From 9d08c558b0d4be00dfd23142250f0056a40114eb Mon Sep 17 00:00:00 2001 From: Bryan Newbold Date: Tue, 20 Sep 2022 20:34:07 -0700 Subject: progress today (tuesday) --- log/2022-09-20.md | 36 ++++++++++++++++++++++++++++++++++++ 1 file changed, 36 insertions(+) create mode 100644 log/2022-09-20.md (limited to 'log') diff --git a/log/2022-09-20.md b/log/2022-09-20.md new file mode 100644 index 0000000..cede4f2 --- /dev/null +++ b/log/2022-09-20.md @@ -0,0 +1,36 @@ + +## Vector Fields + +Struggling a bit to understand what a "vector field" is. Is it a geometric +object? It is described as an operator, which is sort of confusing. Can it be +considered/visualized separately from any specific manifold function? Are +vector fields one-to-one with manifold functions? + +Suspect this because we are using functional terminology, and trying to define +"vector field" as something with a type signature, aka in terms of what it +"takes" and "returns". + +Remember, a manifold function maps (geometric) points on a manifold to real +numbers. + +Didn't understand the "covariantly"/"contravariantly" bit in the "Coordinate +Transformations" section. + +--- + +Ok, so "vector field on manifold" is a mapping from points on the manifold to +vectors. It is often combined with a manifold function to give the directional +derivative of the function at a point, which is a real number. In the language +of differential geometry, it is an operator that takes a real-valued manifold +function, and a point, and returns a real value. + +A "one-form field" is a geometric operator which works on a vector field to +define a mapping from points to real values. + +Maybe I was over-thinking this? + +TODO: look in Spivak to see definitions there + +--- + +Coordinate-basis one-form fields are "dual" to coordinate-basis vector fields. -- cgit v1.2.3