Using chain rule to derive the path equation

In summary, the conversation discussed finding the critical points of a system with equations r' = 4r - rf and f' = -3f + rf and deriving the linearised system about each critical point. The question then moved on to using the chain rule to derive the path equation of the trajectories in the phase plane. The person was unsure of how to proceed with this and asked for ideas. A helpful hint was given to use the linearised system from part (b) to solve for the path equation.
  • #1
helpinghand
39
0
r' = 4r - rf
f' = -3f + rf

In this question, there was three parts:
a) find all the critical points of this system.
b) Derive the linearised system about each critcal point...
c) Use the chain rule to derive the path equation of the trajectories in the phase plane.

I managed to get a and b out.

For c:

dr/df = dr/dt . dt/df

=> dr/df = (4r - rf)/(-3f - rf) ~~ not sure what to do from here.

I'm not sure how to get the path equation... Anyone have ideas?

Cheers
 
Physics news on Phys.org
  • #2
hi helpinghand! :smile:
helpinghand said:
r' = 4r - rf
f' = -3f + rf
…
a) find all the critical points of this system.
b) Derive the linearised system about each critcal point...
c) Use the chain rule to derive the path equation of the trajectories in the phase plane.

hint: for (c), use (b) :wink:
 

Related to Using chain rule to derive the path equation

What is the chain rule?

The chain rule is a mathematical rule used in calculus to find the derivative of a composite function. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

How is the chain rule used to derive the path equation?

The chain rule is used to derive the path equation by first expressing the path as a composite function of two other functions, such as position and time. The derivative of the path is then found by applying the chain rule to the composite function.

Why is the chain rule important in physics?

The chain rule is important in physics because it allows us to calculate the rate of change of a quantity that is dependent on multiple variables. This is crucial in understanding the behavior of physical systems, such as the motion of objects.

Can the chain rule be used for any type of function?

Yes, the chain rule can be used for any type of function, as long as it is a composite function. This means that the function is made up of two or more other functions, where the output of one function becomes the input of the next function.

Are there any common mistakes when using the chain rule to derive the path equation?

Yes, some common mistakes when using the chain rule to derive the path equation include forgetting to apply the chain rule to both parts of the composite function, using incorrect notation, and misidentifying the inner and outer functions. It is important to double check the application of the chain rule to avoid these mistakes.

Similar threads

MHB Applying the Chain Rule to Derive Solutions of the Heat Equation
    • Differential Equations
Replies
12
Views
3K
I The Road to Reality - exercise on scalar product
    • Differential Geometry
Replies
2
Views
682
Finding dE/dx: Using the Chain Rule
    • Differential Equations
Replies
1
Views
2K
I Second derivative, chain rules and order of operations
    • Calculus
Replies
6
Views
2K
Lienard-Wiechert Potential derivation, chain rule
    • Calculus and Beyond Homework Help
Replies
1
Views
657
Solving dy/dt = 0: Chain Rule & y(t)
    • Differential Equations
Replies
8
Views
2K
I Derivative using the chain rule
    • Calculus
Replies
5
Views
1K
I Can the Chain Rule be Applied to Simplify Divergence in Entropy Equation?
    • Calculus
Replies
1
Views
2K
I Exploring the Derivative of y(x,t) in Quantum Mechanics
    • Calculus
Replies
1
Views
928
A Crank-Nicholson method for spherical diffusion
    • Differential Equations
Replies
2
Views
2K

Hot Threads

Recent Insights

Back
Top