Help with Beginner Index Notation

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SUMMARY

The discussion focuses on the application of index notation in vector calculus, specifically addressing the identity curl(fF) = fcurl(F) + (∇f) x F, where f represents a scalar field and F represents a vector field. The user seeks assistance in manipulating the expression using index notation and the curl operator. Key insights include the use of the Levi-Civita symbol (ε) and the chain rule to simplify the expression. Participants emphasize the importance of proper notation, including the distinction between superscripts and subscripts.

PREREQUISITES
  • Understanding of vector calculus concepts, particularly the curl operator.
  • Familiarity with index notation and tensor calculus.
  • Knowledge of scalar and vector fields in physics and mathematics.
  • Basic proficiency in using LaTeX for mathematical expressions.
NEXT STEPS
  • Study the properties of the curl operator in vector calculus.
  • Learn about the Levi-Civita symbol and its applications in tensor calculus.
  • Explore the chain rule in the context of vector fields and scalar fields.
  • Practice converting between index notation and standard vector notation.
USEFUL FOR

Students and professionals in mathematics, physics, and engineering who are learning vector calculus and index notation, particularly those working with vector fields and scalar fields.

fttteotd
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Okay, so I'm learning some basic index notation, and I have a few questions...

Homework Statement


f= scalar field
F = vector field

so, we are supposed to show that curl(fF) = fcurl(F) + ([tex]\nabla[/tex]f) x F

The Attempt at a Solution



curl(fF) = [[tex]\nabla[/tex] x (fF))][tex]_{k}[/tex] = [tex]\in[/tex][tex]_{ijk}[/tex](f[tex]\partial[/tex][tex]_{i}[/tex]F[tex]_{j}[/tex] + F[tex]_{j}[/tex][tex]\partial[/tex][tex]_{i}[/tex]f)

any help??

(all the superscripts are supposed to be subscripts, i dunno)
 
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Welcome to PF!

Hi fttteotd! Welcome to PF! :smile:

(have a curly d: ∂ and a nabla: ∇ and an epsilon: ε and use itex rather than tex in the middle of a line :wink:)

[∇ x (fF)]i = εijk∂j(fFk) …

and now use the chain rule! :wink:
 

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