Skip to main content
Mathematics LibreTexts

2: Vector Fields

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    • 2.1: Definitions and First Examples
      In the last chapter, we studied vector valued functions of a single variable, like, for example, the velocity of a particle at time t. Suppose however that we are interested in a fluid. There is a, possibly different, velocity at each point in the fluid. So the velocity of a fluid is really a vector valued function of several variables. Such a function is called a vector field.
    • 2.2: Optional — Field Lines
      Suppose that we drop a tiny stick into a river with the velocity field of the flowing water being \(\vecs{v} (x,y)\text{.}\) We are assuming, for simplicity, that the velocity field does not depend on time \(t\text{.}\) The stick will move along with the water. When the stick is at \(\vecs{r} \text{,}\) its velocity will be the same as the velocity of the water at \(\vecs{r} \text{,}\) which is \(\vecs{v} (\vecs{r} )\text{.}\) Thus if the stick is at \(\vecs{r} (t)\) at time \(t\text{,}\) we wil
    • 2.3: Conservative Vector Fields
      Not all vector fields are created equal. In particular, some vector fields are easier to work with than others. One important class of vector fields that are relatively easy to work with, at least sometimes, but that still arise in many applications are “conservative vector fields”.
    • 2.4: Line Integrals
      We have already seen one type of integral along curves. We are now going to see a second, that turns out to have significant connections to conservative vector fields. It arose from the concept of “work” in classical mechanics.
    • 2.5: Optional — The Pendulum
      Model a pendulum by a mass \(m\) that is connected to a hinge by an idealized rod that is massless and of fixed length \(\ell\text{.}\) Denote by \(\theta\) the angle between the rod and vertical.

    Thumbnail: A unit sphere with surface vectors ( CC BY-SA 3.0 Unported; Cronholm144 via Wikipedia)

    This page titled 2: Vector Fields is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Joel Feldman, Andrew Rechnitzer and Elyse Yeager via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.