Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 13-18, find div \(\mathbf{F}\) and curl \(\mathbf{F}\). $$ \mathbf{F}(x, y, z)=y z \mathbf{i}+x z \mathbf{j}+x y \mathbf{k} $$

Step-by-Step Solution

Answer

div \( \mathbf{F} = 0 \), curl \( \mathbf{F} = \mathbf{0} \).

Step 1: Understand the Problem

We are given the vector field \( \mathbf{F}(x, y, z) = yz \mathbf{i} + xz \mathbf{j} + xy \mathbf{k} \). We need to find both the divergence (div) and 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 our function, \( P = yz, \ Q = xz, \ R = xy \). We calculate:- \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(yz) = 0 \)- \( \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(xz) = 0 \)- \( \frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(xy) = 0 \)Thus, \( abla \cdot \mathbf{F} = 0 + 0 + 0 = 0 \).

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 \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right) \mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right) \mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k} \).Let's compute each component:- \( \frac{\partial R}{\partial y} = \frac{\partial}{\partial y}(xy) = x \)- \( \frac{\partial Q}{\partial z} = \frac{\partial}{\partial z}(xz) = x \)- \( \frac{\partial P}{\partial z} = \frac{\partial}{\partial z}(yz) = y \)- \( \frac{\partial R}{\partial x} = \frac{\partial}{\partial x}(xy) = y \)- \( \frac{\partial Q}{\partial x} = \frac{\partial}{\partial x}(xz) = z \)- \( \frac{\partial P}{\partial y} = \frac{\partial}{\partial y}(yz) = z \)Thus, the curl is:\( abla \times \mathbf{F} = (x-x) \mathbf{i} + (y-y) \mathbf{j} + (z-z) \mathbf{k} = \mathbf{0} \).

Step 4: Verify and Conclude

Both calculations of divergence and curl lead to zero. This confirms that for the given vector field \( \mathbf{F}(x, y, z) = yz \mathbf{i} + xz \mathbf{j} + xy \mathbf{k} \), the divergence is 0 and the curl is \( \mathbf{0} \). This indicates that \( \mathbf{F} \) is irrotational and divergence-free.

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics focused on vector fields and vector-valued functions. Imagine it as a toolbox used to explore how vectors behave in a multidimensional space. A vector field assigns a vector to each point in space, allowing us to understand physical phenomena like fluid flow or electromagnetic fields.

Key tools in vector calculus include divergence and curl. Divergence measures how much a vector field spreads out from a point. It is calculated using the formula \( abla \cdot \mathbf{F} \), where \( \mathbf{F} \) is the vector field. In the original exercise, the divergence was zero, implying that our vector field does not diverge from the origin point.

• **Divergence:** It indicates whether a point in the field is a source (positive divergence) or a sink (negative divergence), or neither (zero divergence).
• **Curl:** It determines the rotation or spinning of the field at a point, calculated using \( abla \times \mathbf{F} \).

In summary, understanding these concepts in vector calculus can help us decipher how a system behaves on a fundamental level. Calculating divergence and curl gives us insights into whether a system is expanding, contracting, or swirling around.

Irrotational Vector Field

An irrotational vector field is one in which the curl is zero everywhere. In simpler terms, the field does not exhibit any rotation or circulation around any point.

For example, think of water flowing through a straight pipe. The flow is uniform without any swirling or spiraling motion. This lack of spinning means the curl is zero. In the problem, the field \( \mathbf{F}(x, y, z) = yz \mathbf{i} + xz \mathbf{j} + xy \mathbf{k} \) was proven to be irrotational because its curl is \( \mathbf{0} \).

• **Characteristics:** No rotational movement in the field.
• **Significance:** Helps identify fields with simple flow patterns.

These fields are often simpler to analyze as they lack dynamic rotational complexities. The concept is essential in understanding flows and forces that are smooth without any eddies or vortices.

Divergence-Free Vector Field

A divergence-free vector field is characterized by having zero divergence at every point in the field. This means that the field neither sources nor sinks any vectors. It can also be referred to as a solenoidal field.

Imagine a magnetic field wrapping around a magnet. The field lines are closed loops, indicating that no field lines originate or terminate at any point in space, thus exhibiting zero divergence.

In the original exercise, the divergence of the vector field \( \mathbf{F} \) was calculated to be zero, confirming it to be divergence-free.

• **Characteristics:** The field behaves like an incompressible fluid where the flow continues without clumping or spreading out.
• **Implications:** In physics, such behavior is crucial in understanding phenomena where continuity is a key feature, like incompressible fluid dynamics or magnetic fields.

Recognizing a divergence-free field helps in predicting and modeling systems where conservation laws apply, aiding in the simplification of complex dynamics.