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jwsiii
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I'm trying to understand what one-forms are. The book I'm reading says a one-form is a linear map from a vector to a real number. It uses the gradient as an example but isn't the gradient a map from a function to a vector?
mathwonk said:sadly, the terms "covariant" and "contravariant" are extremely unfortunate, as history has crowned with the term "covariant" those objects which behave in a [categorically] contravariant fashion, and vice versa.
mathwonk said:my point here is that there is no difference of opinion whatsoever in the entire world of mathematics
Interesting to see, that if you have an orthonormal frame there really do exist a dual basis which is simply the projection on the ith coordinate.dextercioby said:That reciprocal frame is actually a basis in the cotangent space and is made up of linear functionals over the tangent space (made up of vectors).
You should write
[tex] e^{i}\left(e_{k}\right) =\delta^{i}_{k} [/tex]
Daniel.
A one-form is a mathematical object that maps a vector to a real number. It is a type of linear map that takes in a vector and outputs a single value.
A vector is a mathematical object that has both magnitude and direction, while a one-form only takes in a vector and outputs a single value. Additionally, vectors are represented as columns or rows, while one-forms are represented as rows of numbers.
One-forms are important in science because they allow us to describe and analyze physical systems using mathematical equations. They are particularly useful in fields such as physics, engineering, and economics.
No, one-forms cannot be visualized because they are not geometric objects like vectors. Instead, they are abstract mathematical objects that represent a linear map.
One-forms are used in calculations by taking in a vector and using its components to calculate a single value. This value can then be used in further calculations or analysis of a system.