Problem 248 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\) at the given point. $$ \mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k} \text { at }(1,2,1) $$

Step-by-Step Solution

Answer

The divergence of \(\mathbf{F}\) at (1, 2, 1) is 4.

Step 1: Understanding Divergence

Divergence measures the magnitude of a vector field's source or sink at a given point. It is a scalar field that describes how much the vector field spreads out from a point.

Step 2: Formula for Divergence

The formula for calculating the divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is: \[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \] where \( P \), \( Q \), and \( R \) are the components of \( \mathbf{F} \).

Step 3: Identify Components of \(\mathbf{F}\)

For the given vector field \( \mathbf{F}(x, y, z) = xy z \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), we identify \( P = xyz \), \( Q = y \), and \( R = z \).

Step 4: Calculate Partial Derivative of \( P = xyz \) with respect to \( x \)

Compute \( \frac{\partial}{\partial x}(xyz) = yz \).

Step 5: Calculate Partial Derivative of \( Q = y \) with respect to \( y \)

Compute \( \frac{\partial}{\partial y}(y) = 1 \).

Step 6: Calculate Partial Derivative of \( R = z \) with respect to \( z \)

Compute \( \frac{\partial}{\partial z}(z) = 1 \).

Step 7: Sum the Partial Derivatives

Add the partial derivatives to find the divergence: \( abla \cdot \mathbf{F} = yz + 1 + 1 \).

Step 8: Evaluate at the Given Point (1, 2, 1)

Substitute \( x = 1 \), \( y = 2 \), \( z = 1 \) into the equation: \( abla \cdot \mathbf{F} = (2)(1) + 1 + 1 = 4 \).

Key Concepts

Vector Field

A vector field is a function that assigns a vector to every point in space. Imagine a field of arrows where each arrow represents a vector. The direction of the arrow shows the direction of the vector, and its length describes the magnitude (or strength) at that point. Think of a weather map that shows wind patterns, where each arrow represents the wind direction and speed.
In this exercise, we have the vector field \( \mathbf{F}(x, y, z) = x y z \mathbf{i} + y \mathbf{j} + z \mathbf{k} \), which means at any point \((x, y, z)\), the vector is constructed by its components accompanying the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \).
Each term represents the component of the vector in one direction:

• \(x y z \mathbf{i}\) is along the x-axis.
• \(y \mathbf{j}\) points along the y-axis.
• \(z \mathbf{k}\) aligns with the z-axis.

This makes it clear that depending on where in space you are, the value and direction of the vector will change. A vector field is essentially mapping space into vectors.

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, particularly when dealing with functions of multiple variables. A partial derivative measures how a function changes as each variable changes, while keeping the other variables constant. In simpler terms, it tells us the rate of change of the function in one specific direction.
For example, let's look at the function \( P = xyz \). The partial derivative of \( P \) with respect to \( x \) is calculated by treating \( y \) and \( z \) as constants. This results in \( \frac{\partial}{\partial x}(xyz) = yz \).
Why do we need partial derivatives in finding divergence? Well, divergence relies on these derivatives to assess how the vector field spreads out at a point. In this problem, partial derivatives help us understand how the components of the vector field change along the x, y, and z directions. By summing these, we get information about the vector field's overall behavior at any given point.

Scalar Field

A scalar field is simply a function that assigns a single scalar value to every point in space, unlike a vector field, which attaches a vector. Imagine the contour lines on a topographic map that indicate the elevation levels of a landscape. Those lines represent a scalar field of altitude.
The divergence, mentioned in this exercise, is a perfect example of a scalar field. Once you compute the divergence of a vector field, you are left with a scalar value that indicates whether there is more flow entering or exiting a point. Thus, after performing our calculations, the divergence \( abla \cdot \mathbf{F} \) represents such a scalar field.
This divergence indicates potential sources (positive divergence) or sinks (negative divergence) in the vector field. In our example, calculating the divergence of the vector field at the point (1, 2, 1) produces the scalar field value 4. This tells us about the spread of vectors relative to this particular point.