Question

Mass In Exercises 11 and \(12,\) find the mass of the surface lamina \(S\) of density \(\rho .\) S: z=\sqrt{a^{2}-x^{2}-y^{2}}, \quad \rho(x, y, z)=k z

Short answer

The mass of the lamina is obtained by performing a definite surface integral of the density function. The integral needs to be evaluated using trigonometric identities and methods of definite integration after converting the coordinates to spherical.

Step-by-step solution

Convert the Coordinates

First, convert the coordinates from Cartesian to spherical. This is done via the transformations \(x = r \sin(\theta)\cos(\phi)\), \(y = r \sin(\theta)\sin(\phi)\), and \(z = r \cos(\theta)\). Plug x, y, z in the equation of the surface \(S\), \(z = \sqrt{a^{2} - x^{2} - y^{2}}\). Then the equation becomes \(r \cos(\theta) = \sqrt{a^{2} - r^{2}\sin^{2}(\theta)}\). Square both sides and simplify to obtain \(r^{2} = a^{2} / (1 + \sin^{2}(\theta))\), which simplifies to \(r = a / \sqrt{1 + \sin^{2}(\theta)}\) since \(r \geq 0\). This equation is valid for \(0 \leq \theta \leq \pi/2\) corresponding to the upper hemisphere.

Surface Integral Calculation

The mass of the lamina is found by performing a surface integral of the density function over the surface. The surface element in spherical coordinates is given by \(dS = r^2 \sin(\theta) d\theta d\phi\). Insert into the density function \(\rho(x, y, z) = k z = k r \cos(\theta)\), and the surface element to obtain the surface integral for the mass \(M = \int k r \cos(\theta) \cdot r^2 \sin(\theta) d\theta d\phi\). After substituting \(r = a / \sqrt{1 + \sin^{2}(\theta)}\), it is then necessary to perform the integration over the limits for \(\theta\) from \(0\) to \(\pi / 2\) and \(\phi\) from \(0\) to \(2 \pi\).

Perform Integration

Evaluate the integral for the mass \(M\). This integration will require trigonometric identities and techniques of definite integration. The final evaluation will result in the mass of the surface lamina.

Key concepts

Spherical Coordinates

Understanding spherical coordinates is crucial when describing positions in three-dimensional space, particularly in situations with spherical symmetry, such as those involving spheres or hemispheres. Picture a globe where every location is represented by a set of three values: the radius r, polar angle θ, and azimuthal angle φ.

The radius r measures how far from the origin the point is. The polar angle θ indicates the angle relative to the positive z-axis, ranging from 0 to π. Lastly, the azimuthal angle φ is akin to longitude on the Earth, sweeping around the equator from 0 to 2π.

To convert Cartesian coordinates \( (x, y, z) \) to spherical coordinates \( (r, θ, φ) \) we use the transformations:

• \( x = r \sin(θ)\cos(φ) \)
• \( y = r \sin(θ)\sin(φ) \)
• \( z = r \cos(θ) \)

By understanding these relationships, students can transition between different coordinate systems as required by various mathematical problems, such as calculating the mass of a lamina on a curved surface.

Density Function

In physics and mathematics, a density function describes how mass is distributed over a certain area or volume. For surface laminae, the mass distribution can vary from one point to another depending on certain conditions or characteristics of the material.

A simple yet common example of a density function is \( \rho(x, y, z) = kz \), where ρ is the density at any given point on the surface, and k is a constant that could, for instance, represent the mass per unit area. The variable z signifies that density varies with the surface's height over the x-y plane.

This density function indicates that the mass of the lamina is not spread uniformly. Instead, it increases with the z-coordinate, meaning the higher a point is on the surface, the more mass it has per unit area. When calculating the overall mass of complex shapes, implementing the correct density function within an integral is necessary to account for these variations.

Surface Lamina Mass

The mass of a surface lamina is an integral concept when examining objects with mass distributed over a two-dimensional area. Specifically, in the context of a curved surface, it's vital to consider the three-dimensional geometry while still focusing on the lamina's surface.

To compute the mass M of the surface lamina, one must integrate the density function over the surface area. However, in three-dimensional space, particularly with non-flat surfaces, the element of area dS is a piece of the surface's curved geometry. In spherical coordinates, this is expressed as \( dS = r^2 \sin(θ) dθ dφ \), which covers the 'piece' of the surface area corresponding to the range dθ and dφ.

By integrating the product of the density function and the surface area element - \( \rho(x, y, z) dS \) - across the entire surface, one obtains the lamina's mass. In our example, the integration would account for the varying density with height, leading to a more substantial mass contribution from regions at a higher z-coordinate. This mass calculation is fundamental in fields like materials science and mechanical engineering, where knowing the distribution of mass affects the design and analysis of structures.