Problem 207 - Calculus Volume 3

Question

Determine whether the statement is true or false. \(\nabla \cdot(x \mathbf{i}+y \mathbf{j}+z \mathbf{k})=1\)

Step-by-Step Solution

Answer

False, the divergence is 3, not 1.

Step 1: Identify the vector field

The vector field given is \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). This is a 3-dimensional vector field where each component corresponds to one of the axes: \( x \) for \( \mathbf{i} \), \( y \) for \( \mathbf{j} \), and \( z \) for \( \mathbf{k} \).

Step 2: Recall the formula for the divergence

The divergence of a vector field \( \mathbf{F} = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \) is given by the formula \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).

Step 3: Apply the divergence formula

Substitute \( P = x \), \( Q = y \), and \( R = z \) into the formula for the divergence. Thus, \( abla \cdot \mathbf{F} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{\partial z}{\partial z} \).

Step 4: Calculate the partial derivatives

Calculate each partial derivative: \( \frac{\partial x}{\partial x} = 1 \), \( \frac{\partial y}{\partial y} = 1 \), and \( \frac{\partial z}{\partial z} = 1 \).

Step 5: Sum the partial derivatives

Add the results from the previous step: \( 1 + 1 + 1 = 3 \). Thus, \( abla \cdot (x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) = 3 \).

Step 6: Compare the result to the given statement

The statement given is \( abla \cdot (x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) = 1 \), but our calculation shows \( abla \cdot (x \mathbf{i}+y \mathbf{j}+z \mathbf{k}) = 3 \). Therefore, the statement is false.

Key Concepts

Vector Field

A vector field is a mathematical construct that assigns a vector to every point in a given space. The vector itself represents a quantity that has both magnitude and direction. In three-dimensional space, this means that you can picture the vector as an arrow pointing to some direction with a certain length, which represents the magnitude of the field at that point.

For example, imagine a weather map where each point displays a small arrow indicating wind speed and direction. This is essentially a vector field, where each arrow corresponds to a vector in the field.

In the context of our exercise, the vector field given is \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \). Here, the vector field consists of three components. Each component corresponds to one of the coordinate axes:

• \( x \) is aligned with the \( \mathbf{i} \) direction,
• \( y \) is aligned with the \( \mathbf{j} \) direction,
• \( z \) is aligned with the \( \mathbf{k} \) direction.

Understanding vector fields, particularly in 3D space, is crucial as they form the basis for describing many physical phenomena such as electromagnetic fields, fluid flow, and more.

Partial Derivatives

Partial derivatives are a key concept in calculus that involve taking the derivative of a function with respect to one variable while keeping others constant. This is particularly useful in multivariable calculus, where functions depend on multiple variables.

To better understand, consider a function \( f(x, y, z) \) that depends on three variables \( x \), \( y \), and \( z \). The partial derivative of \( f \) with respect to \( x \), denoted \( \frac{\partial f}{\partial x} \), is the rate at which \( f \) changes as \( x \) changes, with all other variables held fixed.

In the given exercise, the divergence formula requires us to compute partial derivatives of each component of the vector field \( \mathbf{F} \):

• \( \frac{\partial x}{\partial x} = 1 \)
• \( \frac{\partial y}{\partial y} = 1 \)
• \( \frac{\partial z}{\partial z} = 1 \)

Calculating partial derivatives is essential for finding gradients, divergences, and more, making them foundational tools in vector calculus. They allow us to understand how a multivariable function behaves one variable at a time, which is especially useful in physics and engineering.

3-Dimensional Space

The concept of 3-dimensional (3D) space is integral to understanding and visualizing vector fields. In mathematics and physics, 3D space consists of three axes typically labeled as \( x \), \( y \), and \( z \). These axes help define points, lines, planes, and ultimately the position of objects in space.

Each point in 3-dimensional space is represented by an ordered triplet \( (x, y, z) \), where each coordinate indicates the point's position along its respective axis.

Descriptions in 3-dimensional space allow us to model real-world systems more accurately. For instance, when analyzing the motion of an aircraft, it is crucial to determine its position in 3D space; not just in terms of latitude and longitude (2D), but also altitude (giving it the third dimension).

For the exercise we're analyzing, the vector field \( \mathbf{F} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \) represents a field in 3D space. Each vector in this field indicates a position and direction vector determined by the coordinates x, y, and z. Understanding this concept is key to applying various mathematical operations such as divergence, which is concerned with the behavior of vector fields in multiple dimensions.