Derivatives of parametric functions (or whatevs)

Click For Summary
SUMMARY

The discussion centers on calculating the derivative dy/dx for parametric functions x(t,s) and y(t,s) with two parameters. The user seeks clarification on how to express dy/dx in terms of the partial derivatives dx/dt, dx/ds, dy/dt, and dy/ds. The established formula for single-parameter functions, dy/dx = (dy/dt)/(dx/dt), does not directly apply, leading to confusion. The correct approach involves using the chain rule for multivariable calculus, specifically applying the formula dy/dx = (dy/dt * dt/dx + dy/ds * ds/dx) / (dx/dt * dt/dx + dx/ds * ds/dx).

PREREQUISITES
  • Understanding of parametric equations
  • Familiarity with partial derivatives
  • Knowledge of the chain rule in multivariable calculus
  • Basic concepts of contour lines in graphical representations
NEXT STEPS
  • Study the chain rule for multivariable functions
  • Learn about partial derivatives and their applications
  • Explore parametric equations in depth
  • Investigate contour lines and their significance in calculus
USEFUL FOR

Students and professionals in mathematics, particularly those studying calculus, multivariable functions, and parametric equations. This discussion is beneficial for anyone needing to understand derivatives in a multivariable context.

variety
Messages
20
Reaction score
0
Quick question. This is kind of embarrassing actually. Suppose I have functions x(t,s) and y(t,s) (say they're parametric equations of a surface of something) and I want to know what dy/dx is. Specifically, I have x and y in terms of the parameters, which are kind of complicated functions, and I want to know what that derivative is in terms of dx/dt, dx/ds, etc.

I know if they were functions of just 1 parameter, then it would just be dy/dx = (dy/dt)/(dx/dt), but with 2 I'm confused. Can you guys help me out?!?
 
Physics news on Phys.org
hi variety! :smile:
variety said:
Suppose I have functions x(t,s) and y(t,s) … and I want to know what that derivative is in terms of dx/dt, dx/ds, etc.

I know if they were functions of just 1 parameter, then it would just be dy/dx = (dy/dt)/(dx/dt), but with 2 I'm confused.

does it make any sense?

suppose x is height, and t and s are north and south …

then x is a hill, and there are contour lines along which x does not change, but y does change …

what could be the meaning of dy/dx ? :confused:​
 

Similar threads

Calculating 2nd Total Derivative of u w.r.t. t
  • · Replies 28 ·
Replies
28
Views
3K
Finding the second derivative of a given parametric equation
  • · Replies 20 ·
Replies
20
Views
2K
Derivative of a parametric equation
  • · Replies 6 ·
Replies
6
Views
2K
Evaluate the given integrals - line integrals
  • · Replies 2 ·
Replies
2
Views
1K
Relationship between autonomous system and related single equation
  • · Replies 3 ·
Replies
3
Views
2K
Finding the partial derivative from the given information
  • · Replies 2 ·
Replies
2
Views
2K
Exam asks me for a case where ▲x > dx?
  • · Replies 5 ·
Replies
5
Views
1K
Solve the problem involving space curve
  • · Replies 2 ·
Replies
2
Views
2K
Solve the given parametric equation
  • · Replies 3 ·
Replies
3
Views
2K
Solution of a parametric differential equation
  • · Replies 4 ·
Replies
4
Views
2K