Ideas on how to easily solve this long derivation?

  • Context: Graduate 
  • Thread starter Thread starter Kaasen
  • Start date Start date
  • Tags Tags
    Derivation Ideas
Click For Summary
SUMMARY

The discussion focuses on solving the Laplacian operator ∇²E for the function E = exp(ikr)/r, where r = √(x² + y² + z²). Participants suggest utilizing spherical coordinates to simplify the computation of the second derivatives, as the function depends solely on r. This approach reduces the complexity of the calculations involved in deriving the Laplacian, making the process more efficient.

PREREQUISITES
  • Understanding of vector calculus, specifically the Laplacian operator.
  • Familiarity with spherical coordinates and their application in multivariable calculus.
  • Knowledge of exponential functions and their derivatives.
  • Basic proficiency in solving partial differential equations.
NEXT STEPS
  • Research the application of spherical coordinates in vector calculus.
  • Study the properties and applications of the Laplacian operator in physics.
  • Learn techniques for simplifying derivatives of functions in multivariable calculus.
  • Explore computational tools for symbolic differentiation, such as Mathematica or MATLAB.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are tackling problems involving partial differential equations and vector calculus.

Kaasen
Messages
2
Reaction score
0
Mod note: Moved from a Homework section
1. Homework Statement
I want to solve:
∇2E, for E = exp(ikr)/r, where r = √x2+y2+z2

Thus I must solve:
d2E/dx2+d2E/dy2+d2E/dy2

The derivatives themselves are simple, but very time-consuming! Can you think of any tricks I can use to simplify the calculation?
 
Last edited by a moderator:
Physics news on Phys.org
Kaasen said:

Homework Statement


I want to solve:
∇2E, for E = exp(ikr)/r, where r = √x2+y2+z2

Thus I must solve:
d2E/dx2+d2E/dy2+d2E/dy2

The derivatives themselves are simple, but very time-consuming! Can you think of any tricks I can use to simplify the calculation?
Since your function only depends on r, you could try a coordinate system that treats r nicely.
 

Similar threads

Graduate Help needed with derivation: solving a complex double integral
  • · Replies 1 ·
Replies
1
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Understanding how to derive the stress-energy tensor formula
  • · Replies 9 ·
Replies
9
Views
7K
Undergrad Atmospheric Density vs Altitude in an O'Neil Cylinder
  • · Replies 1 ·
Replies
1
Views
3K
How Long Does It Take for a High-Resistance RLC Circuit to Lose Half Its Energy?
  • · Replies 4 ·
Replies
4
Views
3K
Solving for Boyle Temperature in MATLAB
  • · Replies 2 ·
Replies
2
Views
2K
Derive Planck Length in terms of c, G, and h_bar
  • · Replies 3 ·
Replies
3
Views
8K
Graduate Interior volume of a black hole
  • · Replies 23 ·
Replies
23
Views
3K
MHB Contour Integration: Solving $\int_{-\infty}^{\infty}\frac{x - 1}{x^5 - 1}dx$
  • · Replies 2 ·
Replies
2
Views
3K
Graduate External Ballistics: Deriving Equations for Long-Range Shooting
  • · Replies 1 ·
Replies
1
Views
4K