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Chapter 14: Problem 1

Explain how to compute the divergence of the vector field \(\mathbf{F}=\langle f, g, h\rangle\).

Short Answer

Expert verified
Answer: The formula used to calculate the divergence of a vector field \(\mathbf{F} = \langle f, g, h\rangle\) is \(\nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}\), where \(\frac{\partial f}{\partial x}\), \(\frac{\partial g}{\partial y}\), and \(\frac{\partial h}{\partial z}\) are partial derivatives of the components \(f\), \(g\), and \(h\) with respect to their respective variables.

Step by step solution

01

Determine the components of the vector field

The given vector field \(\mathbf{F}\) can be represented as: $$ \mathbf{F} = \langle f, g, h\rangle$$ Here, \(f\), \(g\), and \(h\) are the components of the vector field in the \(x\), \(y\), and \(z\) directions, respectively.
02

Understand the divergence formula

The divergence of a vector field \(\mathbf{F}\) is denoted as \(\nabla \cdot \mathbf{F}\) and is calculated using the following formula: $$ \nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}$$ Here, \(\frac{\partial f}{\partial x}\), \(\frac{\partial g}{\partial y}\), and \(\frac{\partial h}{\partial z}\) are partial derivatives of the components \(f\), \(g\), and \(h\) with respect to their respective variables.
03

Calculate the partial derivatives

Using the divergence formula, we need to find the partial derivatives: 1. \(\frac{\partial f}{\partial x}\): Partial derivative of \(f\) with respect to \(x\). 2. \(\frac{\partial g}{\partial y}\): Partial derivative of \(g\) with respect to \(y\). 3. \(\frac{\partial h}{\partial z}\): Partial derivative of \(h\) with respect to \(z\).
04

Calculate the divergence

After calculating the partial derivatives in Step 3, substitute their values into the divergence formula: $$ \nabla \cdot \mathbf{F} = \frac{\partial f}{\partial x} + \frac{\partial g}{\partial y} + \frac{\partial h}{\partial z}$$ and simplify the expression to find the divergence of the vector field \(\mathbf{F}\).

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Most popular questions from this chapter

Prove the following properties of the divergence and curl. Assume \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields and \(c\) is a real number. a. \(\nabla \cdot(\mathbf{F}+\mathbf{G})=\nabla \cdot \mathbf{F}+\nabla \cdot \mathbf{G}\) b. \(\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})\) c. \(\nabla \cdot(c \mathbf{F})=c(\nabla \cdot \mathbf{F})\) d. \(\nabla \times(c \mathbf{F})=c(\nabla \times \mathbf{F})\)

Let \(\mathbf{F}=\langle z, 0,0\rangle\) and let \(\mathbf{n}\) be a unit vector aligned with the axis of a paddle wheel located on the \(x\) -axis (see figure). a. If the paddle wheel is oriented with \(\mathbf{n}=\langle 1,0,0\rangle,\) in what direction (if any) does the wheel spin? b. If the paddle wheel is oriented with \(\mathbf{n}=\langle 0,1,0\rangle,\) in what direction (if any) does the wheel spin? c. If the paddle wheel is oriented with \(\mathbf{n}=\langle 0,0,1\rangle,\) in what direction (if any) does the wheel spin?

The area of a region \(R\) in the plane, whose boundary is the closed curve \(C,\) may be computed using line integrals with the formula $$\text { area of } R=\int_{C} x d y=-\int_{C} y d x$$ These ideas reappear later in the chapter. Let \(R\) be the rectangle with vertices \((0,0),(a, 0),(0, b),\) and \((a, b),\) and let \(C\) be the boundary of \(R\) oriented counterclockwise. Use the formula \(A=\int_{C} x d y\) to verify that the area of the rectangle is \(a b.\)

For the general rotation field \(\mathbf{F}=\mathbf{a} \times \mathbf{r},\) where \(\mathbf{a}\) is a nonzero constant vector and \(\mathbf{r}=\langle x, y, z\rangle,\) show that curl \(\mathbf{F}=2 \mathbf{a}\).

Prove that for a real number \(p,\) with \(\mathbf{r}=\langle x, y, z\rangle, \nabla\left(\frac{1}{|\mathbf{r}|^{p}}\right)=\frac{-p \mathbf{r}}{|\mathbf{r}|^{p+2}}.\)
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