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Chapter 5: Problem 16

(a) Es ist die Funktion $$ f:] 0, \infty\left[\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}, \quad f(x, y, z)=x^{y}+\sin \left(x y z^{2}\right)\right. $$ gegeben. Berechnen Sie den Gradienten von \(f\) und geben Sie f眉r die Abbildung \(\mathbf{g}(x, y, z)=\operatorname{grad}_{(x, y, z)} f\) den maximalen Definitionsbereich an. (b) Berechnen Sie die Ableitungsmatrix der Abbildung \(\mathrm{g}\).

Short Answer

Expert verified
The gradient is \(\mathbf{g}(x, y, z) = (g_1, g_2, g_3)\) with the domain \(x > 0\). The Jacobian matrix is derived from the gradient components.

Step by step solution

01

Understanding the Function

The function given is \(f(x, y, z) = x^{y} + \sin(xyz^2)\), which takes three inputs \(x\), \(y\), \(z\) with \(x > 0\). Our task is to find the gradient of this function, which involves calculating partial derivatives with respect to each variable.
02

Calculate Partial Derivative with Respect to x

The partial derivative of \(f\) with respect to \(x\) is calculated by differentiating \(x^y\) using the power rule and \(\sin(xyz^2)\) using the chain rule:\[\frac{\partial f}{\partial x} = yx^{y-1} + yz^2\cos(xyz^2)\]
03

Calculate Partial Derivative with Respect to y

For \(\frac{\partial f}{\partial y}\), differentiate \(x^y\) with respect to \(y\) and apply the chain rule to \(\sin(xyz^2)\):\[\frac{\partial f}{\partial y} = x^y\ln(x) + xz^2\cos(xyz^2)\]
04

Calculate Partial Derivative with Respect to z

The derivative of \(f\) with respect to \(z\) is found by applying the chain rule to \(\sin(xyz^2)\):\[\frac{\partial f}{\partial z} = 2xyz\cos(xyz^2)\]
05

Formulate the Gradient

The gradient \(\operatorname{grad}_{(x, y, z)} f\) is a vector consisting of the partial derivatives:\[\mathbf{g}(x, y, z) = \left(yx^{y-1} + yz^2\cos(xyz^2), x^y\ln(x) + xz^2\cos(xyz^2), 2xyz\cos(xyz^2)\right)\]
06

Determine the Maximal Domain

Since \(f\) involves logarithms and power expressions without negative exponents, \(x > 0\) is required for the \(\ln(x)\)\ term. The maximal domain of \(\mathbf{g}(x, y, z)\) is therefore \(x > 0\), \(y \in \mathbb{R}\), \(z \in \mathbb{R}\).
07

Calculate the Jacobian of g

The Jacobian matrix of \(\mathbf{g}\), which is the derivative matrix of the gradient, requires calculating second derivatives and arranging them in a matrix. Let \(\mathbf{g} = (g_1, g_2, g_3)\):\[\begin{bmatrix}\frac{\partial g_1}{\partial x} & \frac{\partial g_1}{\partial y} & \frac{\partial g_1}{\partial z} \\frac{\partial g_2}{\partial x} & \frac{\partial g_2}{\partial y} & \frac{\partial g_2}{\partial z} \\frac{\partial g_3}{\partial x} & \frac{\partial g_3}{\partial y} & \frac{\partial g_3}{\partial z}\end{bmatrix}\]
08

Find Specific Partial Derivatives for the Jacobian

Calculate each element of the Jacobian matrix by taking derivatives of components in \(\mathbf{g}(x, y, z)\). The elements involve more complex derivatives but follow similar differentiation rules and the use of the product and chain rules.
09

Compile the Jacobian Matrix

Combine the specific partial derivatives to construct the Jacobian matrix. Ensure each element corresponds to the proper derivative as identified previously.

