Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { In Problems 1-10, find the gradient } \nabla f \text {. } $$ $$ f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}} $$

Step-by-Step Solution

Answer

The gradient is \( \left( \frac{x}{\sqrt{x^2+y^2+z^2}}, \frac{y}{\sqrt{x^2+y^2+z^2}}, \frac{z}{\sqrt{x^2+y^2+z^2}} \right) \).

Step 1: Understand the Gradient Concept

The gradient of a function \( f(x, y, z) \) is a vector that consists of the partial derivatives of the function with respect to each variable. It points in the direction of the greatest rate of increase of the function.

Step 2: Calculate Partial Derivative with respect to x

To find \( \frac{\partial f}{\partial x} \), apply the chain rule to the function. Let \( u = x^2 + y^2 + z^2 \), then \( f = \sqrt{u} = u^{1/2} \). Therefore, \[ \frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + y^2 + z^2}}. \]

Step 3: Calculate Partial Derivative with respect to y

Similarly, for \( \frac{\partial f}{\partial y} \), using the chain rule: \[ \frac{\partial f}{\partial y} = \frac{1}{2}(x^2 + y^2 + z^2)^{-1/2} \cdot 2y = \frac{y}{\sqrt{x^2 + y^2 + z^2}}. \]

Step 4: Calculate Partial Derivative with respect to z

Finally, for \( \frac{\partial f}{\partial z} \), apply the chain rule to obtain: \[ \frac{\partial f}{\partial z} = \frac{1}{2}(x^2 + y^2 + z^2)^{-1/2} \cdot 2z = \frac{z}{\sqrt{x^2 + y^2 + z^2}}. \]

Step 5: Write the Gradient Vector

Combine the partial derivatives to form the gradient \( abla f \): \[ abla f = \left( \frac{x}{\sqrt{x^2 + y^2 + z^2}}, \frac{y}{\sqrt{x^2 + y^2 + z^2}}, \frac{z}{\sqrt{x^2 + y^2 + z^2}} \right). \]

Key Concepts

Gradient in Vector Calculus

The gradient is a fundamental concept in vector calculus. It is essentially a vector that points in the direction of the steepest ascent of a function. Whenever you hear about gradients, you're dealing with multi-variable calculus and focusing on how a function behaves in multiple dimensions.

Let's take a function of three variables, such as \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \). To find its gradient, we calculate derivatives for each of the function's variables.

• The gradient vector is written as \( abla f \) and combines partial derivatives with respect to each variable.
• Each component of the gradient vector shows how fast the function \( f \) changes in the direction of each variable.

In simpler terms, if you're climbing a hill, the gradient gives you the fastest route uphill. The length of the gradient vector gives the slope at that point.

Remember, calculating a gradient can help find the maximum increase, essential for optimization problems in physics and engineering.

Understanding Partial Derivatives

Partial derivatives are the building blocks of the gradient. They measure how a function changes as you vary just one of its input variables, holding the others constant. For example, in our function \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \), we find the following:

• The partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2 + y^2 + z^2}} \).
• The partial derivative with respect to \( y \) is \( \frac{\partial f}{\partial y} = \frac{y}{\sqrt{x^2 + y^2 + z^2}} \).
• The partial derivative with respect to \( z \) is \( \frac{\partial f}{\partial z} = \frac{z}{\sqrt{x^2 + y^2 + z^2}} \).

Partial derivatives are particularly useful in multivariable functions because they reveal how the function depends on each individual variable.

They help isolate the individual effects of variables and are essential in fields such as economics and thermodynamics where functions depend on many different variables.

The Chain Rule Explained

The chain rule is a powerful tool used in calculus to differentiate compositions of functions. If you think of one function nested within another, the chain rule helps you find the derivative of the combined function.

In our example, to differentiate \( f(x, y, z) = \sqrt{x^2 + y^2 + z^2} \), we see it as a composition:\

• First, define \( u = x^2 + y^2 + z^2 \).
• Next, consider \( f = \sqrt{u} = u^{1/2} \).

For each variable, we use the chain rule:

• The derivative of \( \sqrt{u} \) with respect to \( u \) is \( \frac{1}{2}u^{-1/2} \),
• and multiply by the derivative of \( u \) with respect to the variable, say \( x \), which is \( 2x \).

So you calculate \( \frac{\partial f}{\partial x} = \frac{1}{2}(x^2 + y^2 + z^2)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^2 + y^2 + z^2}} \).

The chain rule helps handle complex derivatives by breaking them into simpler parts, essential in both simple and involved mathematical functions.