# Physics of Differentiable Manifolds: Part I

Often when trying to study general relativity, the most difficult aspect to understand rigorously is the underlying mathematics. Before anyone can do general relativity, the concepts of differentiable manifolds, smooth curves, vectors & co-vectors, tensors, tangent spaces and cotangent spaces, and many more have to be fully fleshed out. There are excellent texts out there that help achieve this goal, such as Sean Carroll’s Spacetime and Geometry. Here, however, we’ll provide a list of definitions with a few nuggets of information towards the end as a useful summary of the topics discussed in this post. Eventually, we will motivate how quantum field theory and general relativity can both be generated with concepts starting with a differentiable manifold with a metric.

# Play With Neural Networks: Tensor Flow

Learn about how Neural Networks evolve and how to control them at TensorFlow. This blog has a through analysis of how neural networks work as well.

# Dimensions of Various Lie Groups

How do you determine the dimensions of a Lie group? Why is the dimension of $O(n)$ the same as $SO(n)$? Similarly for $\text{GL}(n,F)$ and $\text{GL}^+(n,F)$, where $F$ is a field?