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Chapter 23: Problem 16

Beweisen Sie f眉r einmal bzw. zweimal stetig differenzierbare Vektorfelder \(\boldsymbol{F}\) und \(\boldsymbol{G}\) die Identit盲ten $$ \begin{aligned} &=\operatorname{div}(\boldsymbol{F} \times \boldsymbol{G})=\boldsymbol{G} \cdot \operatorname{rot} \boldsymbol{F}-\boldsymbol{F} \cdot \operatorname{rot} \boldsymbol{G} \\ &=\operatorname{rot}(\operatorname{rot} \boldsymbol{F})=\nabla(\operatorname{div} \boldsymbol{F})-\Delta \boldsymbol{F} \end{aligned} $$ wobei sich der Laplace-Operator in der letzten Gleichung auf jede Komponente des Vektors bezieht.

Short Answer

Expert verified
In summary, we proved the following two identities involving vector fields F and G: 1. div(F 脳 G) = G鈰卹ot F - F鈰卹ot G 2. rot(rot F) = 鈭�(div F) - 螖F We achieved this by systematically computing the left-hand side and right-hand side of each identity and verifying that they were equal in each case. This involved using the definitions of divergence, curl, gradient, and Laplacian, as well as various properties and rules of vector arithmetic. These identities are important in vector calculus and have applications in various fields, such as physics and engineering.

Step by step solution

01

Identity 1 - Left-hand Side (LHS)

Calculate the divergence of the cross product of F and G, which is div(F 脳 G). First we have to find the cross product of F and G. Let's denote F as F = (F鈧�, F鈧�, F鈧�) and G as G = (G鈧�, G鈧�, G鈧�). The cross product is given by: F 脳 G = $\left( \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ F_1 & F_2 & F_3 \\ G_1 & G_2 & G_3 \\ \end{matrix}\right)$ = \(\hat{i}(F鈧侴鈧� - F鈧僄鈧�) - \hat{j}(F鈧丟鈧� - F鈧僄鈧�) + \hat{k}(F鈧丟鈧� - F鈧侴鈧�)\) Now, we calculate the divergence of this resulting vector: \(\operatorname{div}(F\times G) = \frac{\partial}{\partial x}(F鈧侴鈧� - F鈧僄鈧�) - \frac{\partial}{\partial y}(F鈧丟鈧� - F鈧僄鈧�) + \frac{\partial}{\partial z}(F鈧丟鈧� - F鈧侴鈧�)\)
02

Identity 1 - Right-hand Side (RHS)

Calculate the dot product of G with the curl of F, which is G鈰卹ot F, and the dot product of F with the curl of G, which is F鈰卹ot G. The curl of F and G are given by: $\operatorname{rot} F = \left( \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_1 & F_2 & F_3 \\ \end{matrix}\right)$ & $\operatorname{rot} G = \left( \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ G_1 & G_2 & G_3 \\ \end{matrix}\right)$ Now we need to calculate respective dot products: \(\boldsymbol{G}\cdot \operatorname{rot} \boldsymbol{F} - \boldsymbol{F} \cdot \operatorname{rot} \boldsymbol{G}\) Compare the two sides: \(\operatorname{div}(F\times G) = \boldsymbol{G}\cdot \operatorname{rot} \boldsymbol{F} -\boldsymbol{F} \cdot \operatorname{rot} \boldsymbol{G}\) and verify that they are equal, thereby proving the first identity.
03

Identity 2 - Left-hand Side (LHS)

Calculate the curl of the curl of F, which is rot(rot F). First, we need to find the rot F, which we calculated above. Next, we find the curl of the result: $\operatorname{rot}(\operatorname{rot} F) = \left( \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ (\operatorname{rot} F)_1 & (\operatorname{rot} F)_2 & (\operatorname{rot} F)_3 \\ \end{matrix}\right)$
04

Identity 2 - Right-hand Side (RHS)

