Problem 245 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\) at the given point. $$ \mathbf{F}(x, y, z)=\mathbf{i}+\mathbf{j}+\mathbf{k} \text { at }(2,-1,3) $$

Step-by-Step Solution

Answer

The divergence of \( \mathbf{F} \) at the point \((2, -1, 3)\) is 0.

Step 1: Understand the Divergence Concept

Divergence is a measure of a vector field's tendency to originate from or converge into a point. For a vector field \( \mathbf{F}(x, y, z) = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the divergence is given by: \[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}. \]

Step 2: Identify the Components of \( \mathbf{F} \)

For the given vector field \( \mathbf{F}(x, y, z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \), the components are: \( P = 1 \), \( Q = 1 \), \( R = 1 \).

Step 3: Compute the Partial Derivatives

Compute the partial derivatives: \( \frac{\partial P}{\partial x} = 0 \) (since \( P \) is constant), \( \frac{\partial Q}{\partial y} = 0 \) (since \( Q \) is constant), \( \frac{\partial R}{\partial z} = 0 \) (since \( R \) is constant).

Step 4: Calculate the Divergence

Substitute the partial derivatives back into the divergence formula: \[ abla \cdot \mathbf{F} = 0 + 0 + 0 = 0. \]

Step 5: Evaluate at the Given Point

Since the divergence is constant and equal to zero, the divergence of \( \mathbf{F} \) at the point \((2, -1, 3)\) is indeed \( 0 \).

Key Concepts

Vector Fields

In the realm of vector calculus, a vector field is a function that assigns a vector to every single point in space. Imagine a gentle breeze flowing over you, with vectors representing the wind's direction and strength at different points. Such fields could include things as diverse as magnetic forces or fluid velocities.

In this exercise, our vector field is defined as \( \mathbf{F}(x, y, z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \). What this means, in simpler terms, is that at every point in space, the field has a vector with a direction and magnitude that remains consistent. This unchanging direction is \( (1, 1, 1) \) throughout, suggesting a balanced and uniform vector field.

Here are some key characteristics of vector fields:

• They can be visualized as arrows in a plot, where each arrow's direction and magnitude indicate the value of the vector at that location.
• Vector fields often arise in physics and engineering to describe the intensity and direction of forces such as gravity or electromagnetism.
• The components of vector fields, \( P \mathbf{i}, Q \mathbf{j}, R \mathbf{k} \), represent the contributions along the \( x, y, \) and \( z \) axes respectively.

Partial Derivatives

Partial derivatives are essential in vector calculus because they enable us to see how a function changes as we vary one of the variables while keeping others constant. Think of it as gently nudging a single aspect of a system to see the effect.

In our vector field \( \mathbf{F}(x, y, z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \), we have individual components: \( P = 1 \), \( Q = 1 \), and \( R = 1 \). Each component represents a constant vector along its respective axis. When computing partial derivatives for each component:

• The partial derivative of \( P \) with respect to \( x \) is \( \frac{\partial P}{\partial x} = 0 \) since \( P \) is constant.
• Similarly, \( \frac{\partial Q}{\partial y} = 0 \) and \( \frac{\partial R}{\partial z} = 0 \) as \( Q \) and \( R \) are constants.

These values are crucial for determining the divergence of the vector field. Simply put, the constants lead to partial derivatives that add up to zero, giving insight into the behavior of the vector field.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and operations on them, such as differentiation and integration. It is often applied in physics and engineering to understand and compute factors like flux and circulation.

Divergence is a core concept in vector calculus. It measures how much a vector field spreads out from a point. Mathematically, for a vector field \( \mathbf{F}(x, y, z) = P \mathbf{i} + Q \mathbf{j} + R \mathbf{k} \), the divergence is calculated using the formula: \[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z}. \] Steps to compute divergence include:

• First, identify the vector field components \( P, Q, R \).
• Then, find the partial derivatives of each with respect to their respective variables.
• Finally, sum these partial derivatives as per the divergence formula.

In this exercise, because \( \mathbf{F}(x, y, z) = \mathbf{i} + \mathbf{j} + \mathbf{k} \) is uniform throughout space, the divergence simplifies to 0, indicating no net 'spreading out' at any point, including at \((2, -1, 3)\). This illustrates a critical application of vector calculus: pinpointing and comprehending vector field behaviors in dynamic systems.