/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 6 A vector field is given by $$ ... [FREE SOLUTION] | 91影视

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Chapter 26: Problem 6

A vector field is given by $$ \begin{aligned} \mathbf{F}=&\left(3 x^{2}-z\right) \mathbf{i}+(2 x+y) \mathbf{j} \\ &+(x+3 y z) \mathbf{k} \end{aligned} $$ Find \(\nabla \cdot \mathbf{F}\) at the point \((1,2,3)\).

Short Answer

Expert verified
The divergence \( \nabla \cdot \mathbf{F} \) at the point \((1,2,3)\) is 13.

Step by step solution

01

Understand the Divergence Formula

The divergence of a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \) is given by \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \). Identify each component of \( \mathbf{F} \): \( P = 3x^2 - z \), \( Q = 2x + y \), and \( R = x + 3yz \).
02

Compute the Partial Derivative with Respect to x

Compute the partial derivative of \( P \) with respect to \( x \): \( \frac{\partial P}{\partial x} = \frac{\partial}{\partial x}(3x^2 - z) = 6x \).
03

Compute the Partial Derivative with Respect to y

Compute the partial derivative of \( Q \) with respect to \( y \): \( \frac{\partial Q}{\partial y} = \frac{\partial}{\partial y}(2x + y) = 1 \).
04

Compute the Partial Derivative with Respect to z

Compute the partial derivative of \( R \) with respect to \( z \): \( \frac{\partial R}{\partial z} = \frac{\partial}{\partial z}(x + 3yz) = 3y \).
05

Calculate the Divergence

Use the results from the partial derivatives to calculate the divergence: \( abla \cdot \mathbf{F} = 6x + 1 + 3y \).
06

Evaluate at the Point (1,2,3)

Substitute \( x = 1 \), \( y = 2 \), and \( z = 3 \) into the divergence equation: \( abla \cdot \mathbf{F} = 6(1) + 1 + 3(2) = 6 + 1 + 6 = 13 \).

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Key Concepts

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

Divergence
Divergence is a fundamental concept in vector calculus that measures how much a vector field spreads out from a point. Contrary to curl, which measures rotation, divergence tells us about the expansion or contraction at a point within a field. To compute divergence, we use the formula:
  • For a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), divergence is \( abla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \).
This formula sums up the partial derivatives of each component function of the vector field with respect to its corresponding variable. Understanding divergence involves recognizing how each variable individually contributes to the field's total spread at any given point. This concept applies in fields as varied as physics, engineering, and data science.
Partial Derivatives
Partial derivatives are central to understanding the behavior of multivariable functions, like those seen in vector calculus. In essence, a partial derivative measures the rate of change of a function with respect to one variable, while other variables are held constant.For example:
  • The partial derivative \( \frac{\partial P}{\partial x} \) tells us how much the function \( P \) changes as \( x \) changes, keeping other variables like \( y \) and \( z \) fixed.
This is essential when computing divergence, as we separately consider the influence of each variable on different components of a vector field. Understanding partial derivatives helps break down complex multivariable functions into simpler one-variable functions, making computations more manageable.
Vector Field
A vector field is a mathematical construction where a vector is assigned to every point in a given space or plane. This approach provides a way to model various physical phenomena, such as fluid flow, magnetic fields, and gravitational fields.The given vector field for this exercise is \( \mathbf{F} = (3x^2 - z) \mathbf{i} + (2x + y) \mathbf{j} + (x + 3yz) \mathbf{k} \), consisting of components \( P \), \( Q \), and \( R \) assigned to the \( x \), \( y \), and \( z \) directions respectively.
  • Understanding each component's role is key to interpreting the overall behavior of the field.
This concept is crucial, as real-world vector fields are often complex and require a thorough understanding of each component to derive meaningful insights.
Mathematical Solution Process
The mathematical solution process involves systematically applying theory to solve problems step-by-step. This not only provides a structured approach to math challenges but also ensures accuracy and understanding. For example, in finding the divergence at a specific point, follow these steps:
  • Identifying each component of the vector field.
  • Computing partial derivatives for these components.
  • Using partial derivatives to compute the divergence.
  • Evaluating the calculated divergence at the given point.
Each step logically follows from the previous one, culminating in a final solution. This rigorous approach ensures that all elements are considered, and results are verified through each stage of the problem-solving process.

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Most popular questions from this chapter

If \(\mathbf{v}=\nabla \phi\), find \(\phi\) when \(\mathbf{v}=\left(2 x-4 y^{2}\right) \mathbf{i}-8 x y \mathbf{j}\)

Given \(\mathbf{v}=2 x \mathbf{i}+3 y z \mathbf{j}+5 x z^{2} \mathbf{k}\) find (a) \(\frac{\partial \mathbf{v}}{\partial x}\) (b) \(\frac{\partial \mathbf{v}}{\partial y}\) (c) \(\frac{\partial \mathbf{v}}{\partial z}\)

For any vector field \(\mathbf{F}\) and any scalar constant \(k\) is \(\nabla \cdot(k \mathbf{F})\) the same as \(k \nabla \cdot \mathbf{F} ?\)

For any vector field \(\mathbf{F}\) is \(\nabla \cdot(-\mathbf{F})\) the same as \(-(\nabla \cdot \mathbf{F}) ?\)

A vector field \(\mathbf{v}\) is given by $$ \mathbf{v}=3 x^{2} y \mathbf{i}+2 y^{3} z \mathbf{j}+x z^{3} \mathbf{k} $$ Find (a) \(v_{x}, v_{y}, v_{z}\) (b) \(\frac{\partial v_{x}}{\partial x}, \frac{\partial v_{y}}{\partial y}, \frac{\partial v_{z}}{\partial z}\) (c) \(\nabla \cdot \mathbf{v}\)
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