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Chapter 14: Problem 3

Berechnen Sie die partiellen Ableitungen \(f_{x x}, f_{x y}, f_{y x}\) und \(f_{y y}\) der Funktion $$ z=R^{2}-x^{2}-y^{2} $$

Short Answer

Expert verified
The second order partial derivatives are \( f_{xx} = -2 \), \( f_{xy} = 0 \), \( f_{yx} = 0 \) and \( f_{yy} = -2 \)

Step by step solution

01

Compute \( f_{x} \)

Differentiate the given function once w.r.t variable \(x\). All the terms that are not involving \(x\) are treated as constants, so \(R^2\) gives 0 and \(-y^2\) gives 0. Differentiation of \(-x^2\) w.r.t \(x\) is \(-2x\). So, \( f_{x} = \frac{d}{dx} (R^2 - x^2 - y^2) = -2x \)
02

Compute \( f_{xx} \)

Differentiate \( f_{x} \) once again w.r.t \(x\) to get \( f_{xx} \). So, \( f_{xx} = \frac{d}{dx} (-2x) = -2 \)
03

Compute \( f_{y} \)

Differentiate the given function once w.r.t variable \(y\). All the terms not involving \(y\) are treated as constants, so \(R^2\) gives 0 and \(-x^2\) gives 0. Differentiation of \(-y^2\) w.r.t \(y\) is \(-2y\). So, \( f_{y} = \frac{d}{dy} (R^2 - x^2 - y^2) = -2y \)
04

Compute \( f_{xy} = f_{yx} \)

Differentiate \( f_{x} \) w.r.t \(y\) to get \( f_{xy} \) and differentiate \( f_{y} \) w.r.t \(x\) to get \( f_{yx} \). As there is no \(y\) term in \( f_{x} \) and no \(x\) term in \( f_{y} \), both these second order partial derivatives are equal to 0. So, \( f_{xy} = f_{yx} = 0 \)
05

Compute \( f_{yy} \)

Differentiate \( f_{y} \) once again w.r.t \(y\) to get \( f_{yy} \). So, \( f_{yy} = \frac{d}{dy} (-2y) = -2 \)

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Most popular questions from this chapter

A Bestimmen Sie die Linien gleicher Höhe, die den Abstand 0,5 von der \(x-y\)-Ebene haben, für die Flächen a) \(z=\sqrt{1-\frac{x^{2}}{4}-\frac{y^{2}}{9}}\) b) \(z=-x-2 y+2\) Geben Sie die Funktionsgleichungen der zugehörigen Höhenlinien an.

A Bilden Sie die partiellen Ableitungen nach \(x, y\) und ggf. nach \(z\) von den Funktionen a) \(f(x, y)=\sin x+\cos y\) b) \(f(x, y)=x^{2} \sqrt{1-y^{2}}\) c) \(f(x, y)=e^{-\left(x^{2}+y^{2}\right)}\) d) \(f(x, y, z)=x y z+x y+z\)

B Berechnen Sie die Steigung der Tangente in \(x\) - und \(y\)-Richtung für die Fläche \(z=x^{2}+y^{2}\) im Punkt \(P=(0,1)\)

B Berechnen Sie das totale Differential für die Funktionen a) \(z=\sqrt{1-x^{2}-y^{2}}\) b) \(z=x^{2}+y^{2}\) c) \(f(x, y, z)=\frac{1}{\sqrt{x^{2}+y^{2}+z^{2}}}\)

Von den skalaren Feldern \(\varphi(x, y)\) sind der Gradient und die Höhenlinien zu berechnen. \(\varphi\) beschreibt eine Fläche. a) \(\varphi=-x-2 y+2\) b) \(\varphi=\sqrt{1-\frac{x^{2}}{4}-\frac{y^{2}}{9}}\) c) \(\varphi=\frac{10}{\sqrt{x^{2}+y^{2}}}\)
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