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I'm confused about why we only differentiate with respect to the X when deriving Snell's law. Can someone explain why we do not also do a differentiation with respect to Y.

For more clarification about the question please see the following screenshots:

simple diagram of ray entering a different medium

enter image description here

My question is why do they only do dt/dx and not dt/dy?

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  • $\begingroup$ The answer is given by Floris. By the way take a look in my answer therein : Why one should follow Snell's law for shortest time?. May be useful. $\endgroup$ – Frobenius Sep 16 '17 at 5:31
  • $\begingroup$ I am confused by this question. I see no y in either the diagram or the equations. $\endgroup$ – Acccumulation Sep 18 '17 at 2:42
  • $\begingroup$ ^by 'y' I was referring to the general Y axis direction. $\endgroup$ – Legend27 Sep 25 '17 at 17:35
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The value of y is constant in your diagram - namely the front face of the glass. If you had a curved surface you would integrate with respect to both x and y - but here y is constant and the derivative is trivially zero.

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