Question

Calculate the gradient \(\nabla f\) of the following functions, \(f(x, y, z):(\mathbf{a}) f=\ln (r),\) (b) \(f=r^{n}\), (c) \(f=g(r),\) where \(r=\sqrt{x^{2}+y^{2}+z^{2}}\) and \(g(r)\) is some unspecified function of \(r .\) [Hint: Use the chain rule.]

Short answer

(a) \(\nabla f = \left(\frac{x}{r^2}, \frac{y}{r^2}, \frac{z}{r^2}\right)\), (b) \(\nabla f = \left(nx r^{n-2}, ny r^{n-2}, nz r^{n-2}\right)\), (c) \(\nabla f = g'(r)\left(\frac{x}{r}, \frac{y}{r}, \frac{z}{r}\right)\).

Step-by-step solution

Understand the Gradient Concept

The gradient of a function is a vector of partial derivatives. For a function \( f(x, y, z) \), the gradient is given by \( abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \). This vector points in the direction of the greatest rate of increase of the function.

Define \( r \) and its Partial Derivatives

Given \( r = \sqrt{x^2 + y^2 + z^2} \), recognize \( r \) as the distance from the origin to a point \( (x, y, z) \). Calculate partial derivatives: \( \frac{\partial r}{\partial x} = \frac{x}{r} \), \( \frac{\partial r}{\partial y} = \frac{y}{r} \), \( \frac{\partial r}{\partial z} = \frac{z}{r} \) using the chain rule.

Calculate \( \nabla f \) for \( f = \ln(r) \)

Using the chain rule, \( \frac{\partial f}{\partial x} = \frac{1}{r} \cdot \frac{\partial r}{\partial x} = \frac{x}{r^2} \), \( \frac{\partial f}{\partial y} = \frac{y}{r^2} \), \( \frac{\partial f}{\partial z} = \frac{z}{r^2} \). Thus, \( abla f = \left( \frac{x}{r^2}, \frac{y}{r^2}, \frac{z}{r^2} \right) \).

Calculate \( \nabla f \) for \( f = r^n \)

Use the chain rule: \( \frac{\partial f}{\partial x} = n r^{n-1} \cdot \frac{\partial r}{\partial x} = n x r^{n-2} \), \( \frac{\partial f}{\partial y} = n y r^{n-2} \), \( \frac{\partial f}{\partial z} = n z r^{n-2} \). Therefore, \( abla f = \left( n x r^{n-2}, n y r^{n-2}, n z r^{n-2} \right) \).

Calculate \( \nabla f \) for \( f = g(r) \)

Using the chain rule: \( \frac{\partial f}{\partial x} = g'(r) \cdot \frac{\partial r}{\partial x} = g'(r) \frac{x}{r} \), \( \frac{\partial f}{\partial y} = g'(r) \frac{y}{r} \), \( \frac{\partial f}{\partial z} = g'(r) \frac{z}{r} \). So, \( abla f = \left( g'(r) \frac{x}{r}, g'(r) \frac{y}{r}, g'(r) \frac{z}{r} \right) \).

Key concepts

Chain Rule

The Chain Rule is a fundamental concept in calculus that helps us find the derivative of composite functions. In simple terms, it's how we deal with functions nested within other functions, kind of like peeling an onion. When we deal with multivariable functions, such as those involving three or more variables, the chain rule provides a pathway to differentiate these functions by connecting their various layers.
For instance, consider a function that depends on another function, like how our exercise involves function values depending on the variable \( r \). Here, \( r = \sqrt{x^2 + y^2 + z^2} \), is an intermediary function that changes the outcome of the final expression \( f(r) \).
Using the chain rule, we determine how our function \( f \) changes as its inner function \( r \) changes, allowing us to find partial derivatives linked with \( r \). These derivatives connect through the chain rule providing:

• \( \frac{\partial f}{\partial x} = \frac{df}{dr} \cdot \frac{\partial r}{\partial x} \)
• \( \frac{\partial f}{\partial y} = \frac{df}{dr} \cdot \frac{\partial r}{\partial y} \)
• \( \frac{\partial f}{\partial z} = \frac{df}{dr} \cdot \frac{\partial r}{\partial z} \)

Mastering the chain rule equips you to systematically tackle complex derivative problems in multivariable calculus.

Gradient Vector

The gradient vector, often denoted as \( abla f \), is an essential tool in vector calculus. It is a vector composed of all the partial derivatives of a function. In layman's terms, it tells you the direction in space where the function increases the most. This concept is crucial in fields like optimization and physics.
For example, with a function \( f(x, y, z) \) dependent on three variables, the gradient \( abla f \) forms the vector:

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

This vector shows us how each variable affects the function's change. It's like having a compass pointing towards the steepest ascent of a hill.
When calculating the gradient in our exercise, we make use of \( r = \sqrt{x^2 + y^2 + z^2} \) to find the individual changes along the \( x, y, \) and \( z \) axes. This solution aids in determining how each coordinate direction affects the function's value.

Partial Derivatives

Partial derivatives provide a way to understand how a function changes when altering only one variable while keeping others constant. They are to multivariable calculus what regular derivatives are to single-variable calculus.
To derive these partial derivatives, we treat the function as dependent solely on one variable at a time. For example, from the function \( r = \sqrt{x^2 + y^2 + z^2} \), calculating the partial derivative with respect to \( x \) gives \( \frac{\partial r}{\partial x} = \frac{x}{r} \). The same goes for finding the derivatives with respect to \( y \) and \( z \).

• \( \frac{\partial r}{\partial y} = \frac{y}{r} \)
• \( \frac{\partial r}{\partial z} = \frac{z}{r} \)

By conducting this operation, you crack open the behavior of our function from various angles, allowing us to compute things like the gradient effectively.
Partial derivatives are not just abstract notions; they represent real changes in systems and are vital in understanding gradients and applying the chain rule in complex scenarios.