Problem 247 - Calculus Volume 3

Question

Find the divergence of \(\mathbf{F}\) at the given point. $$ \mathbf{F}(x, y, z)=e^{-x y} \mathbf{i}+e^{x z} \mathbf{j}+e^{y z} \mathbf{k} \text { at }(3,2,0) $$

Step-by-Step Solution

Answer

The divergence at (3, 2, 0) is \(2(1 - e^{-6})\).

Step 1: Recall the Divergence Formula

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

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

Given \(\mathbf{F}(x, y, z) = e^{-xy}\mathbf{i} + e^{xz}\mathbf{j} + e^{yz}\mathbf{k}\), identify the components: \(P = e^{-xy}\), \(Q = e^{xz}\), and \(R = e^{yz}\).

Step 3: Differentiate \(P\) with respect to \(x\)

Compute \(\frac{\partial P}{\partial x}\). The partial derivative of \(e^{-xy}\) with respect to \(x\) is \(-y e^{-xy}\).

Step 4: Differentiate \(Q\) with respect to \(y\)

Compute \(\frac{\partial Q}{\partial y}\). The partial derivative of \(e^{xz}\) with respect to \(y\) is \(0\) because \(y\) is not present in \(Q\).

Step 5: Differentiate \(R\) with respect to \(z\)

Compute \(\frac{\partial R}{\partial z}\). The partial derivative of \(e^{yz}\) with respect to \(z\) is \(y e^{yz}\).

Step 6: Compute the Divergence

The divergence \(abla \cdot \mathbf{F}\) is calculated as \(-y e^{-xy} + 0 + y e^{yz}\).

Step 7: Substitute the Point into the Divergence

Evaluate the divergence expression at the point \((3, 2, 0)\): substitute \(x = 3\), \(y = 2\), and \(z = 0\) into \(-y e^{-xy} + y e^{yz}\), yielding \(-2 e^{-6} + 2 e^{0} = -2 e^{-6} + 2\).

Step 8: Simplify the Result

Calculate the simplified value: \(2(1 - e^{-6})\).

Key Concepts

Understanding Divergence

Divergence is a key concept in vector calculus that measures how a vector field behaves near a given point. Specifically, it tells us how much a vector field spreads out or diverges from that point. Imagine water flowing through a pipe; where the pipe widens, the water spreads out, indicating positive divergence. On the other hand, if the pipe narrows, negative divergence occurs as the water is funneling in more tightly.
In mathematical terms, for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the divergence \( abla \cdot \mathbf{F} \) is calculated as:

• \( \frac{\partial P}{\partial x} \) - Partial derivative of \( P \) with respect to \( x \)
• \( \frac{\partial Q}{\partial y} \) - Partial derivative of \( Q \) with respect to \( y \)
• \( \frac{\partial R}{\partial z} \) - Partial derivative of \( R \) with respect to \( z \)

In our example, each of these partial derivatives contributes to the final divergence result. By understanding how each component behaves, we gain insight into the overall behavior of the vector field at a specific point.

Exploring Partial Derivatives

Partial derivatives are crucial in determining changes in a function relative to one variable while keeping others constant. They are extensively used in vector calculus to determine divergence, gradient, and curl.
Consider a multivariable function like \( e^{-xy} \). When calculating the partial derivative with respect to \( x \), we treat \( y \) as a constant. Therefore, the partial derivative \( \frac{\partial }{\partial x}(e^{-xy}) \) is computed as \( -y e^{-xy} \).
In the vector field \( \mathbf{F}(x, y, z) = e^{-xy}\mathbf{i} + e^{xz}\mathbf{j} + e^{yz}\mathbf{k} \), partial derivatives help us explore changes in the vector field's components. For instance:

• \( \frac{\partial P}{\partial x} = -y e^{-xy} \) indicates a change in the \( i \)-direction with \( x \)
• \( \frac{\partial Q}{\partial y} = 0 \) reflects no change with \( y \) in the \( j \)-component because \( Q \) is independent of \( y \)
• \( \frac{\partial R}{\partial z} = y e^{yz} \) shows how the \( k \)-component changes with \( z \)

Grasping how to compute and interpret these derivatives facilitates understanding the divergence of a vector field.

Vector Fields in 3D Space

A vector field is essentially a function that assigns a vector to every point in a space, commonly represented in three-dimensional space. Picture each point in space having a tiny arrow (vector) that might indicate things like velocity, force, or field strength. Vector fields are used extensively to model physical phenomena such as wind patterns, electromagnetic fields, and fluid flow.
The vector field \( \mathbf{F}(x, y, z) = e^{-xy}\mathbf{i} + e^{xz}\mathbf{j} + e^{yz}\mathbf{k} \) is defined over three variables \( x, y, \) and \( z \), giving it three dimensions to operate within. Here, each component (\( i, j, \) and \( k \)) of the vector field describes a direction along the corresponding axis.

• The \( i \) vector component affected by \( x \) and \( y \)
• The \( j \) vector component affected by \( x \) and \( z \)
• The \( k \) vector component affected by \( y \) and \( z \)

This gives multifaceted behavior to the field, necessitating robust tools like divergence to decode its spread and concentration properties at any point, such as \( (3, 2, 0) \). Understanding how vector fields are structured and behave helps describe various complex systems in the real world.