Problem 18 - Calculus Early Transcendentals

Question

Find div F and curl F. $$ \mathbf{F}(x, y, z)=x z^{3} \mathbf{i}+2 y^{4} x^{2} \mathbf{j}+5 z^{2} y \mathbf{k} $$

Step-by-Step Solution

Answer

The divergence of \( \mathbf{F} \) is \( z^3 + 8 y^3 x^2 + 10 zy \), and the curl of \( \mathbf{F} \) is \( 5 z^2 \mathbf{i} - 3 x z^2 \mathbf{j} + 4 y^4 x \mathbf{k} \).

Step 1: Understand the Problem

We are given a vector field \( \mathbf{F}(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5 z^2 y \mathbf{k} \). We need to find the divergence (div) and the curl of this vector field.

Step 2: Calculate the Divergence

The divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). For \( \mathbf{F}(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5 z^2 y \mathbf{k} \):- \( P = x z^3 \), \( Q = 2 y^4 x^2 \), \( R = 5 z^2 y \).Calculate each partial derivative:- \( \frac{\partial P}{\partial x} = z^3 \)- \( \frac{\partial Q}{\partial y} = 8 y^3 x^2 \)- \( \frac{\partial R}{\partial z} = 10 z y \)Thus, \( abla \cdot \mathbf{F} = z^3 + 8 y^3 x^2 + 10 z y \).

Step 3: Calculate the Curl

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix} \).Compute the determinant of the matrix:\[abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right) \mathbf{i} - \left(\frac{\partial R}{\partial x} - \frac{\partial P}{\partial z}\right) \mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right) \mathbf{k}\]Calculate each component:- \( \frac{\partial R}{\partial y} = 5 z^2 \), \( \frac{\partial Q}{\partial z} = 0 \), hence the \( \mathbf{i} \)-component is \( 5 z^2 \).- \( \frac{\partial R}{\partial x} = 0 \), \( \frac{\partial P}{\partial z} = 3 x z^2 \), hence the \( \mathbf{j} \)-component is \( -3 x z^2 \).- \( \frac{\partial Q}{\partial x} = 4 y^4 x \), \( \frac{\partial P}{\partial y} = 0 \), hence the \( \mathbf{k} \)-component is \( 4 y^4 x \).Thus, \( abla \times \mathbf{F} = 5 z^2 \mathbf{i} - 3 x z^2 \mathbf{j} + 4 y^4 x \mathbf{k} \).

Key Concepts

Divergence

The divergence of a vector field provides a measure of how much the field spreads out from a given point. Imagine it as checking whether a fluid is expanding or contracting at any particular point in space. The formula to compute divergence for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
\[\mathbf{F}(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5 z^2 y \mathbf{k}\] has components:

• \( P = x z^3 \)
• \( Q = 2 y^4 x^2 \)
• \( R = 5 z^2 y \)

Calculating each partial derivative, we find:

• \( \frac{\partial P}{\partial x} = z^3 \)
• \( \frac{\partial Q}{\partial y} = 8 y^3 x^2 \)
• \( \frac{\partial R}{\partial z} = 10 z y \)

Thus, the divergence is \( abla \cdot \mathbf{F} = z^3 + 8 y^3 x^2 + 10 z y \). This result indicates how the vector field "moves" around each point; if positive, it generally points outward indicating expansion.

Curl

Curl in vector calculus measures the rotation or the "twisting" force of a vector field at a given point. In other words, it tells you how much and in which direction the field tends to spin around that point. The formula to calculate the curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is a determinant:\[abla \times \mathbf{F} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \ P & Q & R \end{vmatrix}\]
When calculated, this gives us the vector \( abla \times \mathbf{F} = 5 z^2 \mathbf{i} - 3 x z^2 \mathbf{j} + 4 y^4 x \mathbf{k} \). To derive this, we find:

• \( \mathbf{i} \)-component as \( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} = 5 z^2 \)
• \( \mathbf{j} \)-component as \( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} = -3 x z^2 \)
• \( \mathbf{k} \)-component as \( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 4 y^4 x \)

This result indicates the amount and direction of the rotational force in the field around that point.

Vector Field Analysis

Vector field analysis encompasses various techniques to understand the behavior of vector fields. This includes studying properties such as divergence and curl, which reveal essential characteristics about the field's flow and rotational tendencies.
A vector field like \( \mathbf{F}(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5 z^2 y \mathbf{k} \) has different behavior in different spatial regions, and tools like divergence and curl help illustrate these behaviors:

• Divergence: Identifies sources and sinks within the field, showing where field lines originate, terminate, or spread out.
• Curl: Describes how the field lines "circulate" or twist around points, crucial for appreciating vortex-like behavior within the field.

By analyzing vector fields using these techniques, we can gain insights not only in theoretical mathematics but also in physical phenomena represented by these fields, such as electromagnetic fields or fluid dynamics. Understanding these properties aids in solving complex problems by visualizing how fields behave and interact with physical spaces.