Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 15: Problem 17
Find div F and curl F. $$ \mathbf{F}(x, y, z)=x z^{3} \mathbf{i}+2 y^{4} x^{2} \mathbf{j}+5 z^{2} y \mathbf{k} $$ $$ \mathbf{F}(x, y, z)=x^{2} \mathbf{i}-2 \mathbf{j}+y z \mathbf{k} $$
Short Answer
Expert verified
\( \nabla \cdot \mathbf{F}_1 = z^3 + 8y^3 x^2 + 10z y \), \( \nabla \times \mathbf{F}_1 = 5z^2 \mathbf{i} + 3x z^2 \mathbf{j} + 4y^4 x \mathbf{k} \); \( \nabla \cdot \mathbf{F}_2 = 2x + y \), \( \nabla \times \mathbf{F}_2 = z \mathbf{i} \).
Step by step solution
01
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.
02
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, \( abla \cdot \mathbf{F}_1 = z^3 + 8y^3 x^2 + 10z y \).
03
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, \( abla \times \mathbf{F}_1 = 5z^2 \mathbf{i} + 3x z^2 \mathbf{j} + 4y^4 x \mathbf{k} \).
04
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, \( abla \cdot \mathbf{F}_2 = 2x + y \).
05
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, \( abla \times \mathbf{F}_2 = z \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = z \mathbf{i} \).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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: \( \mathbf{F}_2(x, y, z) = x^2\mathbf{i} - 2\mathbf{j} + yz \mathbf{k} \)
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.
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.
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.
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.
