This is a follow-on from a previous post on tangent spaces. It is part of a series of posts that I write on the basics of differential or Riemannian geometry, providing the necessary background for reading some of the more advanced textbooks on general relativity. I will introduce the derivative of a smooth map between manifolds and work a little bit with the definition(s) to become familiar with the thicket of mathematical notation. Then I narrow in on the special case where \(f\) is a real-valued function, introduce the notions of dual space and covectors, and finally show the connection to the differential known from basic calculus.
[Read More]Tangent Spaces
This is a follow-on from a previous post on differentiable manifolds. It is part of a series of posts that I write on the basics of differential or Riemannian geometry, providing the necessary background for reading some of the more advanced textbooks on general relativity. The following posts will cover tensors, covariant derivatives and parallel transport. The presentation will be informal, focusing mostly on motivation, definitions and key facts, not on mathematical proofs.
[Read More]Differentiable Manifolds
This is the first post in short series that I write on the basics of differential or Riemannian geometry. Differential geometry is a key mathematical foundation of general relativity, and the terminology presented here is a prerequisite for reading some of the more advanced textbooks, such as the ones by Carroll, MTW, Wald, Straumann or Hawking/Ellis. Here I start with differentiable manifolds. The following posts will cover tangent spaces, tensors, covariant derivatives and parallel transport. The presentation will be informal, focusing mostly on motivation, definitions and key facts, not on mathematical proofs.
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