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Lecture Notes

Below are the lecture notes from my course on multivariable calculus, as well as some handouts I give to students for some of the experiments in the introductory physics lab courses. Note that I update my notes each term, but this website may not link to the latest editions.

Notes on MTH3015: Calculus III

Visualizing 3D Curves with WolframAlpha

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Visualizing 3D Curves with WolframAlpha

Sujeet Akula

In Problem Set #5, several vector-valued functions were listed, and you were asked to sketch the curves that are parametrized by those functions. This is indeed a difficult task, and to this end, I provide a short tutorial on creating plots in WolframAlpha, a sort of free, online front-end to Mathematica, including natural language processing and access to much public data.

  • Direct your browser to WolframAlpha: http://www.wolframalpha.com
  • To see a 2D curve, you can use a nice natural language command. \\ Type something like: "parametric plot (cos t, sin t), t from 0 to 2 pi'', and you should see this.
  • To see a 3D curve, you have to use the Mathematica language. \\ Type (exactly): ParametricPlot3D[{Cos[t],Sin[t],t},{t,0,2 Pi}]
  • These commands will generate plots of the curve as you specified, but especially in the case of 3D curves, you will want to be able to rotate the 'camera' to view the curve from different angles so that you can actually understand what it looks like. Unfortunately, you cannot do this with the website, however, once you create one of these plots, you should see a link towards the bottom of the page to download the result in the 'Live Mathematica' format. This requires software to run. The software, Mathematica, is available on campus to students, and is also accessible view the 'myApps', but in case you cannot get that to work, you can download Mathematica Player for free.

In fact, WolframAlpha is a wonderful tool that is very easy to use and very powerful, so I encourage you all to play around with it. You can use it to check most of your answers to the problem sets that I have assigned (though of course not all). As an example try typing "integrate x from 0 to 5'', or type in "weather in Honolulu 3 days before JFK died''. Also, try, "limit as t approaches 0 of sin t / t''. You should note how ambiguous the way I wrote that last request was--why does it do $\dfrac{\sin{t}}{t}$ and not $\sin{\dfrac{t}{t}}$? This is the magic of natural language processing--it makes the natural choice, and it is pretty good, most of the time.
In order to see the curves from Problem Set #5, I give the commands here:

  • $(\cos{t}, t, \sin{t}) \Rightarrow $ ParametricPlot3D[{Cos[t],Sin[t],t},{t,0,2 Pi}]
  • $(e^t \cos{t}, e^t \sin{t})\Rightarrow $ parametric plot (e^t cos t, e^t sin t), t from 0 to 4 pi
  • $(a\cos{t}, b\sin{t})\Rightarrow $ parametric plot (2 cos t, 5 sin t), t from 0 to 2 pi, axes
  • $(\cosh{t}, \sinh{t})\Rightarrow $ parametric plot (cosh t, sinh t), t from 0 to 10
  • $(t, t^2, t^4)\Rightarrow $ ParametricPlot3D[{t, t^2, t^4},{t,0,5}]

Note that due to a malfunction in the natural language processing, you have to enter the parametric plots for 3D curves in the rigid Mathematica syntax, that is why those commands look different. Also, not that the third curve uses $a=2$, $b=5$, feel free to change these numbers to observe the effects.

Week 1: Introduction to 2D and 3D co-ordinate spaces; introduction to vector, vector spaces, vector operations

Week 2: Parametrization

Week 3: Vector-valued Functions; Curves

Week 4: Functions of Several Variables; Limits and Continuity

Week 5: Partial Derivatives; The Chain Rule

Week 7: Directional Derivative and Gradient

Week 8: Taylor Series and Extrema

Week 9: Double Integrals; Polar Integrals

Week 10: Triple Integrals; Cylindrical and Spherical Co-ordinates

Week 11: The Transformation Theorem

Notes on IPL/CPS Physics Labs

Lab Report Grading Structure

Experiment 9: Maxwell's Wheel

Experiment 12: The Simple Pendulum

Experiment 13: Simple Harmonic Motion

Experiment 14: Standing Waves

Experiment 16: Electric Field and Electric Potential