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Cake day: July 10th, 2023

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  • ltxrtquq@lemmy.mltoScience Memes@mander.xyzSquare!
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    1 month ago

    Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ

    You’re using the derivative of a polar equation as the basis for what a tangent line is. But as the textbook explains, that doesn’t give you a tangent line or describe the slope at that point. I never bothered defining what “tangent” means, but since this seems so important to you why don’t you try coming up with a reasonable definition?


  • ltxrtquq@lemmy.mltoScience Memes@mander.xyzSquare!
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    1 month ago

    I think we fundamentally don’t agree on what “tangent” means. You can use

    x=f(θ)cosθ, y=f(θ)sinθ to compute dydx

    as taken from the textbook, giving you a tangent line in the terms used in polar coordinates. I think your line of reasoning would lead to r=1 in polar coordinates being a line, even though it’s a circle with radius 1.


  • ltxrtquq@lemmy.mltoScience Memes@mander.xyzSquare!
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    1 month ago

    Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ

    I think this part from the textbook describes what you’re talking about

    Instead, we will use x=f(θ)cosθ, y=f(θ)sinθ to compute dydx.

    And this would give you the actual tangent line, or at least the slope of that line.


  • ltxrtquq@lemmy.mltoScience Memes@mander.xyzSquare!
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    1 month ago

    Polar Functions and dydx

    We are interested in the lines tangent a given graph, regardless of whether that graph is produced by rectangular, parametric, or polar equations. In each of these contexts, the slope of the tangent line is dydx. Given r=f(θ), we are generally not concerned with r′=f′(θ); that describes how fast r changes with respect to θ. Instead, we will use x=f(θ)cosθ, y=f(θ)sinθ to compute dydx.

    From the link above. I really don’t understand why you seem to think a tangent line in polar coordinates would be a circle.


  • ltxrtquq@lemmy.mltoScience Memes@mander.xyzSquare!
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    1 month ago

    A straight line in polar coordinates with the same tangent would be a circle.

    I’m not sure that’s true. In non-euclidean geometry it might be, but aren’t polar coordinates just an alternative way of expressing cartesian?

    Looking at a libre textbook, it seems to be showing that a tangent line in polar coordinates is still a straight line, not a circle.