Problem 17 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

\( \nabla \cdot \mathbf{F}_1 = z^3 + 8y^3 x^2 + 10z y \), \( \nabla \times \mathbf{F}_1 = 5z^2 \mathbf{i} + 3x z^2 \mathbf{j} + 4y^4 x \mathbf{k} \); \( \nabla \cdot \mathbf{F}_2 = 2x + y \), \( \nabla \times \mathbf{F}_2 = z \mathbf{i} \).

Step 1: Understand the Vector Fields

We are given two vector fields \( \mathbf{F}_1 \) and \( \mathbf{F}_2 \). The first vector field is \( \mathbf{F}_1(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5 z^2 y \mathbf{k} \). The second vector field is \( \mathbf{F}_2(x, y, z) = x^2 \mathbf{i} - 2 \mathbf{j} + y z \mathbf{k} \). Our task is to calculate both the divergence and the curl for each field.

Step 2: Find Divergence of \( \mathbf{F}_1 \)

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 \( \mathbf{F}_1 \), \( P = x z^3 \), \( Q = 2 y^4 x^2 \), and \( R = 5 z^2 y \). Hence, we compute:- \( \frac{\partial P}{\partial x} = z^3 \)- \( \frac{\partial Q}{\partial y} = 8y^3 x^2 \)- \( \frac{\partial R}{\partial z} = 10z y \)Thus, \( abla \cdot \mathbf{F}_1 = z^3 + 8y^3 x^2 + 10z y \).

Step 3: Find Curl of \( \mathbf{F}_1 \)

The curl of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k} \). For \( \mathbf{F}_1 \), we compute:- \( \frac{\partial R}{\partial y} = 5z^2 \) and \( \frac{\partial Q}{\partial z} = 0 \), so \( (\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}) = 5z^2 \)- \( \frac{\partial P}{\partial z} = 3x z^2 \) and \( \frac{\partial R}{\partial x} = 0 \), so \( (\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}) = 3x z^2 \)- \( \frac{\partial Q}{\partial x} = 4y^4 x \) and \( \frac{\partial P}{\partial y} = 0 \), so \( (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) = 4y^4 x \)Thus, \( abla \times \mathbf{F}_1 = 5z^2 \mathbf{i} + 3x z^2 \mathbf{j} + 4y^4 x \mathbf{k} \).

Step 4: Find Divergence of \( \mathbf{F}_2 \)

For \( \mathbf{F}_2 = x^2 \mathbf{i} - 2 \mathbf{j} + y z \mathbf{k} \), we apply the divergence formula again:- \( \frac{\partial (x^2)}{\partial x} = 2x \)- \( \frac{\partial (-2)}{\partial y} = 0 \)- \( \frac{\partial (y z)}{\partial z} = y \)Thus, \( abla \cdot \mathbf{F}_2 = 2x + y \).

Step 5: Find Curl of \( \mathbf{F}_2 \)

Applying the curl formula for \( \mathbf{F}_2 \):- \( \frac{\partial (y z)}{\partial y} = z \) and \( \frac{\partial (-2)}{\partial z} = 0 \), so \( \left( \frac{\partial (y z)}{\partial y} - \frac{\partial (-2)}{\partial z} \right) = z \)- \( \frac{\partial x^2}{\partial z} = 0 \) and \( \frac{\partial (y z)}{\partial x} = 0 \), so \( \left( \frac{\partial x^2}{\partial z} - \frac{\partial (y z)}{\partial x} \right) = 0 \)- \( \frac{\partial (-2)}{\partial x} = 0 \) and \( \frac{\partial x^2}{\partial y} = 0 \), so \( \left( \frac{\partial (-2)}{\partial x} - \frac{\partial x^2}{\partial y} \right) = 0 \)Thus, \( abla \times \mathbf{F}_2 = z \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = z \mathbf{i} \).

Key Concepts

Vector Fields

Vector fields are fundamental in vector calculus, representing how a vector quantity varies at each point in space. You can imagine a vector field as a kind of map where every point in the space is assigned a vector.
Each vector can represent forces, velocities, or other physical quantities. Mathematically, a vector field in three-dimensional space is often written as \( \mathbf{F}(x, y, z) = P(x, y, z)\mathbf{i} + Q(x, y, z)\mathbf{j} + R(x, y, z)\mathbf{k} \).
Here, \( P, Q, R \) are functions that describe how the components of the array vary with position.

• Example: \( \mathbf{F}_1(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5z^2 y \mathbf{k} \)
• Example: \( \mathbf{F}_2(x, y, z) = x^2\mathbf{i} - 2\mathbf{j} + yz \mathbf{k} \)

Understanding these components helps in finding other properties like divergence and curl, which give us more insight into the behavior of the vector field. They tell us how the field changes over space and provide a deeper understanding of physical phenomena.

Divergence

Divergence is a measure of a vector field's tendency to originate from or converge into certain points. In simpler terms, it tells us how much a vector field spreads out from a point.
Mathematically, for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the divergence is calculated as:\[ abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]
For example, the divergence of the vector field \( \mathbf{F}_1(x, y, z) = x z^3 \mathbf{i} + 2 y^4 x^2 \mathbf{j} + 5 z^2 y \mathbf{k} \) is \( z^3 + 8 y^3 x^2 + 10 z y \). It tells how much the field is expanding or contracting at any point.

• If the divergence is positive, the field is diverging or spreading.
• If it's negative, the field is converging or shrinking.

Understanding divergence is crucial in fields like fluid dynamics and electromagnetism, where it describes how fluids or electric charges behave.

Curl

The curl of a vector field represents its rotational motion. It provides a measure of how much and in which direction the field 'swirls' around a point.
To calculate it for a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), use the formula: \[ abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k} \]
For instance, the curl of \( \mathbf{F}_1 \) is \( 5z^2 \mathbf{i} + 3x z^2 \mathbf{j} + 4y^4 x \mathbf{k} \), indicating how the field loops around different axes.

• A non-zero curl means there is a rotation or a swirl around a point.
• A zero curl signifies no rotational effect in the field at that point.

This concept is widely used in physics, especially when dealing with magnetic fields and fluid rotations.