Differential forms are mathematical objects used to study the properties of smooth functions on manifolds. They are important because they provide a powerful tool for understanding and manipulating multivariate functions, as well as for solving problems in differential geometry and topology.
Writing a differential form in terms of local coordinates means expressing the form as a linear combination of basis forms in a particular coordinate system. This allows us to work with the form in a more concrete and explicit way, making calculations and computations easier.
To convert a differential form from one coordinate system to another, we can use the Jacobian matrix, which relates the coordinates of one system to the coordinates of another. By applying this matrix to the coefficients of the form, we can transform it to the new coordinate system.
Yes, differential forms can be written in terms of non-local coordinates, such as spherical or cylindrical coordinates. However, the form may become more complicated and difficult to work with in these coordinate systems, so using local coordinates is often preferred.
Yes, differential forms have many applications in various fields, including physics, engineering, and computer science. They are used to solve problems involving motion, electromagnetism, fluid dynamics, and more. They are also essential in the study of differential equations and partial differential equations.