There is already an article on physics forums that kind of addresses my issue:(adsbygoogle = window.adsbygoogle || []).push({});

https://www.physicsforums.com/showthread.php?t=107516 and i'm not really satisfied with the wikipedia article.

I am generally confused on what the derivative should be. I'm familiar with Jacobian matrices but am confused by Spivak's definition of a derivative because there is a norm in R^m for the numerator of the limit definition and a norm from R^n in the denominator, so it seems as if the derivative must return a scalar -- unless I am mistaken about the meaning of the norm in this case.

Spivak's definition:

lim h -->0 |f(a+h) - f(a) - λ(h)|/|h| = 0

I'm not really sure if the zeros are vectors or scalars (for example the norm equal to zero?)

On a side note, I'm trying to apply this to solve the problem:

Prove that f:R->R^2 is differentiable at a in R iff f_1 and f_2 are, and then it will follow that

f ' (a) = [ f_1 ' (a) , f_2 ' (a) ]^t

The result seems obvious but having a clear understanding of the definitions would help me prove something. Any help appreciated, thanks

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Total Derivatives of fxns from R^n to R^m

**Physics Forums | Science Articles, Homework Help, Discussion**