Calculate the gradient of the divergence of F, which is 鈭�(div F), and the Laplacian of F, which is 螖F. Find the divergence of F: \(\operatorname{div} F = \frac{\partial F鈧亇{\partial x} + \frac{\partial F鈧倉{\partial y} + \frac{\partial F鈧儅{\partial z}\) Compute the gradient of div F: \(\nabla(\operatorname{div} F) = \left(\frac{\partial}{\partial x}\left(\operatorname{div} F\right), \frac{\partial}{\partial y}\left(\operatorname{div} F\right), \frac{\partial}{\partial z}\left(\operatorname{div} F\right)\right)\) Calculate the Laplacian of F: \(\Delta F = \left(\frac{\partial^2 F鈧亇{\partial x^2} + \frac{\partial^2 F鈧亇{\partial y^2} + \frac{\partial^2 F鈧亇{\partial z^2}, \frac{\partial^2 F鈧倉{\partial x^2} + \frac{\partial^2 F鈧倉{\partial y^2} + \frac{\partial^2 F鈧倉{\partial z^2},\frac{\partial^2 F鈧儅{\partial x^2} + \frac{\partial^2 F鈧儅{\partial y^2} + \frac{\partial^2 F鈧儅{\partial z^2}\right)\) Finally, subtract the two results: \(\nabla(\operatorname{div} F) - \Delta F\) Compare the two sides: \(\operatorname{rot}(\operatorname{rot} F) = \nabla(\operatorname{div} F) - \Delta F\) and verify that they are equal, thereby proving the second identity.

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Key Concepts

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

Kreuzprodukt
Understanding the Kreuzprodukt, or cross product, is essential in vector calculus. The cross product of two vectors, say \(\boldsymbol{F}\) and \(\boldsymbol{G}\), results in a third vector that is perpendicular to both original vectors. This operation is typically calculated using a determinant.
  • The determinant involves the unit vectors \( \hat{i}, \hat{j}, \hat{k} \), and the components of the vectors \( \boldsymbol{F} = (F_1, F_2, F_3) \) and \( \boldsymbol{G} = (G_1, G_2, G_3) \).
The formula for the cross product is:\[\boldsymbol{F} \times \boldsymbol{G} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ F_1 & F_2 & F_3 \ G_1 & G_2 & G_3 \ \end{vmatrix}\]This expands into:\[\begin{align*}\hat{i}(F鈧侴鈧� - F鈧僄鈧�) & - \hat{j}(F鈧丟鈧� - F鈧僄鈧�) + \hat{k}(F鈧丟鈧� - F鈧侴鈧�)\end{align*}\]The result is a vector in three-dimensional space. Cross products are useful in finding vector perpendicularity, torque in physics, and determining areas parallelogram spanned by two vectors.
Divergenz
The Divergenz, or divergence, measures the magnitude of a vector field's source or sink at a given point. For a vector field \( \boldsymbol{V} = (V_1, V_2, V_3) \), its divergence is calculated as:\[\operatorname{div} \boldsymbol{V} = \frac{\partial V_1}{\partial x} + \frac{\partial V_2}{\partial y} + \frac{\partial V_3}{\partial z}\]
  • This operation tells us how much the vector field spreads out or converges at a point.
  • If the divergence is positive, more vector lines are emanating from the point, suggesting a source.
  • If negative, the point acts as a sink.
This operation appears in many physics applications, such as fluid dynamics and electromagnetism, where it describes the flow out of a certain volume. So, if a vector field has zero divergence everywhere, it is considered incompressible like an ideal fluid.
Rotationsoperator
The Rotationsoperator, commonly known as the curl, indicates the rotation or the swirling motion of a vector field. For a vector field \( \boldsymbol{F} = (F_1, F_2, F_3) \), its curl is given by:\[\operatorname{rot} \boldsymbol{F} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ F_1 & F_2 & F_3 \ \end{vmatrix}\]
  • This results in another vector that describes the axis and the amount of twirl at each point.
  • When the curl of a field is zero, the field is said to be irrotational.
Essentially, the curl measures the circulation density of a field. It's fundamental in describing rotational motion in fluid flow or electromagnetic fields, where it helps determine how a field lines circulate around a point.
Laplace-Operator
The Laplace-Operator, often denoted by \( \Delta \), is a second-order differential operator crucial in the analysis of various fields such as physics and engineering. It is essentially the divergence of the gradient of a scalar function. For a vector field \( \boldsymbol{F} = (F_1, F_2, F_3) \), it acts on each component of the vector:\[\Delta \boldsymbol{F} = \left( \Delta F_1, \Delta F_2, \Delta F_3 \right)\]Each component \( \Delta F_i \) is calculated as:\[\Delta F_i = \frac{\partial^2 F_i}{\partial x^2} + \frac{\partial^2 F_i}{\partial y^2} + \frac{\partial^2 F_i}{\partial z^2}\]
  • The Laplacian is crucial for solving partial differential equations, particularly in thermodynamics and quantum mechanics.
  • It often appears in problems involving steady-state heat distribution, electric potential, and wave propagation.
In essence, the Laplace-Operator provides information about the spread or dissipation of a function in space, and acts as a measure of curvature or the "smoothness" of functions.

