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Chapter 16: Problem 1

In Exercises \(1-8,\) find the divergence of the field. $$ \mathbf{F}=(x-y+z) \mathbf{i}+(2 x+y-z) \mathbf{j}+(3 x+2 y-2 z) \mathbf{k} $$

Short Answer

Expert verified
The divergence of the field is 0.

Step by step solution

01

Understanding Divergence

The divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by the formula \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). We need to apply this formula to the given vector field.
02

Identify Components

Identify \( P(x, y, z) \), \( Q(x, y, z) \), and \( R(x, y, z) \) from the given vector field \( \mathbf{F}=(x-y+z) \mathbf{i}+(2 x+y-z) \mathbf{j}+(3 x+2 y-2 z) \mathbf{k} \). In this case, \( P = x - y + z \), \( Q = 2x + y - z \), and \( R = 3x + 2y - 2z \).
03

Calculate Partial Derivatives

Compute the partial derivatives: \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(x-y+z) = 1 \); \( \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(2x+y-z) = 1 \); \( \frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(3x+2y-2z) = -2 \).
04

Compute Divergence

Substitute the partial derivatives into the divergence formula: \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} = 1 + 1 - 2 = 0 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Field
A vector field is an important concept in calculus and physics. It is essentially a function that assigns a vector to each point in a space. In simpler terms, imagine a field consisting of arrows, where each arrow represents a vector and points in a particular direction.
  • Each vector in the field has a magnitude and a direction.
  • The vector components can change depending on their position in space.
For example, the vector field \( \mathbf{F} = (x-y+z) \mathbf{i} + (2x+y-z) \mathbf{j} + (3x+2y-2z) \mathbf{k} \) assigns a vector at every point \( (x, y, z) \) in space.
Partial Derivatives
Partial derivatives are a fundamental tool in calculus when dealing with functions of several variables. A partial derivative measures how a function changes as one specific variable changes while the others are held constant.
  • They are written using the symbol \( \frac{\partial}{\partial x} \) for the variable \( x \), and similarly for other variables.
  • To calculate a partial derivative, only the variable of interest is considered to change.
For instance, in our exercise, the partial derivative \( \frac{\partial P}{\partial x} \) for the function \( P = x - y + z \) evaluates to 1, since only \( x \) contributes to the change.
Divergence Formula
The divergence formula is used to determine the rate at which a vector field spreads out from a point. This scalar quantity offers insights into the behavior of the field, such as whether it is a source or a sink.
  • Mathematically, it is expressed as \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
  • The vector field components \( P, Q, \) and \( R \) define the direction and flow at each point.
Using this formula for our vector field, the divergence is calculated as 0, indicating that the field neither diverges nor converges at any point.
Calculus Problem Solving
Solving calculus problems often involves breaking down seemingly complex problems into understandable parts. Here's an approach to tackling vector field-related problems:
  • **Step 1:** Identify the problem and understand what is being asked.
  • **Step 2:** Break the vector field into its components. For example, recognize \( P, Q, \) and \( R \).
  • **Step 3:** Find the needed derivatives. Calculate required partial derivatives as seen in the steps above.
  • **Step 4:** Apply the appropriate formulas such as the divergence formula.
Using these logical steps in our problem simplifies the process, transforming an intimidating problem into a manageable solution.

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Most popular questions from this chapter

Zero curl, yet the field is not conservative Show that the curl of $$\mathbf{F}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}} \mathbf{j}+z \mathbf{k}$$ is zero but that $$\oint_{C} \mathbf{F} \cdot d \mathbf{r}$$ is not zero if \(C\) is the circle \(x^{2}+y^{2}=1\) in the \(x y\) -plane. Theorem 7 does not apply here because the domain of \(\mathbf{F}\) is not simply connected. The field \(\mathbf{F}\) is not defined along the \(z\) -axis so there is no way to contract \(C\) to a point without leaving the domain of F.)

Conservation of mass Let \(\mathbf{v}(t, x, y, z)\) be a continuously differentiable vector field over the region \(D\) in space and let \(p(t, x)\) \(y, z )\) be a continuously differentiable scalar function. The variable \(t\) represents the time domain. The Law of Conservation of Mass asserts that $$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=-\iint_{S} p \mathbf{v} \cdot \mathbf{n} d \sigma$$ where \(S\) is the surface enclosing \(D\) a. Give a physical interpretation of the conservation of mass law if \(\mathbf{v}\) is a velocity flow field and \(p\) represents the density of the fluid at point \((x, y, z)\) at time \(t\) b. Use the Divergence Theorem and Leibniz's Rule, $$\frac{d}{d t} \iiint_{D} p(t, x, y, z) d V=\iiint_{D} \frac{\partial p}{\partial t} d V$$ to show that the Law of Conservation of Mass is equivalent to the continuity equation, $$\nabla \cdot p \mathbf{v}+\frac{\partial p}{\partial t}=0$$ (In the first term \(\nabla \cdot p \mathbf{v},\) the variable \(t\) is held fixed, and in the second term \(\partial p / \partial t,\) it is assumed that the point \((x, y, z)\) in \(D\) is held fixed.)

Find the area of the surface \(2 x^{3 / 2}+2 y^{3 / 2}-3 z=0\) above the square \(R : 0 \leq x \leq 1,0 \leq y \leq 1,\) in the \(x y\) -plane.

In Exercises \(9-20,\) use the Divergence Theorem to find the outward flux of \(\mathbf{F}\) across the boundary of the region \(D .\) Cylindrical can \(\mathbf{F}=\left(6 x^{2}+2 x y\right) \mathbf{i}+\left(2 y+x^{2} z\right) \mathbf{j}+4 x^{2} y^{3} \mathbf{k}\) \(D :\) The region cut from the first octant by the cylinder \(x^{2}+y^{2}=4\) and the plane \(z=3\)

Evaluate the integral $$\oint_{C} 4 x^{3} y d x+x^{4} d y$$ for any closed path \(C .\)
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