Intuitive / self-apparent derivation of gradient in curvilinear coords

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SUMMARY

The discussion focuses on deriving the gradient in polar and cylindrical coordinates, emphasizing the need for an intuitive approach that resonates with students. The original method using the chain rule and directional derivatives is deemed ineffective for teaching purposes. Participants suggest that understanding the gradient should not be limited to rigorous methods but should instead prioritize clarity and memorability. Visual aids are recommended to enhance comprehension of the concepts involved.

PREREQUISITES
  • Understanding of basic calculus concepts, particularly derivatives.
  • Familiarity with polar and cylindrical coordinate systems.
  • Knowledge of vector calculus, including gradient, divergence, and curl.
  • Experience with visual representation of mathematical concepts.
NEXT STEPS
  • Research intuitive methods for deriving the gradient in polar coordinates.
  • Explore visual aids and diagrams that clarify vector calculus concepts.
  • Study the relationship between coordinate systems and their impact on gradient calculations.
  • Learn about the applications of curl and divergence in various coordinate systems.
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Students, educators, and tutors in mathematics and physics who seek to enhance their understanding of vector calculus and improve their teaching methods for gradient derivation in curvilinear coordinates.

raxAdaam
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Hi there -

I'm looking for a clear and intuitive explanation of how one obtains the gradient in polar / cylindrical / curvilinear coords.

I do a lot of tutoring, but am finding that the method I've been using (basically chain rule + nature of directional derivative) just doesn't roll with students: the directional derivative approach (i.e. defining the unit tangent vector to the curve) seems really unintuitive to students - either because they've not really seen it or because they've not internalized it.

I don't need a rigorous method that applies to any imaginable coord. system, just something to help them a) remember b) derive and c) understand the gradient (and eventually curl, div etc.) in different coordinate systems. I have some vague recollection of seeing something pretty intuitive a long time ago, but it has slipped my memory and I just use the above described method personally. Everything I've found online so far has been a little too vague about details and unclear with its notation, so now I'm here!

Thanks in advance for the help!



Rax
 
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I would stress the difference between what is happening and its description via coordinates. The choice of the coordinate system is only a choice of description, not a choice which affects the gradient. Try to make an image.
 

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