fleeting notes

dr trefor bazett: overview of vector calculus

• vector field: 2 dimensional, 3 dimensional
• what is a vector space?
  • at any point in space we can put a vector
  • example: gravity: the force of gravity working on a particle can be represented as a vector field
  • example: water running through a river
  • vector fields everywhere in physics, math, engineering, etc
  • vector field as curves?
    • suppose that we have a disk sitting in 3d space on the x-y plane. this can be represented as a function
      • however, we can also have another function that lives “above” our function
      • bounded vs unbounded curve
      • surface intervals:
        • example: suppose we
      • relationship between curves and surfaces and vector fields
        • what kinds of questions can we ask?
          • flux: to what degree is the vecotr field dispersing across the boundary?
          • to what degree is the vecotr field aligned in relation to a norm of my boundary?
          • flow: tendency for the vector field to be aligned with the tangents of the curve rather than the pushing out of the boundary?
    • big question: how do local properties relate to global properties?
      • how do the parts relate to the global properties?

week one notes, w2026

• what is the classical derviative?
  • definition, etc
  • implies: f(x) is well approximated by a linear function at x=a (namely, the tangent)
  • taylors theorem:
    • a function is well approximated by a function $P_n(x)$
    • $P_n(x)=a_0+a_1(x-a)+a_2(x-a{)}^2+....+a_n(x-a{)}^n$
    • we can pick our coefficients such that the ‘kth’ derivative at the point a of our polynomial, we want this t be equal to the kth derivative of our function at the point a.
      • why is this relevant?
        • if we’ve already assumed that $P_n$ well approximates $f(x)$.
• basic object of study in this course will be a function that takesa subset from $R^n$, defined as $U$, to $R^m$.
  • what are the valid function operations? because if we have that it is equipped with vector addition and scalar multiplication, it is a vector space
  • where there are n input variables and m output variables
• classical case: m=n=1
  • “traditional” calculus you learn in high school: one input, one output
  • “multivariate scalar functions”
    • multiple inputs, one single output
  • “parametric curves”
    • the curve defined by variables are separate functions of an independent variable, such as time, which is also known as a parameter
  • “vector fields”
    • vector fields versus vector spaces: vector spaces are more generalized and are entire algebraic areas where vectors can be added or scaled, versus a vector field which follows the rules that fields generally follow
      • vector fields assign a vector to every point within a space
      • function built on top of a space
    • vector space is an underlying algebraic set/framework; a vector field is a function or assignment
      • if an analogy were to be made, a vector space is a branch of ethics, such as utlitarianism, and a vector space would be the specific instructions a person creates for how to perform utilitarianism
  • “affine maps” (generalization of a line)
    • linear maps need to send the origin to (0,0), affine maps relax this rule by adding a shift or translation, meaning the origin does not need to stay fixed
      • included operations: rotation, scaling, shearing, reflection, translation
      • preserved properties: straight lines stay straight, and parallel lines remain parallel
        • ratios of distances also remain constant
      • unpreserved properties: angles and absolute lengths do not always stay the same

the topology of $R^n$

• aka properties of $R^n$ that stay the same when the space is bent, twisted, or stretched
  • open ball, closed ball, sphere
  • r: the radius around any given point in space
• in the classical deriavtive, we require the point at which we are differentiating to reside in an open interval within the domain
  • $f(x)$ needs to be defined in a small open interval containing $a$.
  • interior definition
    • basically, if there exists a radius r that is greater than 0 such that the open ball with radius r centered around $a$ is a subset of $U$, which is a subset of $R^n$, then we say that $\vec{a}$ is interior to U.
      • we say that U is open if U = to its interior(i.e. every point in U is interior)