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March 29, 2024, 10:46:01 pm

Author Topic: Multivariable calculus questions  (Read 1389 times)  Share 

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rife168

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Multivariable calculus questions
« on: July 11, 2012, 10:05:52 am »
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I am aware that for a function of two variables a linear approximation of a point close to can be approximated by the tangent plane approximation where and

So is the process the same if you have a function where and are themselves functions of more variables?
For example approximating near

Also, does it scale to an arbitrary number of nested functions?
Thanks
« Last Edit: July 11, 2012, 08:38:46 pm by rife168 »
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enwiabe

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Re: Multivariable linear approximation question
« Reply #1 on: July 11, 2012, 11:02:55 am »
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I am aware that for a function of two variables a linear approximation of a point close to can be approximated by the tangent plane approximation where and

So is the process the same if you have a function where and are themselves functions of more variables?
For example approximating near

Also, does it scale to an arbitrary number of nested functions?
Thanks

It should do. Even if x and y are parametrised in terms of s and t, the values that the approximation take are of x and y. So deltax still exists, deltay still exists and so do the function evaluations near those points. The parametrisation won't change the approximation method, only your way of getting the values to plug in.

Having said that, I haven't touched this stuff for 3 years and that could all be crap. But i'm 95% sure that's correct. TrueTears can probably explain it better though.

rife168

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Re: Multivariable linear approximation question
« Reply #2 on: July 11, 2012, 11:27:42 am »
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I am aware that for a function of two variables a linear approximation of a point close to can be approximated by the tangent plane approximation where and

So is the process the same if you have a function where and are themselves functions of more variables?
For example approximating near

Also, does it scale to an arbitrary number of nested functions?
Thanks

It should do. Even if x and y are parametrised in terms of s and t, the values that the approximation take are of x and y. So deltax still exists, deltay still exists and so do the function evaluations near those points. The parametrisation won't change the approximation method, only your way of getting the values to plug in.

Having said that, I haven't touched this stuff for 3 years and that could all be crap. But i'm 95% sure that's correct. TrueTears can probably explain it better though.

Ok so do you think it would be something like:



or

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rife168

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Re: Multivariable linear approximation question
« Reply #3 on: July 11, 2012, 08:31:35 pm »
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Also, for the double integral   what would the terminals be if I were to change the order of integration - that is, switch from to
« Last Edit: July 11, 2012, 08:39:56 pm by rife168 »
2012: VCE - 99.10
2013: PhB(Sci)@ANU