To apply the chain rule to find \(\frac{\partial U}{\partial s}\), consider the following equations: 1. \(U = f(x, y, z) = x^3 + xyz + z^2\) 2. \(x = g(r, s, t) = e^{2r} \cos(s)\) 3. \(y = h(r, s, t) = e^{2r} \sin(s)\) 4. \(z = m(r, s, t) = e^r \cos(2s)\) To solve for \(\frac{\partial U}{\partial s}\), you will need to compute the partial derivatives of \(U\) with respect to \(x\), \(y\), and \(z\), and then apply the chain rule to incorporate the derivatives of \(x\), \(y\), and \(z\) with respect to \(s\). This involves computing the following derivatives: - \(\frac{\partial U}{\partial x}\), \(\frac{\partial U}{\partial y}\), \(\frac{\partial U}{\partial z}\) - \(\frac{\partial x}{\partial s}\), \(\frac{\partial y}{\partial s}\), \(\frac{\partial z}{\partial s}\) By applying the chain rule: \[ \frac{\partial U}{\partial s} = \frac{\partial U}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial U}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial U}{\partial z} \cdot \frac{\partial z}{\partial s} \] Each partial derivative should be calculated step-by-step to obtain the final result for \(\frac{\partial U}{\partial s}\).
To apply the chain rule to find \(\frac{\partial U}{\partial s}\), consider the following equations: 1. \(U = f(x, y, z) = x^3 + xyz + z^2\) 2. \(x = g(r, s, t) = e^{2r} \cos(s)\) 3. \(y = h(r, s, t) = e^{2r} \sin(s)\) 4. \(z = m(r, s, t) = e^r \cos(2s)\) To solve for \(\frac{\partial U}{\partial s}\), you will need to compute the partial derivatives of \(U\) with respect to \(x\), \(y\), and \(z\), and then apply the chain rule to incorporate the derivatives of \(x\), \(y\), and \(z\) with respect to \(s\). This involves computing the following derivatives: - \(\frac{\partial U}{\partial x}\), \(\frac{\partial U}{\partial y}\), \(\frac{\partial U}{\partial z}\) - \(\frac{\partial x}{\partial s}\), \(\frac{\partial y}{\partial s}\), \(\frac{\partial z}{\partial s}\) By applying the chain rule: \[ \frac{\partial U}{\partial s} = \frac{\partial U}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial U}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial U}{\partial z} \cdot \frac{\partial z}{\partial s} \] Each partial derivative should be calculated step-by-step to obtain the final result for \(\frac{\partial U}{\partial s}\).
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![To apply the chain rule to find \(\frac{\partial U}{\partial s}\), consider the following equations:
1. \(U = f(x, y, z) = x^3 + xyz + z^2\)
2. \(x = g(r, s, t) = e^{2r} \cos(s)\)
3. \(y = h(r, s, t) = e^{2r} \sin(s)\)
4. \(z = m(r, s, t) = e^r \cos(2s)\)
To solve for \(\frac{\partial U}{\partial s}\), you will need to compute the partial derivatives of \(U\) with respect to \(x\), \(y\), and \(z\), and then apply the chain rule to incorporate the derivatives of \(x\), \(y\), and \(z\) with respect to \(s\). This involves computing the following derivatives:
- \(\frac{\partial U}{\partial x}\), \(\frac{\partial U}{\partial y}\), \(\frac{\partial U}{\partial z}\)
- \(\frac{\partial x}{\partial s}\), \(\frac{\partial y}{\partial s}\), \(\frac{\partial z}{\partial s}\)
By applying the chain rule:
\[
\frac{\partial U}{\partial s} = \frac{\partial U}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial U}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial U}{\partial z} \cdot \frac{\partial z}{\partial s}
\]
Each partial derivative should be calculated step-by-step to obtain the final result for \(\frac{\partial U}{\partial s}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe620d41c-dd63-496b-b84c-cb17355be0e7%2Fa9b812aa-69c5-465e-afe3-539af5fd72d4%2Fm6rrs4h.jpeg&w=3840&q=75)
Transcribed Image Text:To apply the chain rule to find \(\frac{\partial U}{\partial s}\), consider the following equations:
1. \(U = f(x, y, z) = x^3 + xyz + z^2\)
2. \(x = g(r, s, t) = e^{2r} \cos(s)\)
3. \(y = h(r, s, t) = e^{2r} \sin(s)\)
4. \(z = m(r, s, t) = e^r \cos(2s)\)
To solve for \(\frac{\partial U}{\partial s}\), you will need to compute the partial derivatives of \(U\) with respect to \(x\), \(y\), and \(z\), and then apply the chain rule to incorporate the derivatives of \(x\), \(y\), and \(z\) with respect to \(s\). This involves computing the following derivatives:
- \(\frac{\partial U}{\partial x}\), \(\frac{\partial U}{\partial y}\), \(\frac{\partial U}{\partial z}\)
- \(\frac{\partial x}{\partial s}\), \(\frac{\partial y}{\partial s}\), \(\frac{\partial z}{\partial s}\)
By applying the chain rule:
\[
\frac{\partial U}{\partial s} = \frac{\partial U}{\partial x} \cdot \frac{\partial x}{\partial s} + \frac{\partial U}{\partial y} \cdot \frac{\partial y}{\partial s} + \frac{\partial U}{\partial z} \cdot \frac{\partial z}{\partial s}
\]
Each partial derivative should be calculated step-by-step to obtain the final result for \(\frac{\partial U}{\partial s}\).
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