Use the Chain Rule to find aw at w = xycos z; x = t6, y = t³, z = arccost
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please please please solve for the problems provided in the photos below
![### Chain Rule Applications in Multivariable Calculus
In this example, we are given a set of functions and asked to apply the chain rule to find specified partial derivatives.
#### Problem 1: Finding \(\frac{\partial w}{\partial t}\)
Given:
\[ w = xy \cos z \]
\[ x = t^6 \]
\[ y = t^3 \]
\[ z = \arccos t \]
Use the chain rule to find \(\frac{\partial w}{\partial t}\).
#### Problem 2: Finding \(\frac{\partial w}{\partial s}\)
Given:
\[ w = x^2 + y^2 + z^2 \]
\[ x = 10 \sin s \]
\[ y = 10 \cos s \]
\[ z = 9st^2 \]
Use the chain rule to find \(\frac{\partial w}{\partial s}\).
#### Problem 3: Finding the Gradient
Given the function:
\[ f(x, y) = 3x^3 - 4x^2y + y^2 \]
Find the gradient, \(\nabla f\), at the point \((1, -1)\).
#### Summary of Topics
1. **Chain Rule in Multivariable Calculus**: This rule helps in differentiating composite functions. It relates the derivative of a composite function to the derivatives of its constituent functions.
2. **Gradient Calculation**: The gradient of a scalar function is a vector consisting of the function's first partial derivatives with respect to each variable.
### Detailed Explanation of Method
**Chain Rule Application:**
- Identify the independent variable with respect to which the differentiation is to be performed.
- Express the dependent variable in terms of intermediate variables, if necessary.
- Apply the chain rule systematically to differentiate the given function.
**Gradient Calculation:**
- Determine the partial derivatives of the function with respect to each variable.
- Combine these partial derivatives into a vector.
These problems exemplify the practical use of the chain rule and gradient in multivariate functions, foundational concepts in higher-level calculus. Understanding these principles is crucial for solving complex differential equations and analyzing multivariable systems in fields such as physics, engineering, and economics.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9981eaaa-92cc-40db-b668-6adab491a4d6%2F39369e48-27d7-4b97-809b-ca348eab22df%2Fithzwj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Chain Rule Applications in Multivariable Calculus
In this example, we are given a set of functions and asked to apply the chain rule to find specified partial derivatives.
#### Problem 1: Finding \(\frac{\partial w}{\partial t}\)
Given:
\[ w = xy \cos z \]
\[ x = t^6 \]
\[ y = t^3 \]
\[ z = \arccos t \]
Use the chain rule to find \(\frac{\partial w}{\partial t}\).
#### Problem 2: Finding \(\frac{\partial w}{\partial s}\)
Given:
\[ w = x^2 + y^2 + z^2 \]
\[ x = 10 \sin s \]
\[ y = 10 \cos s \]
\[ z = 9st^2 \]
Use the chain rule to find \(\frac{\partial w}{\partial s}\).
#### Problem 3: Finding the Gradient
Given the function:
\[ f(x, y) = 3x^3 - 4x^2y + y^2 \]
Find the gradient, \(\nabla f\), at the point \((1, -1)\).
#### Summary of Topics
1. **Chain Rule in Multivariable Calculus**: This rule helps in differentiating composite functions. It relates the derivative of a composite function to the derivatives of its constituent functions.
2. **Gradient Calculation**: The gradient of a scalar function is a vector consisting of the function's first partial derivatives with respect to each variable.
### Detailed Explanation of Method
**Chain Rule Application:**
- Identify the independent variable with respect to which the differentiation is to be performed.
- Express the dependent variable in terms of intermediate variables, if necessary.
- Apply the chain rule systematically to differentiate the given function.
**Gradient Calculation:**
- Determine the partial derivatives of the function with respect to each variable.
- Combine these partial derivatives into a vector.
These problems exemplify the practical use of the chain rule and gradient in multivariate functions, foundational concepts in higher-level calculus. Understanding these principles is crucial for solving complex differential equations and analyzing multivariable systems in fields such as physics, engineering, and economics.
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