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Chapter 14: Problem 49

For what vectors \(\mathbf{n}\) is \((\operatorname{curl} \mathbf{F}) \cdot \mathbf{n}=0\) when \(\mathbf{F}=\langle y,-2 z,-x\rangle ?\)

Short Answer

Expert verified
Answer: \(\mathbf{n} = \langle a, 0, c \rangle\), where \(a\) and \(c\) are any real numbers.

Step by step solution

01

Compute the curl of \(\mathbf{F}\)

To compute the curl of \(\mathbf{F}\), we will use the following formula: $$ \operatorname{curl} \mathbf{F} = (\frac{\partial(-2z)}{\partial y} - \frac{\partial(-x)}{\partial z}, \frac{\partial(y)}{\partial x}-\frac{\partial(-z)}{\partial x}, \frac{\partial(-x)}{\partial x} - \frac{\partial(y)}{\partial x}) $$ Now, find the partial derivatives and plug them into the formula for curl. $$ \operatorname{curl} \mathbf{F} = (\frac{\partial(-2z)}{\partial y} - \frac{\partial(-x)}{\partial z}, \frac{\partial(y)}{\partial x}-\frac{\partial(-z)}{\partial x}, \frac{\partial(-x)}{\partial x} - \frac{\partial(y)}{\partial x}) = (0,2,0) $$ Thus, the curl of \(\mathbf{F}\) is \((0,2,0)\).
02

Set the dot product of curl(\(\mathbf{F}\)) and \(\mathbf{n}\) to zero

We want to find the vectors \(\mathbf{n}\) such that the dot product of the curl of \(\mathbf{F}\) and \(\mathbf{n}\) is zero: $$ (\operatorname{curl} \mathbf{F}) \cdot \mathbf{n} = 0 $$ We found that the curl of \(\mathbf{F}\) is \((0,2,0)\). Let vector \(\mathbf{n} = \langle a,b,c \rangle\). Now, substitute these values into the dot product expression: $$ (0,2,0) \cdot \langle a,b,c \rangle = 0 $$ This simplifies to: $$ 2b = 0 $$
03

Find the vectors \(\mathbf{n}\)

From the previous step, we have the equation \(2b = 0\). This implies that \(b = 0\). The dot product will be zero for any vectors \(\mathbf{n} = \langle a,0,c \rangle\) where \(a\) and \(c\) can be any real numbers. So, for any vectors \(\mathbf{n} = \langle a,0,c \rangle\), where \(a\) and \(c\) are real numbers, the dot product of \((\operatorname{curl} \mathbf{F})\) and \(\mathbf{n}\) is zero.

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Most popular questions from this chapter

Consider the radial vector fields \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p},\) where \(p\) is a real number and \(\mathbf{r}=\langle x, y, z\rangle\) Let \(C\) be any circle in the \(x y\) -plane centered at the origin. a. Evaluate a line integral to show that the field has zero circulation on \(C\) b. For what values of \(p\) does Stokes' Theorem apply? For those values of \(p,\) use the surface integral in Stokes' Theorem to show that the field has zero circulation on \(C\).

Prove that if \(\mathbf{F}\) satisfies the conditions of Stokes' Theorem, then \(\iint_{S}(\nabla \times \mathbf{F}) \cdot \mathbf{n} d S=0\) where \(S\) is a smooth surface that encloses a region.

Use the procedure in Exercise 57 to construct potential functions for the following fields. $$\mathbf{F}=\langle-y,-x\rangle$$

The potential function for the gravitational force field due to a mass \(M\) at the origin acting on a mass \(m\) is \(\varphi=G M m /|\mathbf{r}|,\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of the mass \(m\) and \(G\) is the gravitational constant. a. Compute the gravitational force field \(\mathbf{F}=-\nabla \varphi\). b. Show that the field is irrotational; that is, \(\nabla \times \mathbf{F}=\mathbf{0}\).

Consider the radial field \(\mathbf{F}=\mathbf{r} /|\mathbf{r}|^{p}\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(p\) is a real number. Let \(S\) be the sphere of radius \(a\) centered at the origin. Show that the outward flux of \(\mathbf{F}\) across the sphere is \(4 \pi / a^{p-3} .\) It is instructive to do the calculation using both an explicit and parametric description of the sphere.
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