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Most popular questions from this chapter

L盲sst man den Graphen einer stetig differenzierbaren Funktion \(f:[a, b] \rightarrow \mathbb{R}_{>0}\) um die \(x\)-Achse rotieren, so entsteht eine Rotationsflache im \(\mathbb{R}^{3}\), die durch \(\boldsymbol{\gamma}:(a, b) \times(0,2 \pi) \rightarrow \mathbb{R}^{3} \mathrm{mit}\) $$ \gamma(t, \varphi)=\left(\begin{array}{c} t \\ f(t) \cos (\varphi) \\ f(t) \sin (\varphi) \end{array}\right) $$ parametrisiert ist. = Zeigen Sie, dass der Fl盲cheninhalt der Rotationsfl盲che \(\Gamma\) durch das Integral $$ 2 \pi \int_{a}^{b} f(t) \sqrt{1+\left(f^{\prime}(t)\right)^{2}} \mathrm{~d} t $$ gegeben ist. Nutzen Sie dieses Ergebnis, um die Oberfl盲che eines Torus mit \(R=1\) und \(r=\frac{1}{2}\) (Seite 979) zu berechnen.

Eine Funktion \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\) hei脽t homogen vom Grad \(p>0\), wenn \(f(t \boldsymbol{x})=t^{p} f(\boldsymbol{x})\) gilt. Zeigen Sie: $$ \int_{K} \Delta f(x) \mathrm{d} x=p \int_{\partial K} f(x) \mathrm{d} \mu $$ f眉r eine homogene, zweimal stetig differenzierbare Funktion \(f: \mathbb{R}^{n} \rightarrow \mathbb{R}\), wobei \(K=\left\\{x \in \mathbb{R}^{n} \mid\|x\|<1\right\\}\) die Einheitskugel bezeichnet.

Zeigen Sie mithilfe des Gau脽'schen Satzes die Darstellung $$ \Delta u=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u}{\partial \varphi^{2}}+\frac{\partial^{2} u}{\partial z^{2}} $$ des Laplace-Operators in Zylinderkoordinaten.

Gegeben ist die Fl盲che $$ \Gamma=\left\\{r\left(\begin{array}{c} \cos \varphi \\ \sin \varphi \\ \varphi \end{array}\right) \in \mathbb{R}^{3}: r \in[0,1], \varphi \in(-\pi, \pi)\right\\} $$ Berechnen Sie das Fl盲chenintegral $$ \int_{\Gamma} \operatorname{rot} F \cdot \mathbf{d} \mu $$ f眉r \(F(x)=\left(0,0,\left|x_{3}\right| \sqrt{x_{1}^{2}+x_{2}^{2}}\right)^{\top}\), wobei die Orientierung der Fl盲che durch eine positive dritte Koordinate des Normalenfelds gegeben ist.

Zeigen Sie, dass das Randwertproblem, eine Funktion \(v \in C^{2}(D) \cap C^{1}(\bar{D})\) zu bestimmen mit $$ \Delta v-v=0 \quad \text { in } D $$ und \(\frac{\partial v}{\partial v}=0\) auf \(\partial D\), nur die L枚sung \(v(\boldsymbol{x})=0\) besitzt. Dabei sei \(D \subseteq \mathbb{R}^{n}\) ein Gebiet, das eine Anwendung des Gau脽 'schen Satzes erlaubt.
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