Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Total differential for a function of two variables is known as

$df = f_x(x_0, y_0)dx + f_y(x_0,y_0)dy$

It's not clear how to derive this. Thomas' Calculus 11th Ed pg 1021 says to use the definition of linearization L(x,y). The definition of linearization for a function f(x) at page 223 is only for one variable

$f(x) \approx L(x) = f(x_0) + f'(x_0)(x-a)$

or

$L(x) = f(x_0) + f'(x_0)∆x$

I had the idea to rewrite the formula $L(x)$ for $L(x,y)$, so I had

$L(x,y) = f(x_0,y) + f_x(x_0,y)∆x$

Then I thought I could maybe linearize $f(x_0,y)$ to get

$L(x,y) = L(x_0,y_0) + f_y(x_0,y_0)∆y + f_x(x_0,y_0)∆x$

So

$∆L = L(x,y) - L(x_0,y_0) = f_y(x_0,y_0)∆y + f_x(x_0,y_0)∆x$

Would this be right?

I also found this page from Google

http://www.physicsforums.com/showthread.php?p=3109168

share|improve this question
1  
Looks good to me. –  Grumpy Parsnip Feb 9 '11 at 23:23

2 Answers 2

Sorry, the "comments" section is disabled, so I will post in here. My apologies if this is against the protocol.

Yes; the total differential is what you just described: the equation of the plane ( as a 2-d linear object) tangent to your function at a point; this tangent plane is the linearization of f, or, in a precise del-eps. sense, the best linear approximation to f, in a small 'hood (neighborhood) of a point.

share|improve this answer
up vote 0 down vote accepted

It seems the derivation is correct, thanks Jim Conant.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.