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Key Concepts

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

Partial Derivatives
Partial derivatives are a fundamental tool in calculus for dealing with functions of multiple variables. They represent how a function changes as one particular variable changes while the others remain constant. This is important when working with functions like \(f(x, y, z) = x^{y} + \sin(xyz^2)\). To find the gradient of this function, we need to compute partial derivatives with respect to \(x\), \(y\), and \(z\).
  • For \(x\), use the power rule on \(x^y\) and the chain rule on \(\sin(xyz^2)\): \(\frac{\partial f}{\partial x} = yx^{y-1} + yz^2\cos(xyz^2)\).
  • For \(y\), differentiate using the natural logarithm and chain rule: \(\frac{\partial f}{\partial y} = x^y \ln(x) + xz^2 \cos(xyz^2)\).
  • For \(z\), apply the chain rule: \(\frac{\partial f}{\partial z} = 2xyz \cos(xyz^2)\).
These derivatives are then compiled into a vector, known as the gradient, providing essential insights into the behavior of \(f\) around any point \((x, y, z)\).
Jacobian Matrix
The Jacobian matrix is a mathematical framework used to describe the derivative of a function that outputs a vector. It is particularly useful for transformations between coordinate systems or for analyzing dynamic systems. For the function \(\mathbf{g}(x, y, z)\), which is the gradient of \(f\), the Jacobian is a matrix of partial derivatives of \(g_1, g_2, g_3\) components.
  • Each entry in the Jacobian matrix represents a partial derivative of one component with respect to one input variable.
  • The matrix is structured as follows: \[ \begin{bmatrix} \frac{\partial g_1}{\partial x} & \frac{\partial g_1}{\partial y} & \frac{\partial g_1}{\partial z} \ \frac{\partial g_2}{\partial x} & \frac{\partial g_2}{\partial y} & \frac{\partial g_2}{\partial z} \ \frac{\partial g_3}{\partial x} & \frac{\partial g_3}{\partial y} & \frac{\partial g_3}{\partial z} \end{bmatrix} \]
Constructing the Jacobian requires computing these partial derivatives and arranging them into this matrix form. This matrix is crucial because it provides a linear approximation of the function \(\mathbf{g}\) around each point, facilitating the study of its behavior and effects between inputs and outputs.
Maximal Domain
When examining functions like the given \(f(x, y, z) = x^{y} + \sin(xyz^2)\), it's essential to define where the function is valid 鈥� its domain. Especially when dealing with logarithmic expressions and powers, these considerations become crucial.
  • In this function, \(x > 0\) is crucial because \(\ln(x)\) only exists for positive \(x\). This condition ensures that the logarithmic operation within \(\mathbf{g}(x, y, z)\) is valid.
  • Other variables, \(y\) and \(z\), can take any real number values, given \(y \in \mathbb{R}\) and \(z \in \mathbb{R}\), since no restrictions like those on \(x\) apply.
Thus, the maximal domain for the gradient \(\mathbf{g}(x, y, z)\) becomes \(x > 0\), \(y \in \mathbb{R}\), \(z \in \mathbb{R}\). Understanding the domain is crucial as it defines the range of acceptable inputs for the calculations and analyses involving these functions, emphasizing specific limitations intrinsic to the function's structure.

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Most popular questions from this chapter

Berechnen Sie das TAYLOR-Polynom 2. Grades der Funktion \(f(x, y, z)=x y z^{3}\) am Entwicklungspunkt \(\mathbf{x}_{0}=(1,1,1)^{T}\).

Approximieren Sie die Abbildung \(\mathbf{f}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}\) $$ \mathbf{f}(\mathbf{x})=\left(\begin{array}{c} x y \sin z \\ x+y^{3} z \end{array}\right) $$ durch eine lineare Abbildung um den Punkt \(\mathbf{x}_{0}=(0,0,0)^{T}\).

Betrachten Sie die Funktion $$ f: \mathbb{R}^{2} \rightarrow \mathbb{R}, \quad f(x, y)=x^{2}+x+x y+y^{3} $$ (a) Bestimmen Sie alle lokalen und globalen Extrem- bzw. Sattelstellen der Funktion auf \(\mathbb{R}^{2}\) und geben Sie die TAYLOR-Polynome 2. Grades von \(f\) an den gefundenen Stellen an. (b) Charakterisieren Sie s盲mtliche lokalen wie globalen Extremalstellen von \(f\) auf der Menge \(H=\left\\{(x, y) \in \mathbb{R}^{2} \mid 2 x+y \leq 0\right\\}\).

Gegeben sei eine Funktion $$ f: \mathbb{R}^{2} \rightarrow \mathbb{R} \text { mit } \quad f^{\prime}(x, y)=\left(\sin \left(x^{2}\right) y\right) $$ sowie die Abbildung $$ \left.\mathbf{g}: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \quad \text { mit } \quad \mathbf{g}(x, y)=(x y, x+2 y)^{T}\right) $$ Ermitteln Sie f眉r die Funktion $$ h: \mathbb{R}^{2} \rightarrow \mathbb{R} \text { mit } h=f \circ \mathbf{g} $$ im Punkt \((1,0)\) die Richtung des st盲rksten Anstiegs.

(a) Gegeben ist die Matrix $$ M=\left(\begin{array}{ll} 2 & 1 \\ 1 & 3 \\ 2 & 0 \end{array}\right), \text { der Vektor } \mathbf{y}=\left(\begin{array}{l} 1 \\ 3 \\ 2 \end{array}\right) $$ und die Funktion $$ F: \mathbb{R}^{2} \rightarrow \mathbb{R} ;, \quad F(x, y)=\left\|M\left(\begin{array}{c} x \\ y \end{array}\right)-\mathbf{y}\right\|_{2}^{2} $$ Berechnen Sie den Gradienten und die HESSE-Matrix von \(F .\) (b) Bestimmen Sie s盲mtliche lokalen Extremalstellen und entscheiden Sie, ob es sich um Maxima oder Minima handelt.
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