- #1

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## Homework Statement

Please see the attached pdf file, which is the bottom of page 365 from the 3rd edition of the book. This is a lesson about (generalized) waves, and f(z,t) is the vertical displacement of the medium at point z, time t. F is net force and T is the tension on the string in the picture.

I don't understand how we get from the tangents of angles to partial derivatives in this derivation. I've asked a professor about this before, and he said something about tangent lines to curves being connected to derivatives, but it didn't make sense because the tangent line to a curve is a different use of tangent than what I have here, tangent the trig function. Is there some connection between the two?

Also, the last step (from first to second partial derivatives) seems too hand wavy to be legitimate.

## Homework Equations

None for my first question.

For my second question, I understand that it involves the definition of the derivative:

[tex]\lim_{\Delta z \to 0} \frac{\frac{\partial f(z+ \Delta z)}{\partial t} -

\frac{\partial f(z)}{\partial t}}{\Delta z} = \frac{\partial^{2} f}{\partial t^{2}}[/tex]

What I'm not sure about is if it's mathematically sound to just multiply both sides of that equation by [tex]\Delta z[/tex], seeing as how one side has a limit.

## The Attempt at a Solution

I have no idea.