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when I solve a volume problem in calc3 the hardest part is to visualise the solid in 3d but I recently noticed that after solving many problems I can visualise solids more clearly and I figured that might be a skill that I can train myself to be better at , so my question is : is there a way to train myself to visualise 3d objects , projection and Cross section of them ( something like problem book or training plan )

pie
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    Playing around with a 3D graphing calculator is a good start. Just try out different functions and see what they do. Then the “3D conics” are good to just have memorized. – moboDawn_φ Jun 11 '23 at 01:13
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    Machines won't help you. It is better to learn how to use level curves like we use intercepts in two dimensional space. We can set each of $x, y$ or $z$ to zero to see what the three views are in three space. When we set $x$ to zero we get the front view, when we set $y$ to zero we get the side view and when we set $z$ to zero we get the top view. Architects and engineers have been doing this for centuries. There is no short cut. You must practice. – John Douma Jun 11 '23 at 02:55
  • @John Douma so what and where should I practice this ?do you have some book or something? – pie Jun 11 '23 at 02:59
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    Get an undergraduate calculus book that covers third semester calculus. The first section on multi-variable functions and partial derivatives should start with being able to visualize surfaces in three dimensions. – John Douma Jun 11 '23 at 03:04
  • @John Douma I solved these problems in thomas' book, larson's book and stewart's book , so do you have any other recommendations ? – pie Jun 11 '23 at 03:08
  • No. In fact, I learned from Thomas and Finney. What did you find inadequate about the books you have used? – John Douma Jun 11 '23 at 03:14
  • @John Douma they are great books but already solved most of their problems and I want another book or source that have more difficult visualising problems – pie Jun 11 '23 at 03:16
  • @J. W. Tanner do you have any suggestions ? – pie Jun 12 '23 at 08:21
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    Why are you wanting to aquire these skills? Are you wanting to study in serious detail any higher levels of mathematics? Just as an offhand comment, the desire to visualize can often hold one back, even in two and three dimensions, and entire fields such as measure theory were in part created to replace geometric intuition and visualization with rigor and abstraction, due to the serious limitations of using the former as crutches. – Derek H. Jun 13 '23 at 03:22

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I've been a mentor for first year engineering students and I've noticed that the students are very (and I mean VERY) different when it comes to visual abilities. Some of them can imagine things in 3d in a matter of milliseconds, while others really struggle to do it. But why is that?

This question has hunted me for at least two years, but I think I've got some ideas about it.

  1. Better know what we are dealing with: What you've described is known as "Spatial Visualization Ability". This is a Wikipedia article about it.

  2. It is probably not about how smart you are: I've seen students with all levels of IQ experiencing the same problem. Although, some references consider this visual ability as a subcategory of intelligence. See this Wikipedia article for example.

  3. How to test this ability: The most common test for this ability is the "Mental Rotation Test". You are given a 3D object (printed in 2D) and you are asked to choose which of the 4 options is actually the result of rotating the object. This is a very interesting test. Just so you know, there is a significant difference between male and female participants. You can search which gender performs better on this test, but all sexuality-related debates aside, there is some good news:

  4. This ability is improved by practicing: Don't believe me? See this. If you are not good at it yet, don't worry. You can do great after some exercising.

  5. Where to start: I think mental rotation test problems are a good place to start. Also, you can use technical drawing books. See for example, "Basic Technical Drawing Problems" by Spencer.

Fish_n_Chips
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