Besides mathematical analysis, the upper and lower envelopes have other applications. For example, in the engineering field, the upper and lower envelope lines can be used to describe the strength and durability of materials. In the field of physics, the upper and lower envelopes can be used to study the characteristics of wave phenomena such as light waves and sound waves. In the field of data analysis, upper and lower envelopes can be used to diagnose the shape and abnormal situation of data distribution. Therefore, the upper and lower envelopes are widely used in different fields.

Generally speaking, the upper and lower envelope is a very useful mathematical tool, which not only plays an important role in mathematical analysis, but also can be applied to different fields such as engineering, physics and data analysis. Although this concept seems abstract, as long as we master its basic principles and application methods, we can play a great role in solving various practical problems. Therefore, it is very necessary for students and engineers in the field of mathematics to master the concept of upper and lower envelopes.