Partial derivative chain rule proof

Click For Summary
SUMMARY

The discussion focuses on proving the equation d²u/dx² + d²u/dy² = e⁻²s[d²u/ds² + d²u/dt²] for the function u = f(x, y) where x = e^s cos(t) and y = e^s sin(t). Participants express confusion regarding the interpretation of the operator d/ds, clarifying that it is not simply the second partial derivative of s. The conversation highlights the need for a clearer understanding of the transformation from Cartesian to polar coordinates and the resulting terms in the equation.

PREREQUISITES
  • Understanding of partial derivatives and their notation
  • Familiarity with the chain rule in multivariable calculus
  • Knowledge of transformations between Cartesian and polar coordinates
  • Basic proficiency in calculus, specifically second derivatives
NEXT STEPS
  • Study the application of the chain rule in multivariable calculus
  • Learn about transformations between Cartesian and polar coordinates
  • Review the properties and applications of second partial derivatives
  • Explore examples of deriving equations involving multiple variables
USEFUL FOR

Students and educators in calculus, particularly those focusing on multivariable calculus and differential equations, as well as anyone seeking to deepen their understanding of the chain rule and partial derivatives.

platinumrice
Messages
5
Reaction score
0

Homework Statement



If u=f(x,y) where x=escost and y=essint

show that d2u/dx2+d2u/dy2 = e-2s[d2u/ds2+d2u/dt2

Homework Equations



http://s11.postimage.org/sjwt1wkvl/Untitled.jpg

The Attempt at a Solution



ok i don't understand how they got to that


i don't know what d/ds is supposed to do exactly, i thought it was the second partial derivative of s, but it seems that it's not if we check the answer.


i basically got e^s cost (d^2/dx^2) + e^s sint (d^2u/dy^2), but they got 6 terms. i am kinda lost, can you explain to me what d/ds is supposed to do?
 
Physics news on Phys.org
Maybe you should try typing the problem a little clearer. I think I could help you with this.
 

Similar threads

Why Does the Multidimensional Chain Rule Lead to Euler's Theorem Proof?
  • · Replies 2 ·
Replies
2
Views
2K
Lienard-Wiechert Potential derivation, chain rule
  • · Replies 1 ·
Replies
1
Views
2K
Finding the partial derivative from the given information
  • · Replies 2 ·
Replies
2
Views
2K
Help explaining the chain rule please
  • · Replies 12 ·
Replies
12
Views
3K
Understanding the Chain Rule in Multivariable Calculus
  • · Replies 9 ·
Replies
9
Views
3K
General relativity, geodesic, KVF, chain rule covariant derivatives
  • · Replies 2 ·
Replies
2
Views
3K
Using Chain Rule to Find Partial Derivatives of a Multivariable Function
  • · Replies 2 ·
Replies
2
Views
2K
Multivariable Chain Rule Question
  • · Replies 4 ·
Replies
4
Views
2K
Partial Derivatives Using Chain Rule
  • · Replies 7 ·
Replies
7
Views
2K
Differential Integration Problem
  • · Replies 4 ·
Replies
4
Views
2K