/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Gegeben ist das Gebiet \(D \subs... [FREE SOLUTION] | 91影视

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Chapter 25: Problem 3

Gegeben ist das Gebiet \(D \subseteq \mathbb{R}^{3}\), das als Schnitt der Einheitskugel mit der Menge \(\left\\{x \in \mathbb{R}^{3} \mid x_{1}, x_{2}, x_{3}>0\right\\}\) entsteht. Beschreiben Sie dieses Gebiet in kartesischen Koordinaten, Zylinderkoordinaten und Kugelkoordinaten.

Short Answer

Expert verified
In this exercise, we're given a region D, which is the intersection of the unit sphere and the set {x 鈭� 鈩澛� | x鈧�, x鈧�, x鈧�>0}. The region D is described as follows: 1. Cartesian Coordinates: \((x_1, x_2, x_3) \in \mathbb{R}^3: x_1^2 + x_2^2 + x_3^2 = 1, x_1 > 0, x_2 > 0, x_3 > 0\) 2. Cylindrical Coordinates: \((r, \theta, z) \in \mathbb{R}^3: r^2 + z^2 = 1, 0 < \theta < \frac{\pi}{2}, z > 0\) 3. Spherical Coordinates: \((\rho, \theta, \phi) \in \mathbb{R}^3: \rho = 1, 0 < \theta < \frac{\pi}{2}, 0 < \phi < \frac{\pi}{2}\)

Step by step solution

01

Cartesian Coordinates

First, we describe the region D in cartesian coordinates. The unit sphere can be described by the equation: \(x_1^2 + x_2^2 + x_3^2 = 1\) As all coordinates x鈧�, x鈧� and x鈧� are greater than 0, we have the restriction that x鈧� > 0, x鈧� > 0, and x鈧� > 0. Therefore, in cartesian coordinates, region D can be described as: { \((x_1, x_2, x_3) \in \mathbb{R}^3: x_1^2 + x_2^2 + x_3^2 = 1 \quad \text{and} \quad x_1 > 0, x_2 > 0, x_3 > 0\) }
02

Cylindrical Coordinates

Now, let's describe the region D in cylindrical coordinates. To do this, we need to use the following transformation equations: $r^2 = x_1^2 + x_2^2 \\ \theta = \arctan\left(\frac{x_2}{x_1}\right) \\ z = x_3$ The unit sphere equation in cylindrical coordinates becomes: \(r^2 + z^2 = 1\) Since we have x鈧� > 0 and x鈧� > 0, we have 0 < 胃 < 蟺/2. So, the region D in cylindrical coordinates is given by: { \((r, \theta, z) \in \mathbb{R}^3: r^2 + z^2 = 1 \quad \text{and} \quad 0 < \theta < \frac{\pi}{2}, z > 0\) }
03

Spherical Coordinates

Finally, we describe the region D in spherical coordinates. To do this, we need to use the following transformation equations: $\rho = \sqrt{x_1^2 + x_2^2 + x_3^2} \\ \theta = \arctan\left(\frac{x_2}{x_1}\right) \\ \phi = \arccos\left(\frac{x_3}{\rho}\right)$ The unit sphere equation in spherical coordinates becomes: \(\rho = 1\) Since we have x鈧� > 0, x鈧� > 0, and x鈧� > 0, we have 0 < 胃 < 蟺/2 and 0 < 蠁 < 蟺/2. Thus, the region D in spherical coordinates is described as: { \((\rho, \theta, \phi) \in \mathbb{R}^3: \rho = 1 \quad \text{and} \quad 0 < \theta < \frac{\pi}{2}, 0 < \phi < \frac{\pi}{2}\) }

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Most popular questions from this chapter

Bestimmen Sie f眉r die folgenden Gebiete \(D\) je eine Transformation \(\psi: B \rightarrow D\), bei der \(B\) ein Quader ist: (a) \(D=\left\\{x \in \mathbb{R}^{2} \mid 00, x_{1}^{2}+x_{2}^{2}+x_{3}^{2}<1\right\\}\) (c) \(D=\left\\{x \in \mathbb{R}^{2} \mid 00, x_{1}^{2}<9-x_{2}^{2}\right\\}\)

Die Menge all derjenigen Punkte \(x \in \mathbb{R}^{3}\), die L枚sungen einer Gleichung der Form $$ a x_{1}^{2}+b x_{2}^{2}+c x_{3}^{2}=r^{2} $$ bei gegebenem \(a, b, c\) und \(r>0\) sind, nennt man ein Ellipsoid. F眉r \(a=b=c\) erh盲lt man den Spezialfall einer Kugel. Bei Kugelkoordinaten erh盲lt man f眉r konstantes \(r\) und variable Winkelkoordinaten eine Kugelschale. Modifizieren Sie die Kugelkoordinaten so, dass bei konstantem \(r\) ein Ellipsoid entsteht. Wie lautet die Funktionaldeterminante der zugeh枚rigen Transformation?

Wir n盲hern die Erde durch eine Kugel mit Radius \(R=7000 \mathrm{~km}\) an. Entlang des 脛quators soll rund um die Erde eine Stra脽e der Breite \(B=60 \mathrm{~m}\) gebaut werden. Welches Volumen \(V\) hat die abgetragene Planetenmasse, wenn die Stra脽enoberfl盲che genau die Mantelfl盲che eines Zylinders bildet (siehe Abb. 25.26)? Wie gro脽 ist das Volumen, wenn die Stra脽e auf dem Mond \((R=1700 \mathrm{~km})\) gebaut wird?

Die Halbkugel \(B=\left\\{x \in \mathbb{R}^{3} \mid\|x\|0\right\\}\) besteht aus einem Material mit der Dichte \(\varrho(x)=a x_{3}, a>0\). Berechnen Sie die Masse und die dritte Koordinate des Schwerpunkts der Halbkugel.

Berechnen Sie die folgenden Gebietsintegrale: (a) \(J=\int_{D} \frac{\sin \left(x_{1}+x_{3}\right)}{x_{2}+2} \mathrm{~d} x\) mit \(D=\left[-\frac{\pi}{4}, 0\right] \times[0,2] \times\left[0, \frac{\pi}{2}\right]\) (b) \(J=\int_{D} \frac{2 x_{1} x_{3}}{\left(x_{1}^{2}+x_{2}^{2}\right)^{2}} \mathrm{~d} x\) mit \(D=\left[\frac{1}{\sqrt{3}}, 1\right] \times[0,1] \times[0,1]\)
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