Proving d<r>/dt = <dr/dt>: A Case Study

  • Context: Graduate 
  • Thread starter Thread starter Silva_physics
  • Start date Start date
  • Tags Tags
    Study
Click For Summary
SUMMARY

The discussion centers on proving the equation d/dt = using the running average formula (t)/dt = 1/2Pi * Integrate[r(t)*dt,{r,t-Pi,t+Pi}]. The user seeks assistance in demonstrating that the order of differentiation and time-averaging does not affect the outcome. The application of Leibniz's rule and the symmetry of the integration interval are suggested as potential methods for solving the problem.

PREREQUISITES
  • Understanding of calculus, specifically differentiation and integration.
  • Familiarity with Leibniz's rule for differentiation under the integral sign.
  • Knowledge of sinusoidal functions and their properties.
  • Basic concepts of running averages in mathematical analysis.
NEXT STEPS
  • Study the application of Leibniz's rule in calculus.
  • Research the properties of sinusoidal functions and their integrals.
  • Explore the concept of running averages in mathematical contexts.
  • Practice problems involving differentiation and integration of oscillatory functions.
USEFUL FOR

Students studying calculus, particularly those tackling problems involving differentiation and integration of oscillatory functions, as well as anyone preparing for advanced mathematics coursework.

Silva_physics
Messages
10
Reaction score
0
Let <r> (t)/dt= 1/2Pi * Integrate[r(t)*dt,{r,t-Pi,t+Pi}] denote the running average of r over one cycle of the sinosoidal oscillation.

I have to show that d<r>/dt = <dr/dt>, it does not matter whether we differentiate or time-average first.


Should I work with Leibniz rule? Can I use some symmetry of interval, I don't really know, something I tried to do but not successfully.

Can anybody, please, show solution?
 
Physics news on Phys.org
PLEASE, help me, the homework's deadline is approaching!
 

Similar threads

MHB Proving Equation (1): Let r(t) be a Vector in $\mathbb{R^3}$
  • · Replies 2 ·
Replies
2
Views
2K
MHB Are $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ Parallel When m+n=1?
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Derivatives in Action, Change in radius per time of circle.
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Can I apply L'Hopital's rule to this integral expression?
  • · Replies 7 ·
Replies
7
Views
4K
High School Having trouble deriving the volume of an elliptical ring toroid
  • · Replies 4 ·
Replies
4
Views
4K
Deriving Cross Vectors: Understanding d/dt[a * (v x r)] = a (v x r)
  • · Replies 16 ·
Replies
16
Views
3K
Undergrad Einstein brother-in-law elevator
  • · Replies 11 ·
Replies
11
Views
2K
Dynamical Systems: how to find equation for Poincare map?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Proving that this integral is divergent
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Frequency in Maxwell's equations
  • · Replies 19 ·
Replies
19
Views
3K