Problem 28
Question
Suppose \(f\) is a differentiable function of \(x, y\), and \(z\) and \(u=f(x, y, z) .\) Then if \(x=r \sin \phi \cos \theta, y=r \sin \phi \sin \theta\), and \(z=r \cos \phi\), express \(\partial u / \partial r, \partial u / \partial \phi\), and \(\partial u / \partial \theta\) in terms of \(\partial u / \partial x, \partial u / \partial y\), and \(\partial u / \partial z\).
Step-by-Step Solution
Verified Answer
\( \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \sin \phi \cos \theta + \frac{\partial u}{\partial y} \sin \phi \sin \theta + \frac{\partial u}{\partial z} \cos \phi \ \ \ \frac{\partial u}{\partial \phi} = \frac{\partial u}{\partial x} r \cos \phi \cos \theta + \frac{\partial u}{\partial y} r \cos \phi \sin \theta + \frac{\partial u}{\partial z} (-r \sin \phi) \ \ \ \frac{\partial u}{\partial \theta} = \frac{\partial u}{\partial x} (-r \sin \phi \sin \theta) + \frac{\partial u}{\partial y} r \sin \phi \cos \theta \)
1Step 1 - Recognize the transformation
Identify the given coordinate transformation: \[ x = r \sin \phi \cos \theta, \ y = r \sin \phi \sin \theta, \ z = r \cos \phi \]
2Step 2 - Apply the chain rule for partial derivatives
Use the chain rule to express the partial derivatives of \(u\) with respect to \(r\), \(\phi\), and \(\theta\) in terms of partial derivatives of \(u\) with respect to \(x\), \(y\), and \(z\):\[ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial r} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial r} \ \frac{\partial u}{\partial \phi} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial \phi} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial \phi} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial \phi} \ \frac{\partial u}{\partial \theta} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial \theta} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial \theta} \]
3Step 3 - Compute partial derivatives of \(x\), \(y\), and \(z\) with respect to the new coordinates
Calculate the following:\[ \frac{\partial x}{\partial r} = \sin \phi \cos \theta \ \ \ \frac{\partial y}{\partial r} = \sin \phi \sin \theta \ \ \ \frac{\partial z}{\partial r} = \cos \phi \]\[ \frac{\partial x}{\partial \phi} = r \cos \phi \cos \theta \ \ \ \frac{\partial y}{\partial \phi} = r \cos \phi \sin \theta \ \ \ \frac{\partial z}{\partial \phi} = -r \sin \phi \]\[ \frac{\partial x}{\partial \theta} = -r \sin \phi \sin \theta \ \ \ \frac{\partial y}{\partial \theta} = r \sin \phi \cos \theta \ \ \ \frac{\partial z}{\partial \theta} = 0 \]
4Step 4 - Substitute the partial derivatives
Substitute the calculated partial derivatives into the chain rule expressions:\[ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \sin \phi \cos \theta + \frac{\partial u}{\partial y} \sin \phi \sin \theta + \frac{\partial u}{\partial z} \cos \phi \ \]\[ \frac{\partial u}{\partial \phi} = \frac{\partial u}{\partial x} r \cos \phi \cos \theta + \frac{\partial u}{\partial y} r \cos \phi \sin \theta + \frac{\partial u}{\partial z} (-r \sin \phi) \]\[ \frac{\partial u}{\partial \theta} = \frac{\partial u}{\partial x} (-r \sin \phi \sin \theta) + \frac{\partial u}{\partial y} r \sin \phi \cos \theta \]
Key Concepts
Partial DerivativesChain RuleCoordinate Transformation
Partial Derivatives
In multivariable calculus, we often deal with functions that depend on multiple variables. To understand how these functions change, we use partial derivatives. A partial derivative measures the rate at which a function changes as one of its input variables changes while keeping the other input variables constant.
For instance, if we have a function \( u = f(x, y, z) \), the partial derivative of \( u \) with respect to \( x \) is denoted as \( \frac{\partial u}{\partial x} \). This tells us how \( u \) changes as \( x \) changes, with \( y \) and \( z \) remaining unchanged.
Partial derivatives are foundational in understanding how multidimensional functions behave. They are also essential in optimizing functions of multiple variables and in calculating gradients, divergence, and curl in vector calculus.
For instance, if we have a function \( u = f(x, y, z) \), the partial derivative of \( u \) with respect to \( x \) is denoted as \( \frac{\partial u}{\partial x} \). This tells us how \( u \) changes as \( x \) changes, with \( y \) and \( z \) remaining unchanged.
