Problem 1 - Thomas Calculus

Question

Find the gradient fields of the functions $$ f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} $$

Step-by-Step Solution

Answer

The gradient field is \( \nabla f = \left(-\frac{x}{(x^2 + y^2 + z^2)^{3/2}}, -\frac{y}{(x^2 + y^2 + z^2)^{3/2}}, -\frac{z}{(x^2 + y^2 + z^2)^{3/2}} \right) \).

Step 1: Understand the Gradient of a Function

The gradient of a function \(f\) in three dimensions is a vector field given by the vector of its partial derivatives with respect to its variables \((x, y, z)\). It can be represented as:\[ abla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]

Step 2: Compute the Partial Derivative with Respect to \(x\)

For the function \(f(x, y, z) = \left(x^2 + y^2 + z^2\right)^{-1/2}\), compute the partial derivative with respect to \(x\):Use the chain rule:\[ \frac{\partial f}{\partial x} = \frac{d}{dx}\left(x^2 + y^2 + z^2\right)^{-1/2} = -\frac{1}{2}(x^2 + y^2 + z^2)^{-3/2} \cdot 2x = -\frac{x}{(x^2 + y^2 + z^2)^{3/2}} \]

Step 3: Compute the Partial Derivative with Respect to \(y\)

Similarly, compute the partial derivative of \(f\) with respect to \(y\):\[ \frac{\partial f}{\partial y} = -\frac{y}{(x^2 + y^2 + z^2)^{3/2}} \]

Step 4: Compute the Partial Derivative with Respect to \(z\)

Finally, compute the partial derivative of \(f\) with respect to \(z\):\[ \frac{\partial f}{\partial z} = -\frac{z}{(x^2 + y^2 + z^2)^{3/2}} \]

Step 5: Write the Gradient Field

After obtaining all the partial derivatives, write down the gradient vector \( abla f \):\[ abla f = \left( -\frac{x}{(x^2 + y^2 + z^2)^{3/2}}, -\frac{y}{(x^2 + y^2 + z^2)^{3/2}}, -\frac{z}{(x^2 + y^2 + z^2)^{3/2}} \right) \]

Key Concepts

Partial Derivatives

Partial derivatives are essential tools in calculus, particularly when dealing with functions of multiple variables. Here, we're working with a function that takes three variables: \( x \), \( y \), and \( z \). To find how the function changes with respect to one of these variables, we hold the others constant while differentiating just one variable. This is what we call a partial derivative.

In our example, the function is \( f(x, y, z) = (x^2 + y^2 + z^2)^{-1/2} \). To find \( \frac{\partial f}{\partial x} \), we treat \( y \) and \( z \) as constants and focus on differentiating with respect to \( x \).

Keep in mind, partial derivatives give us the rate of change of a function's output along just one direction in the variable's space, which is crucial for understanding the behavior of multi-variable functions.

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It helps us break down complex functions into simpler parts so we can understand how each part influences the function's overall rate of change.

In our function \( f(x, y, z) = (x^2 + y^2 + z^2)^{-1/2} \), the expression inside the power is a sum of squares: \( x^2 + y^2 + z^2 \). To differentiate this using the chain rule, we look at the outer function, which is the power of \(-1/2\), and the inner function, which is the sum of squares.

When applying the chain rule, we first differentiate the outer function while keeping the inner function intact, and then multiply the result by the derivative of the inner function. This dual-step process is crucial whenever the function you're working with is composed of other functions.

Vector Calculus

Vector calculus is a branch of mathematics focused on differentiating and integrating vector fields, which represent quantities having both magnitude and direction. It is highly relevant in physics and engineering.

The gradient, which is what we've calculated in this exercise, is a vector field that shows the direction of the greatest rate of increase of a scalar function. The magnitude of this vector represents how quickly the function increases in that direction.

For a scalar function \( f(x, y, z) \), the gradient vector \( abla f \) is given by \( \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right) \). This vector points in the direction of the most rapid increase of the function and is perpendicular to the level surfaces of the function. Understanding vector calculus helps us unravel complex phenomena in multi-dimensional spaces.