Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the gradient \(\nabla f\). $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}} $$

Step-by-Step Solution

Answer

\( \nabla f = \left( \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}} \right) \)

Step 1: Recognize the formula for the gradient

The gradient of a function \( f(x, y, z) \) is a vector of partial derivatives of \( f \) with respect to \( x \), \( y \), and \( z \). It is represented as \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \).

Step 2: Differentiate with respect to x

To find \( \frac{\partial f}{\partial x} \), apply the chain rule: \( f(x, y, z) = \left( x^2 + y^2 + z^2 \right)^{1/2} \). The derivative of \( \sqrt{x^2+y^2+z^2} \) with respect to \( x \) is \( \frac{x}{\sqrt{x^2+y^2+z^2}} \).

Step 3: Differentiate with respect to y

Similarly, for \( \frac{\partial f}{\partial y} \), apply the derivative and chain rule: \( \frac{y}{\sqrt{x^2+y^2+z^2}} \).

Step 4: Differentiate with respect to z

For \( \frac{\partial f}{\partial z} \), the derivative is \( \frac{z}{\sqrt{x^2+y^2+z^2}} \).

Step 5: Combine to write the gradient

Now collect all partial derivatives into the gradient vector: \( abla f = \left( \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}} \right) \).

Key Concepts

Partial Derivatives

Partial derivatives are fundamental concepts in calculus, especially when dealing with functions of multiple variables. They represent the rate at which a function changes as one of the variables changes, while keeping the other variables constant.
For a function \( f(x, y, z) \), the partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \). This is essentially the derivative of \( f \) concerning \( x \), treating \( y \) and \( z \) as constants.

• To compute \( \frac{\partial f}{\partial x} \) for the function \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \), we apply standard differentiation rules while considering \( y \) and \( z \) as constants.
• Do the same process for \( y \) and \( z \) to find \( \frac{\partial f}{\partial y} \) and \( \frac{\partial f}{\partial z} \).

Understanding partial derivatives is crucial because they form the basis for more complex operations in multivariable calculus, such as gradients and optimization problems.

Chain Rule

The chain rule is a powerful tool in calculus that allows us to differentiate compositions of functions. In the context of multivariable functions, it helps find partial derivatives when one variable is a function of other variables.
Consider the function \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \). When differentiating this function with respect to \( x \), the chain rule is utilized.

• The outer function is \( u^{1/2} \) where \( u = x^2 + y^2 + z^2 \).
• The derivative of the outer function \( u^{1/2} \) with respect to \( u \) is \( \frac{1}{2}u^{-1/2} \).
• We then multiply this by the inner derivative \( \frac{\partial}{\partial x} (x^2 + y^2 + z^2) = 2x \).

This integration of the chain rule allows us to accurately compute the rate of change, displaying its utility in simplifying complex differentiations.

Vector Calculus

Vector calculus involves mathematical operations on vector fields, an essential part of physics and engineering. It deals with quantities that have both magnitude and direction, extending the tools of calculus to vector fields.
One significant operation in vector calculus is finding the gradient, a vector field representing the rate and direction of the quickest increase of a scalar field, like our function \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \).

• The gradient is denoted as \( abla f \), a vector that gives the partial derivatives of the function.
• For our function, the gradient is calculated as \( abla f = \left(\frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}}\right) \).

The gradient direction indicates the direction of the steepest ascent in the function, while its magnitude tells us the rate of increase in that direction.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus into higher dimensions, dealing with functions of several variables. It's essential in fields like physics, engineering, and economics where systems are often described by more than one variable.
In multivariable calculus, we explore the behavior of functions that have multiple inputs and outputs. This includes:

• Partial derivatives, which help us understand how a function changes as each variable changes independently.
• The gradient, which provides a vector pointing in the direction of the greatest rate of increase of a function.
• Using tools like the chain rule to handle composite functions with multiple variables.

These concepts allow for the analysis and problem solving of complex systems, leveraging mathematical principles to tackle real-world scenarios involving multiple interdependent variables.