Partial derivatives are foundational in understanding how multidimensional functions behave. They are also essential in optimizing functions of multiple variables and in calculating gradients, divergence, and curl in vector calculus.
Chain Rule
The chain rule is a formula used to compute the derivative of a composite function. When dealing with multivariable functions, the chain rule helps to find how a function changes with respect to new variables that are themselves functions of other variables.
Given \( u = f(x, y, z) \) and new variables \( x = g(r, \phi, \theta) \), \( y = h(r, \phi, \theta) \), and \( z = k(r, \phi, \theta) \), the chain rule tells us how to find \( \frac{\partial u}{\partial r} \), \( \frac{\partial u}{\partial \phi} \), and \( \frac{\partial u}{\partial \theta} \) in terms of \( \frac{\partial u}{\partial x} \), \( \frac{\partial u}{\partial y} \), and \( \frac{\partial u}{\partial z} \).
The chain rule formula for \( r \) is:
\[ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial r} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial r} \]
Similar formulas apply for \( \phi \) and \( \theta \). This method involves taking partial derivatives of the new variables \( x, y, \) and \( z \) with respect to \( r, \phi, \) and \( \theta \), and then multiplying and summing them appropriately.
Given \( u = f(x, y, z) \) and new variables \( x = g(r, \phi, \theta) \), \( y = h(r, \phi, \theta) \), and \( z = k(r, \phi, \theta) \), the chain rule tells us how to find \( \frac{\partial u}{\partial r} \), \( \frac{\partial u}{\partial \phi} \), and \( \frac{\partial u}{\partial \theta} \) in terms of \( \frac{\partial u}{\partial x} \), \( \frac{\partial u}{\partial y} \), and \( \frac{\partial u}{\partial z} \).
The chain rule formula for \( r \) is:
\[ \frac{\partial u}{\partial r} = \frac{\partial u}{\partial x} \frac{\partial x}{\partial r} + \frac{\partial u}{\partial y} \frac{\partial y}{\partial r} + \frac{\partial u}{\partial z} \frac{\partial z}{\partial r} \]
Similar formulas apply for \( \phi \) and \( \theta \). This method involves taking partial derivatives of the new variables \( x, y, \) and \( z \) with respect to \( r, \phi, \) and \( \theta \), and then multiplying and summing them appropriately.
Coordinate Transformation
A coordinate transformation involves changing from one set of coordinates to another. Common transformations include Cartesian to polar or spherical coordinates. These transformations are crucial in simplifying problems involving symmetry or integrating functions over complex regions.
In the given exercise, we transform Cartesian coordinates \( (x, y, z) \) to spherical coordinates \( (r, \phi, \theta) \). The transformations are:
These formulas relate the spherical coordinates to the Cartesian ones. To find partial derivatives in the new coordinate system, we use these transformations in conjunction with the chain rule.
By computing partial derivatives such as \( \frac{\partial x}{\partial r} \), \( \frac{\partial y}{\partial r} \), and \( \frac{\partial z}{\partial r} \), we can express derivatives like \( \frac{\partial u}{\partial r} \) in terms of \( \frac{\partial u}{\partial x} \), \( \frac{\partial u}{\partial y} \), and \( \frac{\partial u}{\partial z} \).
This approach simplifies complex integrals and differential equations by leveraging the symmetry and structure provided by the new coordinate system.
In the given exercise, we transform Cartesian coordinates \( (x, y, z) \) to spherical coordinates \( (r, \phi, \theta) \). The transformations are:
- \( x = r \sin \phi \cos \theta \)
- \( y = r \sin \phi \sin \theta \)
- \( z = r \cos \phi \)
These formulas relate the spherical coordinates to the Cartesian ones. To find partial derivatives in the new coordinate system, we use these transformations in conjunction with the chain rule.
By computing partial derivatives such as \( \frac{\partial x}{\partial r} \), \( \frac{\partial y}{\partial r} \), and \( \frac{\partial z}{\partial r} \), we can express derivatives like \( \frac{\partial u}{\partial r} \) in terms of \( \frac{\partial u}{\partial x} \), \( \frac{\partial u}{\partial y} \), and \( \frac{\partial u}{\partial z} \).
This approach simplifies complex integrals and differential equations by leveraging the symmetry and structure provided by the new coordinate system.
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