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Chapter 9: Problem 18

Answer analogous questions for the mapping defined by $$ u=x^{2}-y^{2}, \quad v=2 x y . $$

Short Answer

Expert verified
Answer: By analyzing the differences in u and v under various changes in x and y, we observe how the mapping defined by \(u = x^2 - y^2\) and \(v = 2xy\) behaves. To summarize, under the given transformation, different combinations of increases and decreases in x and y values result in distinct alterations of the u and v values. Understanding this relationship is crucial for answering various analogous questions associated with the given transformation.

Step by step solution

01

Determine what happens when both x and y increase.

We will begin by analyzing what happens to \(u\) and \(v\) when both \(x\) and \(y\) increase by the same amount (let's call it \(\Delta\)). So now, we have: $$ u = (x+\Delta)^2 - (y+\Delta)^2 \\ v = 2(x+\Delta)(y+\Delta) $$ Let's calculate the difference between the original and transformed values for \(u\) and \(v\): $$ \Delta u = u' - u = (x+\Delta)^2 - (y+\Delta)^2 - (x^2 - y^2) \\ \Delta v = v' - v = 2(x+\Delta)(y+\Delta) - 2xy $$
02

Determine what happens when x increases and y decreases.

Now, we will analyze the case where \(x\) increases by \(\Delta\) and \(y\) decreases by \(\Delta\). We have: $$ u = (x+\Delta)^2 - (y-\Delta)^2 \\ v = 2(x+\Delta)(y-\Delta) $$ Again, let's compute the difference between the original and transformed values for \(u\) and \(v\): $$ \Delta u = u' - u = (x+\Delta)^2 - (y-\Delta)^2 - (x^2 - y^2) \\ \Delta v = v' - v = 2(x+\Delta)(y-\Delta) - 2xy $$
03

Determine what happens when x decreases and y increases.

Next, we will investigate the case in which \(x\) decreases by \(\Delta\) while \(y\) increases by \(\Delta\). We have: $$ u = (x-\Delta)^2 - (y+\Delta)^2 \\ v = 2(x-\Delta)(y+\Delta) $$ As before, we will calculate the difference between the original and transformed values for \(u\) and \(v\): $$ \Delta u = u' - u = (x-\Delta)^2 - (y+\Delta)^2 - (x^2 - y^2) \\ \Delta v = v' - v = 2(x-\Delta)(y+\Delta) - 2xy $$
04

Determine what happens when both x and y decrease.

Finally, we will look at the case where both \(x\) and \(y\) decrease by \(\Delta\). We have: $$ u = (x-\Delta)^2 - (y-\Delta)^2 \\ v = 2(x-\Delta)(y-\Delta) $$ Now, calculate the difference between the original and transformed values for \(u\) and \(v\): $$ \Delta u = u' - u = (x-\Delta)^2 - (y-\Delta)^2 - (x^2 - y^2) \\ \Delta v = v' - v = 2(x-\Delta)(y-\Delta) - 2xy $$ By comparing the results obtained in the various steps, we can understand the mapping defined by \(u = x^2 - y^2\) and \(v = 2xy\). This will help us to answer various analogous questions related to the given transformation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transformation Mapping
In the realm of complex analysis, transformation mapping plays a pivotal role. It is essentially a function that takes points from one space and translates them into another, often preserving some structure. For the given exercise, the mapping is defined by two functions: \( u = x^2 - y^2 \) and \( v = 2xy \).

This mapping transforms points in the \(xy\)-plane to points in the \(uv\)-plane. By examining different changes in \(x\) and \(y\), one can discern how transformations affect the entire coordinate system. With shifts like simultaneous increase or decrease, and opposing movements, we evaluate the resultant changes in \(u\) and \(v\).

Transformation mappings help in visualizing how specific coordinates translate into another complex plane, providing insights into potential geometric distortions or symmetries.
Analytic Functions
An analytic function in complex analysis is one that is complex differentiable in a neighborhood of each point in its domain. The concept pertains closely to the idea of a function being well-behaved and possessing a derivative.

In the context of the exercise, although the functions for \(u\) and \(v\) are given as real-valued functions, they can be seen as parts of a single complex function \( f(z) = z^2 \), where \( z = x + iy \) and \( \bar{z} = x - iy \).

Analytic functions are vital in complex analysis because they allow the use of powerful tools such as contour integration and complex power series. They also insist on locally smooth behaviors and transformations, making predictions and calculations more robust and tractable.
Complex Variables
Complex variables are the building blocks of complex analysis. These variables consist of a real component and an imaginary component, expressed as \(z = x + iy\). The real and imaginary parts, \(x\) and \(y\), respectively, make up the axes of the complex plane.

In our mapping scenario, studying complex variables involves observing how changes in the real parts \(x\) and \(y\) affect the functions \(u\) and \(v\). The resulting transformations reflect how complex numbers behave under various operations, such as squaring in this particular exercise.

Understanding complex variables helps simplify otherwise tricky calculus problems into more approachable forms, fostering deeper comprehension in topics like transformation and integration.
Coordinate Transformation
Coordinate transformation is a fundamental concept where an entire coordinate system is converted into another, often to simplify a problem or provide a clearer geometric interpretation.

In this exercise, the transformation from \(xy\)-coordinates to \(uv\)-coordinates provides a new perspective by simplifying the expressions into functions easier to analyze or visualize. Specifically, \( u = x^2 - y^2 \) and \( v = 2xy \) are a choice that reflects radial symmetry and scaling within specific symmetric shapes like hyperbolas.

