Problem 64 - Calculus Early Transcendentals

Question

Consider the function \(F(x, y, z)=e^{x y z}\) a. Write \(F\) as a composite function \(f^{\circ} g,\) where \(f\) is a function of one variable and \(g\) is a function of three variables. b. Relate \(\nabla F\) to \(\nabla g\)

Step-by-Step Solution

Answer

Question: Write the function \(F(x, y, z) = e^{xyz}\) as a composite function of two other functions. Relate the gradients of F and the function you used in the composition. Solution: Part a: The function F can be written as a composite function with \(f(u) = e^u\) and \(g(x, y, z) = x y z\). Part b: The gradient of F is related to the gradient of g by \(\nabla F(x, y, z) = e^{x y z}\nabla g(x, y, z)\).

Step 1: Part a: Writing F as a composite function

Firstly, let's define the function \(F(x, y, z) = e^{xyz}\). A composite function can be written as \(F=f\circ g\), in which f is a function of one variable and g is a function of three variables. The idea is to express F(x, y, z) in terms of two functions: one function that handles the exponentiation and one function that manages the multiplication of the three variables. We can define the functions f(u) and g(x, y, z) as follows: - \(f(u) = e^u\), where u is a single variable - \(g(x, y, z) = x y z\), a function of three variables Now, by composing f and g, we get: \(F(x, y, z) = f \circ g (x, y, z) = f(g(x, y, z)) = f(x y z) = e^{x y z}\) Thus, F can be written as a composite function with \(f(u) = e^u\) and \(g(x, y, z) = x y z\).

Step 2: Part b: Relating the gradients of F and g

Gradient, denoted by \(\nabla\), is a vector containing the partial derivatives of a multivariable function. We need to find the gradient of F and g and then relate them. First, we will calculate the gradient of F(x, y, z): \(\nabla F(x, y, z) = \left(\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z}\right)\) Calculating the partial derivatives: - \(\frac{\partial F}{\partial x} = e^{x y z} yz\) (as \(e^{xyz}\) will be treated as constant with respect to x) - \(\frac{\partial F}{\partial y} = e^{x y z} xz\) (as \(e^{xyz}\) will be treated as constant with respect to y) - \(\frac{\partial F}{\partial z} = e^{x y z} xy\) (as \(e^{xyz}\) will be treated as constant with respect to z) So, the gradient of F is: \(\nabla F(x, y, z) = (e^{x y z}yz, e^{x y z}xz, e^{x y z}xy)\) Now, let's calculate the gradient of g(x, y, z): \(\nabla g(x, y, z) = \left(\frac{\partial g}{\partial x},\frac{\partial g}{\partial y},\frac{\partial g}{\partial z}\right)\) Calculating the partial derivatives: - \(\frac{\partial g}{\partial x} = yz\) (as xy will be treated as constant with respect to x) - \(\frac{\partial g}{\partial y} = xz\) (as xy will be treated as constant with respect to y) - \(\frac{\partial g}{\partial z} = xy\) (as xy will be treated as constant with respect to z) So, the gradient of g is: \(\nabla g(x, y, z) = (yz, xz, xy)\) Comparing the gradients of F and g, we can observe the following relationship: \(\nabla F(x, y, z) = e^{x y z}\nabla g(x, y, z)\) That is, the gradient of F is equal to the gradient of g times the exponential of the product of x, y, and z.

Key Concepts

Composite Function

A composite function is a function that is made by combining two or more functions. In the context of multivariable calculus, it can be seen when we have a function of several variables expressed through other functions. For example, consider the function

• F(x, y, z) = e^{xyz}

To express F as a composite function, we need to identify two functions: one that performs an operation on the input variables (in this case, multiplying them together) and another that applies another operation on the result of the first function (exponentiating the product). Here's how we achieve this:

• Define a function of three variables: g(x, y, z) = xyz
• Define a function of one variable: f(u) = e^u, where u = xyz
• Combine them: F(x, y, z) = f(g(x, y, z)) = e^{xyz}

By structuring F in this way, we recognize the composite nature of the function, with one function feeding into another.

Gradient

The gradient is an important concept in multivariable calculus that provides information about the direction and rate of change of a function. It's essentially a vector that contains the partial derivatives of a function with respect to its variables. For a function F(x, y, z), the gradient, abla F, has the following components:

• The partial derivative with respect to x, \( \frac{\partial F}{\partial x} \)
• The partial derivative with respect to y, \( \frac{\partial F}{\partial y} \)
• The partial derivative with respect to z, \( \frac{\partial F}{\partial z} \)

In our specific example, for F(x, y, z) = e^{xyz}, we determined:

• \( \frac{\partial F}{\partial x} = e^{xyz} yz \)
• \( \frac{\partial F}{\partial y} = e^{xyz} xz \)
• \( \frac{\partial F}{\partial z} = e^{xyz} xy \)

This sets up the gradient as a vector: \( abla F(x, y, z) = (e^{xyz} yz, e^{xyz} xz, e^{xyz} xy) \).
The gradient points in the direction of the steepest ascent of the function F and its magnitude gives the rate of that increase.

Partial Derivatives

Partial derivatives are used to see how a multivariable function changes as one of the variables is varied, while the others are held constant. They are a fundamental aspect of the gradient and are crucial for understanding the behavior and geometry of multivariable functions.When we calculate the partial derivative of a function like \( F(x, y, z) = e^{xyz} \), we take each variable in isolation and compute how F changes when just that variable changes:

• \( \frac{\partial F}{\partial x} = e^{xyz} yz \) This derivative examines the change of F with respect to x, keeping y and z constant.
• \( \frac{\partial F}{\partial y} = e^{xyz} xz \) This derivative focuses on the change with respect to y.
• \( \frac{\partial F}{\partial z} = e^{xyz} xy \) And this one looks at the change with respect to z.

Understanding partial derivatives allows us to construct the gradient and further analyze the behavior of complex functions, whether for optimization, finding extrema, or mapping the function's behavior inside its domain.