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What is the geometric significance of substitution? For instance substituting $x = r\cos(\theta)$ and $y = r\sin(\theta)$ in the following problem to find the limit. $\lim\limits_{(x,y) \to (0,0)} \dfrac{\tan(x^2 + y^2)}{x^2+y^2}$. Of course, this manipulation/substitution makes it easier to solve, but geometrically what does the above substitution mean. I hope the question is clear. Thank you in advance.

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I see the geometrical interpretation as follows:

As the definition of limit requires the convergence for any sequence, the change to polar coordinates parametrizes in an easy way all linear (in the sense, along a line) approaches to the point, the angle parametrizes the slope at which we approach the point.

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  • $\begingroup$ Thank you. Further sir, we make substitutions everywhere in calculus - integrations (usually indefinite integrals), differentiation etc. What is the geometric interpretation in these cases. $\endgroup$ Commented Aug 27, 2016 at 18:04
  • $\begingroup$ my geometric interpretation in these cases, is to parametrize the space with circular level lines of different radii instead of horizontal and vertical $\endgroup$
    – b00n heT
    Commented Aug 27, 2016 at 18:23

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