Vector calculus generally means multivariate calculus.
A function f(x,y) = z denotes a surface in 3-dimensional space.
IF we FIX z to a specific value, then note that we just have two variables now! Then, we are looking at a 2D-CURVE. This is equivalent to “looking at the surface from the top”, or fixing the height to a specific value, and seeing which points on the surface correspond to that height. Or you can view it as taking slices of the surface, for each setting of the height that you are interested in.
This is why these level curves are also called height curves, contours, etc.
Relationship to convexity: If the multivariable function in non-convex, then we will have non-overlapping, non-encapsulating sets of level curves. That is, if we graph a few values of the level curves, then, invariably, we will find that one “cluster” exists for a given value k. Yet this closed level curve will also appear again elsewhere.
Takeaway: A function with domain of n variables (and codomain in the single, final variable) represents a surface in n+1 dimensional space.
In more math-y terms: A function of n variables represents a manifold in n+1 dimensional space. This is equivalently called a hypersurface.
For a real-valued function defined over a space with ambient dimension n+1, its manifold or hypersurface is the collection of all points satisfying that function.
A real valued function in an ambient space of dimension n+1 defines a hypersurface or manifold in n dimensions.
Some takeaways: manifold == curve/surface (there are some caveats here; a manifold is a very precise type of mathematical object; strictly more restrictive than just a surface) ambient space == what context, or dimensional space we are studying the object in Ambient space in particular concretizes a powerful paradigm: we have a mathematical concept, yet it depends on what we are studying wrt In particular: when we have expectation of a random variable, it depends on the underlying (joint) distribution and probability space it is defined over. The RV may be embedded in p_x, or it may be embedded in p_x,p_y,… (infinite dimensional ambient space)
We can have a line. but is the line in 2D space or 3D space? It matters!
(Image not working due to subsite structure of the blog…)
https://johntiger1.github.io/blog/static/vector_calc/ambient_space.PNG
Ok, so the directory structure is as follows: \blog\quickstart\static\static\vector_calc. We need the extra static, since everything is copied over to the root directory.
So when we had \blog\quickstart\static\vector_calc_dead_end, the folder vector_calc_dead_end was being copied over to the ROOT of the github site!