This transformation not only caters to finding solutions efficiently but also aids in understanding interactions between different coordinate systems and how they describe geometric figures or physical phenomena seamlessly.

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Most popular questions from this chapter

Let \(f \in \mathscr{f}^{(m)}(E)\), where \(E\) is an open subset of \(R^{n}\). Fix a \(\in E\), and suppose \(x \in R^{n}\) is so close to 0 that the points $$ \mathbf{p}(t)=\mathbf{a}+\boldsymbol{r} \mathbf{x} $$ lie in \(E\) whenever \(0 \leq t \leq 1\). Define $$ h(t)=f(\mathrm{p}(t)) $$ for all \(t \in R^{1}\) for which \(\mathrm{p}(t) \in E\). (a) For \(1 \leq k \leq m\), show (by repeated application of the chain rule) that $$ h^{(k)}(t)=\sum\left(D_{i_{1} \ldots i_{k}} f\right)(\mathrm{p}(t)) x_{i_{1} \ldots} x_{i_{k}} . $$ The sum extends over all ordered \(k\) -tuples \(\left(i_{1}, \ldots, i_{k}\right)\) in which each \(i\), is one of the integers \(1, \ldots, n\). (b) By Taylor's theorem (5.15), $$ h(1)=\sum_{k=0}^{m-1} \frac{h^{(k)}(0)}{k !}+\frac{h^{(m)}(t)}{m !} $$ for some \(t \in(0,1) .\) Use this to prove Taylor's theorem in \(n\) variables by showing that the formula $$ f(\mathbf{a}+\mathbf{x})=\sum_{k=0}^{m-1} \frac{1}{k !} \sum\left(D_{i 1} \ldots, f\right)(\mathbf{a}) x_{i_{1}} \ldots x_{1,}+r(\mathbf{x}) $$ represents \(f(\mathrm{a}+\mathbf{x})\) as the sum of its so-called "Taylor polynomial of degree \(m-1, "\) plus a remainder that satisfies $$ \lim _{x \rightarrow 0} \frac{r(\mathbf{x})}{|x|=-1}=0 $$ Each of the inner sums extends over all ordered \(k\) -tuples \(\left(i_{1}, \ldots, i_{k}\right)\), as in part \((a)\); as usual, the zero-order derivative of \(f\) is simply \(f\), so that the constant term of the Taylor polynomial of \(f\) at a is \(f(\mathbf{a})\). (c) Exercise 29 shows that repetition occurs in the Taylor polynomial as written in part ( \(b\) ). For instance, \(D_{113}\) occurs three times, as \(D_{113}, D_{131}, D_{311}\). The sum of the corresponding three terms can be written in the form $$ 3\left(D_{1}^{2} D_{3} f\right)(a) x_{1}^{2} x_{3} $$ Prove (by calculating how often each derivative occurs) that the Taylor polynomial in \((b)\) can be written in the form $$ \sum \frac{\left(D_{1}^{21} \cdots D_{n}^{s n} f\right)(a)}{s_{1} ! \cdots s_{n} !} x_{1}^{21} \cdots x_{n}^{s_{n}} $$ Here the summation extends over all ordered \(n\) -tuples \(\left(s_{1}, \ldots, s_{n}\right)\) such that each \(s_{1}\) is a nonnegative integer, and \(s_{1}+\cdots+s_{n} \leq m-1\).

Let \(f=\left(f_{1}, f_{2}\right)\) be the mapping of \(R^{2}\) into \(R^{2}\) given by $$ f_{1}(x, y)=e^{x} \cos y, \quad f_{2}(x, y)=e^{x} \sin y $$ (a) What is the range of \(f ?\) (b) Show that the Jacobian of \(f\) is not zero at any point of \(R^{2}\). Thus every point of \(R^{2}\) has a neighborhood in which \(f\) is one-to-one. Nevertheless, \(f\) is not one-toone on \(R^{2}\). (c) Put \(a=(0, \pi / 3), b=f(a)\), let \(g\) be the continuous inverse of \(\mathbf{f}\), defined in a neighborhood of \(\mathrm{b}\), such that \(\mathrm{g}(\mathrm{b})=\mathrm{a} .\) Find an explicit formula for \(\mathrm{g}\), compute \(\mathbf{f}^{\prime}(\mathbf{a})\) and \(\mathbf{g}^{\prime}(\mathbf{b})\), and verify the formula (52). ( ) What are the images under \(\mathrm{f}\) of lines parallel to the coordinate axes?

Prove that to every \(A \in L\left(R^{n}, R^{1}\right)\) corresponds a unique \(y \in R^{n}\) such that \(A \mathrm{x}=\mathbf{x} \cdot \mathrm{y}\). Prove also that \(\|A\|=|\mathbf{y}|\). Hint: Under certain conditions, equality holds in the Schwarz inequality.

Suppose \(\mathrm{f}\) is a differentiable mapping of \(R^{1}\) into \(R^{3}\) such that \(|\mathbf{f}(t)|=1\) for every \(t\). Prove that \(f^{\prime}(t) \cdot f(t)=0\). Interpret this result geometrically.

Suppose \(A \in L\left(R^{n}, R^{*}\right)\), let \(r\) be the rank of \(A\). (a) Define \(S\) as in the proof of Theorem 9.32. Show that \(S A\) is a projection in \(R^{\circ}\) whose null space is \(\mathcal{N}(A)\) and whose range is \(\mathscr{R}(S) .\) Hint: By (68), SASA \(=S A\). (b) Use (a) to show that $$ \operatorname{dim} \mathcal{N}(A)+\operatorname{dim} \mathscr{A}(A)=n $$
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