Trying to get my head around some basic exterior calculus concepts

In summary, exterior calculus involves concepts such as 1-forms, covectors, gradients, divergence, and wedge products. The grad is a covector that takes a vector to a scalar, while the curl is a dual 1-form that can be written as a wedge product. This allows for a correspondance between the two concepts.
  • #1
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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  • #2
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:
 

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

1. What is exterior calculus?

Exterior calculus is a branch of mathematics that deals with the geometric properties of vector fields, differential forms, and multilinear algebra. It is used to study and understand the behavior of objects in the physical world, such as surfaces, curves, and volumes.

2. What are some basic concepts in exterior calculus?

Some basic concepts in exterior calculus include vector fields, differential forms, the exterior derivative, and integration of differential forms. These concepts are used to describe and analyze the behavior of objects in space.

3. How is exterior calculus different from traditional calculus?

In traditional calculus, we study functions of one or more variables and their derivatives. In exterior calculus, we study differential forms and their derivatives, which are more general and can be applied to objects in any dimension. Additionally, exterior calculus allows for the use of multivariable calculus in a more elegant and concise manner.

4. Why is exterior calculus important in science and engineering?

Exterior calculus is important in science and engineering because it provides a powerful mathematical framework for describing and understanding the physical world. It allows us to analyze and model complex systems, such as fluid dynamics, electromagnetism, and general relativity.

5. How can I improve my understanding of exterior calculus concepts?

To improve your understanding of exterior calculus, it is important to have a solid foundation in multivariable calculus and linear algebra. It can also be helpful to practice solving problems and working through proofs to gain a deeper understanding of the concepts. Additionally, seeking help from a tutor or professor can also be beneficial.

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