Skip to main content

Calculus Small Changes, Big Ideas

Bill Shillito

Section 17.1 Vector fields

Introduction goes here.
🔗
  • This is another way to visualize a multivariable function of multiple variables — vector input gives vector output.
    🔗
    🔗
  • 2D vector fields:
    🔗
    \begin{align*} \bv{F}(x,y) \amp = (P(x,y),Q(x,y))\\ \amp = P(x,y)\,\uv{x}+Q(x,y)\,\uv{y} \end{align*}
    Good examples are \((x,y)\) and \((-y,x)\)
    🔗
    🔗
  • 3D vector fields:
    🔗
    \begin{align*} \bv{F}(x,y,z) \amp = (P(x,y,z),Q(x,y,z),R(x,y,z))\\ \amp = P(x,y,z)\,\uv{x}+Q(x,y,z)\,\uv{y}+R(x,y,z)\,\uv{z} \end{align*}
    🔗
  • Inverse square fields
    🔗
    🔗
  • Interpreting as a force field vs a velocity field
    🔗
    🔗
  • The Pólya vector field of a complex function \(f(z)=u(x,y)+\ii v(x,y)\) is given by:
    \begin{equation*} \bv{F}(x,y) = u(x,y)\,\uv{x} - v(x,y)\,\uv{y}\text{.} \end{equation*}
    Note the minus sign in the second component. We’ll see that minus sign throughout, and it’ll become clear why it’s useful.
    🔗
    🔗
🔗
PrevTopNext