Unit 13: Work

The Concept

Work is the measurement of a force on an object along a line through a 2D or 3D vector field. An intuitive example would be kayaking upstream in a river. The river or water source would be the vector field [latex]\vec{F}[/latex] and the object would be the kayak, the path would be the route we take upstream with the kayak, and, lastly, the work is effort used to overcome the current. Understand that the current will have different magnitudes and directions at different points–this is why we represent this with a vector field.

Mathematically, the definition of a vector field [latex]\vec{F}[/latex] in 2D or 3D is given as

[latex]\vec{F}(x,y) = \langle P(x, y) , Q(x, y) \rangle[/latex] or [latex]\vec{F}(x,y,z)= \langle P(x, y,z) , Q(x, y,z) , R(x,y,z)\rangle[/latex]

Looking to answer the question of how we can compute the work done by the river of moving the kayak along route [latex]C[/latex], we can calculate the work [latex]W[/latex] done by force field [latex]\vec{F}[/latex] along the curve [latex]C[/latex] as the following equation

[latex]W = \int_C \vec{F} \cdot dr= \int_a^b F(r(t)) r'(t)\,dt[/latex]

The Plot

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

  1. Fill in [latex]P(x, y)[/latex] and [latex]Q(x, y)[/latex] (i.e., the first and second compartments of the vector field function).
  2. Adjust xmin, xmax, ymin and max and they are the lower and upper bounds for the [latex]x[/latex]-axis and [latex]y[/latex]-axis.
  3. Input the function for curve C, [latex]y=f(x)[/latex] (i.e., the trajectory that the object travels along).
  4. Adjust [latex]a[/latex] and [latex]b[/latex] (i.e., the [latex]x[/latex]-coordinates of the start and end points of curve C).
  5. The result of the work is shown.

Self-Checking Questions

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

  1. Find the work done by vector field [latex]\vec{F}(x,y)=y\vec{i}+2x\vec{j}[/latex] in moving an object along path [latex]C[/latex], which joins points (1,0) and (0,1).

Use the graph to find the answer to this question.


  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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