# Dx as change in distance vs dx as infinitesimal x?

• jaredvert
In summary, the notation dx can represent both a change in distance and an infinitesimal x, and they are considered the same thing in the context of Newton's Law and Calculus. However, to avoid confusion, an actual change in distance is often denoted by δx or Δx.
jaredvert
dx as change in distance vs dx as infinitesimal x?

Why are they the same notation?Sent from my iPhone using Physics Forums

Because they're the same thing. What meaning exactly would $dx$ have otherwise?

If at time $t$ you have some position $x(t)$ then Newton's Law tells you how a small change in position is related to the velocity of the particle - namely $x + dx = x + v(t)dt$, where $v(t)$ satisfies the equation $m\frac{dv}{dt} = F$.

An actual change in distance is often denoted by δx or Δx, to distinguish it from the infinitesimal dx, which is part of Calculus. dy/dx really means the limit of δt/δx as δx approaches zero. In Science, we are often too sloppy about these notations as there may be pitfalls when you don't stick to the 'rules' precisely.

## 1. What is the difference between dx as change in distance and dx as infinitesimal x?

The difference lies in the interpretation of the notation. When dx is used as a change in distance, it represents a finite change in the distance between two points. On the other hand, when dx is used as infinitesimal x, it represents an infinitely small change in the distance.

## 2. Which one is more accurate in representing a change in distance?

Both representations have their own advantages and limitations. When dealing with real-world scenarios, dx as change in distance is more accurate as it takes into account the finite nature of distance. However, in mathematical and theoretical applications, dx as infinitesimal x is more accurate as it allows for easier calculations and simplification of equations.

## 3. How are these two representations related?

Dx as change in distance and dx as infinitesimal x are related through the concept of limits in calculus. As the change in distance (dx) becomes infinitely small, it approaches the value of infinitesimal x. Therefore, in the limit, the two representations are equivalent.

## 4. Can you provide an example of using dx as change in distance vs dx as infinitesimal x?

Sure, let's consider the distance travelled by a car. If we want to calculate the total distance travelled by the car, we would use dx as change in distance. However, if we want to calculate the instantaneous speed of the car at a specific point, we would use dx as infinitesimal x in the velocity equation.

## 5. Why is it important to understand the difference between dx as change in distance and dx as infinitesimal x?

Understanding the difference between these two representations is crucial in accurately interpreting and solving mathematical and scientific problems. It also helps in understanding the concept of limits and how they are used in calculus. Additionally, it allows for a better understanding of the relationship between the real world and mathematical models.

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