Understanding Multivariable Taylor Expansions with Vector Components

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SUMMARY

The discussion focuses on the application of multivariable Taylor expansions involving vector components, specifically in the context of gravitational equations. The user attempts to expand the expression involving terms like GMR/R^3 and GMr/R^3, while also incorporating the dot product of vectors. The key takeaway is the importance of correctly applying the chain rule and understanding the expansion of the denominator, which is expressed as Re^3(1 + 2Re·r/Re^2 + o(r^2))^(3/2). This highlights the complexity of vector calculus in Taylor expansions.

PREREQUISITES
  • Understanding of multivariable calculus
  • Familiarity with vector operations, including dot products
  • Knowledge of Taylor series expansions
  • Basic principles of gravitational equations in physics
NEXT STEPS
  • Study the derivation of Taylor series for multivariable functions
  • Learn about vector calculus and its applications in physics
  • Explore the chain rule in the context of multivariable functions
  • Investigate gravitational equations and their mathematical representations
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are dealing with multivariable calculus and vector analysis, particularly in the context of gravitational modeling and expansions.

roeb
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Homework Statement


I'm having a hard time following a taylor expansion that contains vectors...

http://img9.imageshack.us/img9/9656/blahz.png
http://g.imageshack.us/img9/blahz.png/1/

Homework Equations





The Attempt at a Solution



Here's how I would expand it:

-GMR/R^3 - GMr/R^3 + 3GMR/R^5*(R + r) + 3GMR/R^5(r + R)
So you take d/dR(R^2 + r^2 + 2r dot R)^5/2 and then take d/dr ?
I am basing my knowledge of multidimensional expansions off of what wikipedia is telling me, but I can't quite see how that r dot R term comes about

Anyone have any ideas?
 
Last edited by a moderator:
Physics news on Phys.org
Hi roeb! :smile:

Hint: the denominator is Re3(1 + 2Re.r/Re2 + o(r2))3/2 :wink:
 

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