surface area formula is: $A(S) = 2\pi \int_{\Gamma} \rho \, ds.$ For the torus, the radial distance $\rho = R + r\cos(\theta)$, and we are integrating over the circular arc of length $2\pi r$. Thus, the surface area of the torus is: $A_{\text{torus}} = 2\pi \cdot R \cdot 2\pi r = 4\pi^2 Rr.$ This confirms the result.

Would you like a more detailed explanation or any clarifications on these concepts?

5 related questions:

1. How is the arc length differential $ds$ computed for a given curve $\Gamma$?
2. How does the surface of revolution formula change for different axes of rotation?
3. What is the difference between a solid torus and a surface torus in mathematical terms?
4. How would you compute the volume of the torus using the same rotational principles?
5. Can this surface area formula be generalized to other coordinate systems, such as spherical coordinates?

Tip:

When dealing with surfaces of revolution, always check the radial distance $\rho$ to the axis of rotation, as it directly impacts the surface area formula.