Unit 11 : Vector Fields in 2D and 3D

The Concept

A vector field is an assignment of a vector to each point in a subset of space. In other words, if we are given a vector [latex]\langle x,y\rangle[/latex], then the vector is simply the mapping in 2D of each point.

Vector fields can be written in two equivalent notations shown below for both 2D and 3D:

  • 2D Notation: [latex]\vec{F}(x, y) = P(x, y)\vec{i} + Q(x, y)\vec{j} = \langle P(x, y) , Q(x, y)\rangle[/latex]
  • 3D Notation: [latex]\vec{F}(x,y,z) =  P(x, y,z)\vec{i}+ Q(x,y,z)\vec{j} + R(x,y,z)\vec{k} = \langle P(x, y,z) , Q(x, y,z) , R(x,y,z)\rangle[/latex]

Where [latex]\vec{i} = \langle 1 , 0 \rangle[/latex] and [latex]\vec{j} = \langle 0,1\rangle[/latex] represent unit vectors in 2D, and [latex]\vec{i} = \langle1,0,0\rangle ,\, \vec{j} = \langle 0,1,0 \rangle[/latex] and [latex]\vec{k} = \langle 0,0,1\rangle[/latex] are unit vectors in 3D. A real life example that can be modeled as a vector field would be a fluid dynamics problem such as a river, where the velocity of the liquid is a vector at any given point. The magnitude (i.e., amplitude) of the vector represents the speed and the direction represents the direction of the flow at any given point.

The Plot

Now, you should engage with the 2D and 3D plots below to understand 2D and 3D vector fields[1]. Follow the steps below to apply changes to the plot and observe the effects:

  1. The vector definition is done using [latex]P[/latex] and [latex]Q[/latex].
  2. Y Grid and X Grid control the number of arrows that will appear in the 2D plot.
  3. Xmin and Ymin set the minimum boundaries for the plot.
  4. YMax and Xmax set the maximum boundaries for the plot.

2D Vector Field Plot

3D Vector Field Plot

Self-Checking Questions

Check your understanding by solving the following questions[2]:

  1. Draw the following vector field [latex]\vec{F}(x,y)=x\vec{i} + y\vec{j}[/latex]
  2. Draw the following vector field [latex]\vec{F}(x, y, z) = 2x\vec{i} − 2y\vec{j} − 2z\vec{k}[/latex]

Use the graphs to find the answers to these questions.

  1. Made with GeoGebra, licensed Creative Commons CC BY-NC-SA 4.0.
  2. Gilbert Strang, Edwin “Jed” Herman,  OpenStax, Calculus Volume 3,  Calculus Volumes 1, 2, and 3 are licensed under an Attribution-NonCommercial-Sharealike 4.0 International License (CC BY-NC-SA).


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3D Interactive Plots for Multivariate Calculus Copyright © 2022 by Dr. Na Yu, Ryerson University is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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