Derivatives and Linear transformations

[raghad]

Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer.

I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?

 

[Svein]

I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?

No, but observe: [itex] f'(x) = (\frac{\partial f}{\partial x_{1}}, \frac{\partial f}{\partial x_{2}}, \dotso , \frac{\partial f}{\partial x_{n}})[/itex] and since A is linear, [itex] A=(a_{1}, a_{2}, \dotso , a_{n})[/itex]. Therefore [itex] \frac{\partial f}{\partial x_{1}}=a_{1}[/itex] etc. SInce all a_k are constants, ...

 

[mathwonk]

is a function uniquely determined by its derivative?

 

[Svein]

is a function uniquely determined by its derivative?

Of course not - g(x) and g(x)+C have the same derivatives. I am a mathematician - I leave the details as an exercise for the student.

 

[HallsofIvy]

In general, with f a function from, say, Rⁿ to R^m, we can define the derivative of f, at point p in Rⁿ as "the linear transformation, from Rⁿ to R^m that best approximates f in some neighborhood of p".

To make that "best approximates f" more precise, note that we can write any function, f, as [tex]f(x)= A(x- p)+ D(x)[/tex] where A is a linear transformation and D(x) has the property that the limit, as x approaches p, of D(x)/||x- p|| is 0. Then A is the "derivative of f at p".

 

[WWGD]

In general, with f a function from, say, Rⁿ to R^m, we can define the derivative of f, at point p in Rⁿ as "the linear transformation, from Rⁿ to R^m that best approximates f in some neighborhood of p".

To make that "best approximates f" more precise, note that we can write any function, f, as [tex]f(x)= A(x- p)+ D(x)[/tex] where A is a linear transformation and D(x) has the property that the limit, as x approaches p, of D(x)/||x- p|| is 0. Then A is the "derivative of f at p".

That's where it gets confusing: some call it the differential.

 

[Xiuh]

That's where it gets confusing: some call it the differential.

Really? I've never heard someone call it the differential. As far as I know, the linear transformation that best approximates a function at a given point is always called the derivative (http://en.wikipedia.org/wiki/Fréchet_derivative)

To answer the OP, it is almost true, you're just missing the constant. The answer is [itex]f(x) = Ax + c[/itex]. The linear transformation that best approximates this [itex]f[/itex] is clearly [itex]A[/itex], in other words [itex]f'(a) = A[/itex] for every element in [itex]G[/itex]. And since [itex]G[/itex] is connected, any other function with derivative equal to [itex]A[/itex] in [itex]G[/itex], must differ only by a constant.

 

[WWGD]

Really? I've never heard someone call it the differential. As far as I know, the linear transformation that best approximates a function at a given point is always called the derivative (http://en.wikipedia.org/wiki/Fréchet_derivative)

To answer the OP, it is almost true, you're just missing the constant. The answer is [itex]f(x) = Ax + c[/itex]. The linear transformation that best approximates this [itex]f[/itex] is clearly [itex]A[/itex], in other words [itex]f'(a) = A[/itex] for every element in [itex]G[/itex]. And since [itex]G[/itex] is connected, any other function with derivative equal to [itex]A[/itex] in [itex]G[/itex], must differ only by a constant.

I think that A as a function is usually called, in my experience, the differential. Once you plug in numbers , it is called the derivative in this layout, so that, e.g., for ##f(x)=x^2##, ##2x## is the differential, but the derivative at a fixed ##x_0## is ##2x_0##.

 

[HallsofIvy]

No. If f(x)= x², the derivative is [itex]df/dx= 2x[/itex]. The "differential" is [itex]df= 2xdx[/itex].
And, as you say, the "derivative at fixed x₀" is 2x₀.

 

[WWGD]

It would help if you quoted actual definitions: the differential is the best linear map approximating the local change of the function near the point. The derivative is the rate of change (modulo higher dimensions).

 

[Mark44]

I think that A as a function is usually called, in my experience, the differential. Once you plug in numbers , it is called the derivative in this layout, so that, e.g., for ##f(x)=x^2##, ##2x## is the differential, but the derivative at a fixed ##x_0## is ##2x_0##.

No. If f(x)= x², the derivative is [itex]df/dx= 2x[/itex]. The "differential" is [itex]df= 2xdx[/itex].
And, as you say, the "derivative at fixed x₀" is 2x₀.

I agree with Halls here, and would add only "differential of f."

If f is a function of two variables, say z = f(x, y), then the differential of f (also called the total differential of f) is ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy##.

 

[WWGD]

I agree with Halls here, and would add only "differential of f."

If f is a function of two variables, say z = f(x, y), then the differential of f (also called the total differential of f) is ##df = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy##.

Yes, I corrected myself in my post after that one.

 

[mathwonk]

what used to be called the Frechet derivative some 50 years ago, e.g. in Dieudonne's Foundations of modern analysis, is now usually called the differential.