Differentiation of Vector Valued Functions - Browder, Proposition 8.12 ....

[Math Amateur]

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 8: Differentiable Maps and am specifically focused on Section 8.2 Differentials ...

I need help in proving Proposition 8.12 ... ...

Proposition 8.12 and the definitions, remarks and propositions leading up to it read as follows:
View attachment 7467
https://www.physicsforums.com/attachments/7468Although Browder states that Proposition 8.12 is easy to prove I am unable to make an effective start on the proof ...

Can someone please demonstrate how Proposition 8.12 is proved ...

[Note that I am unsure about the definition ... and the nature ... of the \(\displaystyle f'_i (p)\) ... ... ]

Help will be much appreciated ...

Peter

 

[HallsofIvy]

When you are "not sure" about definitions, look at simple examples. Start with a two dimensional case, say \(\displaystyle \vec{f}(x, y)= \begin{pmatrix} f_1(x, y) \\ f_2(x, y)\end{pmatrix}= \begin{pmatrix}x^2y \\ e^x sin(y)\end{pmatrix}\).

The "rows" referred to are \(\displaystyle \begin{pmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y}\end{pmatrix}= \begin{pmatrix}2xy & x^2\end{pmatrix}\) and \(\displaystyle \begin{pmatrix}\frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{pmatrix}= \begin{pmatrix} e^x sin(y) & e^x cos(y)\end{pmatrix}\) so that the 2 by 2 "derivative matrix" is
\(\displaystyle \begin{pmatrix}\frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\partial f_2}{\partial x} & \frac{\partial f_2}{\partial y}\end{pmatrix}= \begin{pmatrix} 2xy & x^2 \\ e^x sin(y) & e^x cos(y)\end{pmatrix}\)

 

[math771]

Hi Peter,

I am also currently reading Browder's book and am at the same section as you. I have not yet attempted to prove Proposition 8.12, but I can offer some tips and insights that may help you in your proof.

Firstly, let's clarify the definitions and notations used in the proposition. The proposition is about differentiable maps, which are functions that have a derivative at every point in their domain. In this section, we are specifically looking at differentiable maps between two Euclidean spaces: f: U --> V where U and V are open subsets of R^n and R^m respectively.

The notation f'_i (p) refers to the partial derivative of f with respect to the i-th variable at the point p. This means that if we have a function f(x,y,z) and we want to find the partial derivative with respect to y at the point (a,b,c), then f'_2 (a,b,c) would be the notation used.

Now, onto the proof of Proposition 8.12. The proposition states that if f: U --> V is a differentiable map at p in U, then f is continuous at p. To prove this, we need to show that for any sequence of points in U that converge to p, the corresponding sequence of points in V also converges to f(p).

To do this, we can use the definition of differentiability at p. This means that the limit of the difference quotient (f(x)-f(p))/(x-p) as x approaches p exists. This limit is equal to the derivative of f at p, which we can denote as f'(p). This means that for any ε>0, there exists a δ>0 such that for all x in U, if |x-p|<δ, then |(f(x)-f(p))/(x-p) - f'(p)|<ε.

Now, let's consider a sequence (x_n) in U that converges to p. This means that for any ε>0, there exists a N such that for all n>N, |x_n-p|<δ. Using the above definition, we can see that for all n>N, |(f(x_n)-f(p))/(x_n-p) - f'(p)|<ε. This means that the sequence (f(x_n)) also converges to f(p), since the difference quotient is approaching f'(p).

Therefore,