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I've a doubt about the total derivative or differntial of an application. If $F : U \subset \mathbb{R}^n \to \mathbb{R}^m$ has differential $dF_{p}$ at each point $p \in U$ then the differential acts on a vector giving the instantaneous rate of change of $F$ in that direction or the increment in the linearization in that direction?

This question arose when I thought on the case $n = m = 1$, i.e.: a function $f : U \subset \mathbb{R} \to \mathbb{R}$. In that case, the differential matrix will have only one entry $f'(x)$ and a vector in $U$ will be simply a scalar $v = x_f - x$. In that case, the action of the differential in the vector will be: $df_{x}(v) = f'(x)(x_f - x)$ and in that case the matrix multiplication produces an increment on the linearization of $f$ instead just the derivative of $f$.

However, if the vector is the unit vector in that direction, then it'll be just $v = 1$ and $df_{x}(v) = f'(x)$, and it is indeed the derivative. So, when the vector is a unit vector de differential produces a rate of change and when it's not a unit vector it produces an increment to the linearization?

Sorry if I've said something silly, and if I wasn't clear enough ask me to explain better. Thanks in advance.

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Those two things read the same for me at first. $dF_p$ acts on a curve through $p$ by giving it "the increment in the linearization in the direction". This is a term I've never heard of but I know what you mean. –  Tunococ Aug 31 '12 at 1:43

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I think you have two concepts confused:

1.) the differential $df_a$ is actually a linear transformation which takes vectors at the tangent space at $a$ and, in the case of $f$ being real-valued, gives the rate of change of $f$ in the direction. However, to make a standard rate of change it is customary to consider only unit-vectors. In advanced calculus texts this is often dropped and the differential takes vectors of arbitrary length as inputs.

2.) the derivative at a point is the matrix of the differential. In the case of a real valued function this reduces to $df_a(v) = (\nabla f)(a) \cdot v$. In the case of functions on $\mathbb{R}$ it reduces to $df_a(v) = f'(a)v$. This can be cutely written as $df_a(dx)=f'(a)dx = \frac{dy}{dx}dx$, so the notation is consistent with the idea $dy=df_a(dx)$.

The "linearization" is related to the differential by the following simple formula:

$$L(a+h) = f(a)+df_a(h)$$

The linearization or differential in standard calculus are really just different notations for the same underlying idea: the tangent line (or space in general) is the best linear approximation to the curve (or curved space in general). To approximate the function itself you can use the linearization. To approximate a change in the function you use the differential.

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