Problem 17 - Calculus Early Transcendentals: Pearson New International Edition

Question

Show that $$ \nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}} $$

Step-by-Step Solution

Answer

The gradient of \( \frac{f}{g} \) is \( \frac{g \nabla f - f \nabla g}{g^2} \), confirmed via the quotient rule.

Step 1: Understand the Formula

The exercise asks us to demonstrate a formula involving the gradient of a quotient of two functions. We are given the expression \( \frac{f}{g} \) and need to find its gradient \( abla \left( \frac{f}{g} \right) \) using the provided formula.

Step 2: Apply the Quotient Rule

The formula can be understood by analogy to the derivative of a quotient, where the gradient of a quotient can be expressed as: \( abla \left( \frac{f}{g} \right) = \frac{g abla f - f abla g}{g^2} \). This is similar to the derivative of a quotient \( \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v u' - u v'}{v^2} \).

Step 3: Define the Gradient Components

If \( f \) and \( g \) are scalar fields, then their gradients \( abla f \) and \( abla g \) are vector fields composed of partial derivatives with respect to \( x, y, z, \ldots \). Let \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \ldots \right) \) and \( abla g = \left( \frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \ldots \right) \).

Step 4: Substitute the Gradients into the Formula

Plug in the gradients of \( f \) and \( g \) into the given formula. Replace \( abla f \) and \( abla g \) with their respective vector components, resulting in the expression: \( \frac{g abla f - f abla g}{g^2} = \frac{g \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \ldots \right) - f \left(\frac{\partial g}{\partial x}, \frac{\partial g}{\partial y}, \ldots \right)}{g^2} \).

Step 5: Verify Each Component of the Gradient

Each component of the gradient of \( \frac{f}{g} \) is a derivative, so verify the formula for each component \( x, y, z, \ldots \). For each dimension, the partial derivative will be \( \frac{g \frac{\partial f}{\partial x} - f \frac{\partial g}{\partial x}}{g^2} \), similar for other dimensions. This confirms the formula is applied correctly.

Step 6: Conclude the Validity

Once the component-wise verification is done, it is confirmed that the provided formula \( abla \left( \frac{f}{g} \right) = \frac{g abla f - f abla g}{g^2} \) is mathematically correct for the gradient of a quotient.

Key Concepts

Gradient vector field

The concept of a gradient vector field is essential in understanding how a function changes in space. For a multivariable function, the gradient is a vector that points in the direction of the greatest rate of increase of the function and whose magnitude is the rate of increase in that direction. In simple terms, if you were hiking on a mountain, the gradient points uphill and tells you how steep the climb is.

A gradient vector field consists of gradients of a scalar field at each point in space, forming a vector field. If the scalar function is denoted by \(f\), its gradient is expressed as \(abla f\), often written as:

• \(abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}, \ldots \right)\)

Here, every partial derivative is a component of the gradient vector, describing how \(f\) changes concerning each coordinate. Understanding gradient fields helps in visualizing how functions behave and guides in maximizing or minimizing them in fields like physics and engineering.

Quotient rule in calculus

The quotient rule in calculus is a technique for finding the derivative (or rate of change) of a quotient of two functions. It is vital when dealing with problems involving division of functions, as it tells us how the ratio of two functions varies.

For a quotient \( \frac{u}{v} \) where both \(u\) and \(v\) are functions, the derivative is given by:

• \(\frac{d}{dx}\left( \frac{u}{v} \right) = \frac{v \cdot u' - u \cdot v'}{v^2} \)

Similarly, when applying this to gradients, a similar rule holds when computing the gradient of a quotient \( abla \left( \frac{f}{g} \right) \). It follows an analogous form:

• \(abla \left( \frac{f}{g} \right) = \frac{g abla f - f abla g}{g^2} \)

This rule is incredibly useful for vector fields, making it simpler to handle complex expressions involving scalar fields that are divided. It relies on understanding each component's partial derivatives and ensuring that the order of differentiation is correctly handled.

Partial derivatives

Partial derivatives are a cornerstone of calculus for functions of several variables. They measure how a function changes as its input variables are varied individually, holding the other variables constant. This concept is handy when analyzing functions with multiple inputs, providing insights into each variable's influence.

For a function \( f(x, y, z, \ldots) \), the partial derivative with respect to a variable, say \( x \), is denoted as \( \frac{\partial f}{\partial x} \). It captures the rate of change of \( f \) with respect to \( x \), treating other variables like \( y \) and \( z \) as constants.

• \( \frac{\partial f}{\partial x} \) implies how \( f \) shifts when \( x \) is altered a small amount
• \( \frac{\partial f}{\partial y} \) looks at the change when \( y \) varies

In gradient computations, such partial derivatives are key components of the gradient vector field \( abla f \). Mastering partial derivatives lets us smoothly explore and interpret multidimensional spaces.