数学代写|复分析作业代写Complex function代考|The Jacobian Matrix

数学代写|复分析作业代写Complex function代考|The Jacobian Matrix

Consider [4.4]. As discussed, the direction of the illustrated curve through $\mathrm{q}$ is being described with an infinitesimal vector $\left(\begin{array}{l}d x \ d y\end{array}\right)$; the infinitesimal image vector $\left(\begin{array}{l}d u \ d v\end{array}\right)$ gives the direction of the image curve through $Q$. We can determine the component du of $\left(\begin{array}{l}d u \ d v\end{array}\right)$ as follows:
\begin{aligned} & d u=\quad \text { total change in } u \text { due to moving along }\left(\begin{array}{l} d x \ d y \end{array}\right) \ & =\text { (change in u produced by moving } \mathrm{d} x \text { in the } x \text {-direction) } \ & \text { (change in } u \text { produced by moving } d y \text { in the y-direction) } \ & =\text { (rate of change of } u \text { with } x) \cdot(\text { change } d x \text { in } x) \ & \text { (rate of change of } u \text { with } y \text { ). (change dy in } y \text { ) } \ & =\left(\partial_x u\right) d x+\left(\partial_y u\right) d y, \ & \end{aligned}
where $\partial_x=\partial / \partial x$ etc. Likewise, we find that the vertical component is given by the formula
$$d v=\left(\partial_x v\right) d x+\left(\partial_y v\right) d y$$
Since these expressions are linear in $d x$ and $d y$, it follows (assuming that not all the partial derivatives vanish) that the infinitesimal vectors are carried to their images by a linear transformation. The general significance of this will be discussed later, but for the moment it means that the local effect of our mapping is completely described by a matrix J called the Jacobian.

数学代写|复分析作业代写Complex function代考|The Amplitwist Concept

We have just seen that the local effect of $z \mapsto z^2$ on infinitesimal vectors is to expand them and to rotate them. Transformations of this type (i.e., whose local effect is produced in these two steps) will play a dominating role from now on, and it will be very much to our advantage to have vivid new words specifically to describe them.

If all the infinitesimal vectors $(\overrightarrow{q p}$ etc.) emanating from $q$ merely undergo an equal enlargement to produce their images at $Q$, then we shall say that the local effect of the mapping is to amplify the vectors, and that the magnification factor involved is the amplification of the mapping at the point q. If, on the other hand, they all undergo an equal rotation, then we shall say that the local effect of the mapping is to twist the vectors, and that the angle of rotation involved is the twist of the mapping at the point q. More generally, the kind of mapping that will concern us will locally both amplify and twist infinitesimal vectors-we say that such a transformation is locally an amplitwist. Thus “an amplitwist” is synonymous with “a (direct) similarity”, except that the former refers to the transformation of infinitesimal vectors, whereas “a similarity” has no such connotation.
[We remind the reader of the discussion in both Prefaces: “infinitesimal” is being used here in a definite, technical sense-small and ultimately vanishing, the relationship to other infinitesimals being expressed via Newtonian “ultimate equalities $\left.{ }^{\prime \prime}.\right]$

We can illustrate the new terminology with reference to the concrete case we have just analysed: The mapping $z \mapsto z^2$ is locally an amplitwist with amplification $2 \mathrm{r}$ and twist $\theta$. See [4.5]. Quite generally, this figure makes it clear that if a mapping is locally an amplitwist then it is automatically conformal-the angle $\phi$ between the infinitesimal complex numbers is preserved.

Returning to [4.1] and [4.2], we now understand why infinitesimal squares were mapped to infinitesimal squares. Indeed, an infinitesimal region of arbitrary shape located at $z$ will be “amplitwisted” (amplified and twisted) to a similar shape at $z^2$. Note that here we are extending our terminology still further: henceforth we will freely employ the verb “to amplitwist”, meaning to amplify and to twist an infinitesimal geometric object.

All we really have at the moment is one simple mapping that turned out to be locally an amplitwist. In order to appreciate how truly fundamental this amplitwist concept is, we must return to $\mathbb{C}$ and begin from scratch to develop the idea of complex differentiation.

复分析代考

数学代写|复分析作业代写Complex function代考|The Jacobian Matrix

$d u=$ total change in $u$ due to moving along ( $d x d y) \quad=$ (change in u prod

$$d v=\left(\partial_x v\right) d x+\left(\partial_y v\right) d y$$

数学代写|复分析作业代写Complex function代考|The Amplitwist Concept

[我们提醒读者注意两篇前言中的讨论：“无穷小”在这里是在明确的技术意义上使用的——微小且最终消失，与其他无穷小的关系通过牛顿的“终极等式”表达′′.]

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