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Chapter 14: Problem 4

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

Short Answer

Expert verified
Given the vector field \(\mathbf{F}=\langle x^2 + 3y, xy - z^2, 2x + y^3\rangle\), compute the curl of the vector field. Solution: Step 1: Write down the given vector field and the curl formula The given vector field is \(\mathbf{F}=\langle x^2 + 3y, xy - z^2, 2x + y^3\rangle\). The curl of the vector field is given by the formula: $$ \text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \left\langle\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}, \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}, \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\right\rangle $$ Step 2: Compute the partial derivatives 1. \(\frac{\partial f}{\partial x} = 2x\), \(\frac{\partial f}{\partial y} = 3\), and \(\frac{\partial f}{\partial z} = 0\) 2. \(\frac{\partial g}{\partial x} = y\), \(\frac{\partial g}{\partial y} = x\), and \(\frac{\partial g}{\partial z} = -2z\) 3. \(\frac{\partial h}{\partial x} = 2\), \(\frac{\partial h}{\partial y} = 3y^2\), and \(\frac{\partial h}{\partial z} = 0\) Step 3: Plug the partial derivatives into the curl formula $$ \text{curl } \mathbf{F} = \left\langle\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}, \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}, \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\right\rangle = \left\langle 3y^2 - (-2z), 0-2, y-3\right\rangle $$ Step 4: Simplify the expression $$ \text{curl } \mathbf{F} = \left\langle 3y^2 + 2z, -2, y-3 \right\rangle $$ The curl of the given vector field \(\mathbf{F}=\langle x^2 + 3y, xy - z^2, 2x + y^3\rangle\) is \(\text{curl } \mathbf{F} = \left\langle 3y^2 + 2z, -2, y-3 \right\rangle\).

Step by step solution

01

Write down the given vector field and the curl formula

We are given the vector field \(\mathbf{F}=\langle f, g, h\rangle\). The curl of the vector field is given by the formula: $$ \text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \left\langle\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}, \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}, \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\right\rangle $$
02

Compute the partial derivatives

Compute the partial derivatives of the components of the vector field: 1. \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\), and \(\frac{\partial f}{\partial z}\) 2. \(\frac{\partial g}{\partial x}\), \(\frac{\partial g}{\partial y}\), and \(\frac{\partial g}{\partial z}\) 3. \(\frac{\partial h}{\partial x}\), \(\frac{\partial h}{\partial y}\), and \(\frac{\partial h}{\partial z}\)
03

Plug the partial derivatives into the curl formula

Substitute the computed partial derivatives into the curl formula: $$ \text{curl } \mathbf{F} = \left\langle\frac{\partial h}{\partial y} - \frac{\partial g}{\partial z}, \frac{\partial f}{\partial z} - \frac{\partial h}{\partial x}, \frac{\partial g}{\partial x} - \frac{\partial f}{\partial y}\right\rangle $$
04

Simplify the expression

Simplify the expression, if possible, to get the curl of the vector field: $$ \text{curl } \mathbf{F} = \left\langle A, B, C \right\rangle $$ The final result will be a vector field that represents the curl of the given vector field \(\mathbf{F}=\langle f, g, h\rangle\).

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Most popular questions from this chapter

Consider the sphere \(x^{2}+y^{2}+z^{2}=4\) and the cylinder \((x-1)^{2}+y^{2}=1,\) for \(z \geq 0\) a. Find the surface area of the cylinder inside the sphere. b. Find the surface area of the sphere inside the cylinder.

Let \(R\) be a region in a plane that has a unit normal vector \(\mathbf{n}=\langle a, b, c\rangle\) and boundary \(C .\) Let \(\mathbf{F}=\langle b z, c x, a y\rangle\) a. Show that \(\nabla \times \mathbf{F}=\mathbf{n}\) b. Use Stokes' Theorem to show that $$\operatorname{area} \text { of } R=\oint_{C} \mathbf{F} \cdot d \mathbf{r}$$ c. Consider the curve \(C\) given by \(\mathbf{r}=\langle 5 \sin t, 13 \cos t, 12 \sin t\rangle\) for \(0 \leq t \leq 2 \pi .\) Prove that \(C\) lies in a plane by showing that \(\mathbf{r} \times \mathbf{r}^{\prime}\) is constant for all \(t\) d. Use part (b) to find the area of the region enclosed by \(C\) in part (c). (Hint: Find the unit normal vector that is consistent with the orientation of \(C\).)

Consider the potential function \(\varphi(x, y, z)=G(\rho),\) where \(G\) is any twice differentiable function and \(\rho=\sqrt{x^{2}+y^{2}+z^{2}} ;\) therefore, \(G\) depends only on the distance from the origin. a. Show that the gradient vector field associated with \(\varphi\) is \(\mathbf{F}=\nabla \varphi=G^{\prime}(\rho) \frac{\mathbf{r}}{\rho},\) where \(\mathbf{r}=\langle x, y, z\rangle\) and \(\rho=|\mathbf{r}|\) b. Let \(S\) be the sphere of radius \(a\) centered at the origin and let \(D\) be the region enclosed by \(S\). Show that the flux of \(\mathbf{F}\) across \(S\) is $$\iint_{S} \mathbf{F} \cdot \mathbf{n} d S=4 \pi a^{2} G^{\prime}(a) $$ c. Show that \(\nabla \cdot \mathbf{F}=\nabla \cdot \nabla \varphi=\frac{2 G^{\prime}(\rho)}{\rho}+G^{\prime \prime}(\rho)\) d. Use part (c) to show that the flux across \(S\) (as given in part (b)) is also obtained by the volume integral \(\iiint_{D} \nabla \cdot \mathbf{F} d V\). (Hint: use spherical coordinates and integrate by parts.)

The heat flow vector field for conducting objects is \(\mathbf{F}=-k \nabla T,\) where \(T(x, y, z)\) is the temperature in the object and \(k>0\) is a constant that depends on the material. Compute the outward flux of \(\mathbf{F}\) across the following surfaces S for the given temperature distributions. Assume \(k=1\). \(T(x, y, z)=100 e^{-x-y} ; S\) consists of the faces of the cube \(|x| \leq 1,|y| \leq 1,|z| \leq 1\).

Let \(S\) be a surface that represents a thin shell with density \(\rho .\) The moments about the coordinate planes (see Section 13.6 ) are \(M_{y z}=\iint_{S} x \rho(x, y, z) d S, M_{x z}=\iint_{S} y \rho(x, y, z) d S\) and \(M_{x y}=\iint_{S} z \rho(x, y, z) d S .\) The coordinates of the center of mass of the shell are \(\bar{x}=\frac{M_{y z}}{m}, \bar{y}=\frac{M_{x z}}{m}, \bar{z}=\frac{M_{x y}}{m},\) where \(m\) is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible. The constant-density half cylinder \(x^{2}+z^{2}=a^{2},-h / 2 \leq y \leq h / 2, z \geq 0\)
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