# In-frame time derivative proof

• yakattack
In summary, the conversation discusses how to prove that in non-inertial frames, the in-frame time derivative D obeys a specific equation involving a scalar function f and a vector a. The participants mention using the equations Df=df/dt and Da= the sum of the derivatives of the components of a times the relevant basis vector, but are struggling to apply them to prove the question. The non-inertial observer also introduces the concept of a time-dependent a vector, leading to the conclusion that D \vec{a} = a \cdot \frac{d \vec{e_t}}{dt}.
yakattack

For non-inertial frames show that the in-frame time derivative D obeys:
D(fa)=fDa+df/dta

Where f is a scalar function and a is a vector.

I know that Df=df/dt and that Da= the sum of the derivatives of the components of a times the relavant basis vector. But can't seem to apply this to prove the question.

Well for the non inertial observer, the a vector is a time dependent function. So actually you get $$\vec{a} = a \cdot \vec{e_t}$$. Hence the $$D \vec{a} = a \cdot \frac{d \vec{e_t}}{dt}$$.

marlon

Last edited:

## 1. What is an in-frame time derivative?

An in-frame time derivative is a mathematical concept that describes the rate of change of a variable within a specific reference frame. It takes into account the motion and orientation of the frame itself, rather than just the overall change of the variable.

## 2. Why is it important to include in-frame time derivatives in scientific calculations?

In-frame time derivatives are important because they provide a more accurate representation of the dynamics of a system. By considering the motion and orientation of a reference frame, we can better understand the behavior of variables and make more precise predictions.

## 3. How is the in-frame time derivative calculated?

The in-frame time derivative is calculated by taking the total derivative of a variable with respect to time and then subtracting the contribution from the motion and orientation of the reference frame. This can be done using mathematical techniques such as the chain rule and the product rule.

## 4. Can the in-frame time derivative be negative?

Yes, the in-frame time derivative can be negative. This indicates that the variable is decreasing in value within the given reference frame. However, it is important to note that the overall change of the variable may still be positive if the reference frame itself is also changing.

## 5. In what fields of science is the concept of in-frame time derivatives commonly used?

The concept of in-frame time derivatives is commonly used in fields such as physics, engineering, and astronomy. It is especially relevant in areas where the motion and orientation of reference frames play a significant role in the behavior of variables, such as in fluid dynamics and celestial mechanics.

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