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I'm trying to understand linear approximation in calculus 3 ( I think). And I need to understand delta epsilon limit definition from calculus 1. I have the equation \[\Delta z = dz + \epsilon _1 \Delta x + \epsilon _2 \Delta y\] Can anyone help me understand this with pictures or links to helpful sights or anything? I think I understand some of delta epsilon limit with 1 variable. If you have a domain of \[x-\delta ,x +\delta\] then you have a range of\[[f(x-\delta ,x+\delta]\].

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But I don't exactly know what epsilon is here...|dw:1351625107802:dw|
|dw:1351625442880:dw|Here,\[\Delta y_1 > \Delta y2\]\[\Delta y_1 \ne \Delta y_2\] So the range is not anything like \[[f(x)-\epsilon, f(x)+\epsilon]\]
the essence of it is that for some delta surrounding x, the function will be located in some epsilon surrounding f(x). That isn't worded perfectly, but maybe it will help a little. Where this idea is taking you is that you don't want to have to approximate the function with a stairstep-like approach... you would really like to have a way to describe f(x) over a range of x values where you don't have to treat the curve as something like a staircase (imagine a curve pixelated... would look like steps, not smooth).

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These videos explain it better than I can :)
I'm not sure this will help with Calc 3 though :( But it's a good review from Calc 1
Thank you for your help! :) Not a problem!
sorry, I didn't correctly read your question at first.. missed the fact the variables included z... then I noticed it said Calc 3 :) didn't mean to give you too simplistic a starting point. I do like those videos though :) good luck!
Thank you! Part of the issue is that I don't think I understand calculus 1 well enough! Lots of things from calculus 1 can be applied to calculus 3. Therefore, I am glad you missed the calculus 3 part :) Take care! And thanks for the video links!
I learned something... SUCCESS!! :D

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