From Tensor Analysis
5.4 Elements of the Theory of Surfaces
The reader is familiar with many standard surfaces: the sphere, the cone, the cylinder, etc. These are easy to visualize, and in Cartesian coordinates are described by simple equations. For example, the equation of a sphere reflects the definition of a sphere: all points (x, y, z) of a sphere have the same distance R from the center ( x 0, y 0, z 0):
| (5.11) |  |
An infinite circular cylinder with generator parallel to the z-axis is given by the equation
in which the variable z does not appear. We can obtain other types of cylindrical surfaces by considering a set of spatial points satisfying the equation
This equation does not depend on z, and thus drawing a generator parallel to the z-axis through the point ( x, y, 0) we get a more general cylindrical surface. A paraboloid is an example of a more complex surface:
| (5.12) |  |
where a, b, c, d, e, f are numerical coefficients, Depending on the values of a, b, c, the paraboloid can be elliptic, hyperbolic or parabolic. The reader may wish to take the time to review these terms from analytic geometry, since we shall use them to characterize the shape of a surface at a point. The areas and volumes associated with all of these standard surfaces (or portions thereof) can be calculated by integration or, sometimes, by the use of elementary formulas.
Products & Services
Hitch pins and their variations are simple forms of hardware used to temporarily mount or conjoin mating components. Linch pins are specifically designed to retain wheels or other rotating devices on their axles, but can be used as a fastener as well. Both of these types of pins require mating holes and some form of a lock to be effective.
Topics of Interest
5.5 The Second Fundamental Form of a Surface Using the Taylor expansion we can find the change of r( u 1, u 2) to a higher degree of approximation: Here we apply the o-notation to vector...
Rivets and pins can be used as inexpensive alternatives to threaded fasteners in automated assembly. Pins are used where ease of disassembly is desired, while disassembly usually destroys rivets.
The taperlock gage is so-called because of the tapered shank, which locks into the tapered hole in the end of the handle. It is the time honored standard of the American Gage Design Committee, for...
Pins are typically used under primarily shear loading. They are separated into two groups: semipermanent and quick release. These general design rules apply to all types of semipermanent pins: Removal...
Designing vehicles to accommodate their complex electrical wiring systems and hardware requires careful planning and engineering development. Interconnections within distribution boxes and...
