What exactly allows a differential relation form of an equation?

In summary, the conversation discusses the use and understanding of the differential relation in equations, specifically in the context of optics. The small-angle approximation is used to find the differential relation known as the plate scale, which relates the variables y and θ. There is a question about the rules behind switching from one form of the equation to the other, and if this can be applied to any two variables.
  • #1
rexregisanimi
43
6
What exactly allows a "differential relation" form of an equation?

I understand this in a superficial way but I'd really like some more clarification. If anybody can provide a little better understanding on this subject, please feel free to post anything at all. Even a sentence or two would be helpful.

I am reading through the optics section of my Stellar Astrophysics book and I came across the following sentences:

Using the small-angle approximation, tan(θ) ≈ θ, for θ expressed in radians, we find

y = fθ.​

This immediately leads to the differential relation known as the plate scale, dθ/dy,

[itex]\frac{dθ}{dy}[/itex] = [itex]\frac{1}{f}[/itex].​

What I don't completely understand is how and why one can simply go from the y=fθ form to the dθ/dy = 1/f form. I understand what the equation means but I don't understand the rules behind switching from one form to the other. Are there any or can I change any two variables to differential form to get a new relation? Any help or guidance at all would be appreciated.
 
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  • #2


For f not a function of y or theta,

dy= fdθ

Then its just symbol pushing.
 

1. What is a differential relation form of an equation?

A differential relation form of an equation is an equation that describes the relationship between two or more variables in terms of their rates of change. It is typically written using derivatives and is used to model dynamic systems in mathematics and science.

2. How is a differential relation form of an equation different from a regular equation?

A regular equation expresses a relationship between variables in terms of their values, while a differential relation form expresses the relationship in terms of their rates of change. This allows for a more dynamic and precise understanding of the system being modeled.

3. What are the advantages of using a differential relation form of an equation?

Using a differential relation form of an equation allows for a more accurate and detailed representation of dynamic systems, as it takes into account the rates of change of the variables involved. This is particularly useful in fields such as physics, chemistry, and engineering, where systems are constantly changing and evolving.

4. How do you solve a differential relation form of an equation?

The process of solving a differential relation form of an equation involves finding the function or functions that satisfy the equation. This is typically done by using mathematical techniques such as integration, substitution, and separation of variables. In some cases, numerical methods may also be used.

5. Can a differential relation form of an equation be used to predict future behavior?

Yes, a differential relation form of an equation can be used to make predictions about the future behavior of a system. By understanding the rates of change of the variables involved, it is possible to make informed predictions about how the system will evolve over time. However, these predictions are only as accurate as the initial conditions and assumptions used in the equation.

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