Trying to get my head around some basic exterior calculus concepts

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SUMMARY

This discussion centers on the fundamental concepts of exterior calculus, specifically the relationship between covectors, gradients, and their applications in vector calculus. It establishes that the gradient is indeed a covector or 1-form that, when combined with a vector, yields a scalar, exemplified by the divergence operation. Additionally, it confirms that curl can be expressed as a wedge product, highlighting the correspondence between curl and the dual 1-form of the wedge product 2-form.

PREREQUISITES
  • Understanding of 1-forms and covectors in differential geometry
  • Familiarity with vector calculus operations such as gradient, divergence, and curl
  • Knowledge of wedge products and their role in exterior algebra
  • Basic concepts of dual spaces and dual forms
NEXT STEPS
  • Study the properties and applications of 1-forms in differential geometry
  • Explore the relationship between curl and wedge products in more depth
  • Learn about the dual space and its significance in exterior calculus
  • Investigate the implications of pseudovectors in vector calculus
USEFUL FOR

Mathematicians, physicists, and students of advanced calculus who are looking to deepen their understanding of exterior calculus and its applications in vector analysis.

BWV
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Trying to get my head around some basic exterior calculus concepts


So #1 - 1-form or covector the grad is a covector / 1-form defined as something that combined with a vector takes it to a scalar, for example divergence is a scalar which is the dot-product of a function with its grad (e.g. a 1-form or covector) - correct?


#2 - is there an correspondance between curl and a wedge product? can curl be written as a wedge product?
 
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Hi BWV! :smile:
BWV said:
#1 - 1-form or covector the grad is a covector / 1-form defined as something that combined with a vector takes it to a scalar, for example divergence is a scalar which is the dot-product of a function with its grad (e.g. a 1-form or covector) - correct?

#2 - is there an correspondance between curl and a wedge product? can curl be written as a wedge product?

#1 - yes! :smile:

#2 - a cross-product of two three-dimensional vectors is a pseudovector, not a vector, and it's a dual 1-form, the dual of the wedge product 2-form …

same with curl (so it can be written as a wedge product starred) :wink:
 

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