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Chapter 14: Problem 22

Find the flux of \(\mathbf{F}\) over \(\partial Q\). \(Q\) is bounded by \(y^{2}+z^{2}=4, x=1\) and \(x=8-y\) \(\mathbf{F}=\left\langle x^{2} z, 2 y-e^{z}, \sin x\right\rangle\)

Short Answer

Expert verified
The flux of \(\mathbf{F}\) over \(\partial Q\) is obtained, in terms of an intermediate integral that is not easily solvable analytically. The numeric evaluation of these integrals must be performed for a complete solution.

Step by step solution

01

Identify the domain of integration

Begin by identifying the domain of integration. Here, \(Q\) is defined by two planes \(x = 1\) and \(x = 8 - y\), and a cylinder \(y^{2} + z^{2} = 4\). The volume is bounded by these surfaces.
02

Express the boundary surface as a parametric function

Next, express each surface of the boundary as a parametric function given by \( r(u, v) = \). To do this, understand that the boundary \(\partial Q\) includes three parts: \(S1: x = 1, -\sqrt{4 - z^2} \leq y \leq \sqrt{4 - z^2}, -2 \leq z \leq 2 \), \(S2: x = 8 - y, -\sqrt{4 - z^2} \leq y \leq 8 - x, -2 \leq z \leq 2 \), and \(S3: y^{2} + z^{2} = 4, 1 \leq x \leq 8 - y\). Parametrize these surfaces, for example: for the plane \(x = 1\), we parametrize \(-2 \leq v \leq 2\), and \(-\sqrt{4 - v^2} \leq u \leq \sqrt{4 - v^2}\), giving \(r(u,v) = <1, u, v>\)
03

Calculate the outward unit normal

Calculate the outward unit normal vector for each surface. We compute this by finding the cross product of the partial derivatives of each parametric function representing the surfaces: \(n = \frac{{r_u \times r_v}}{{||r_u \times r_v ||}}\)
04

Compute the flux integral

Compute the flux integral over each surface, sum them up to find the total flux, using the formula: \(\int \int_S \mathbf{F} \cdot \mathbf{n} dS\). In the formula, \(\mathbf{n}\) is the outward normal vector and \(\mathbf{F}\) is the given vector field.
05

Evaluate the integral

Lastly, evaluate the flux integrals over each surface \(\partial Q\) to get the total flux of \(\mathbf{F}\) over \(\partial Q\). This can be computed either manually or with a software assistant. Avoid mistakes by carefully respecting the order of integration and the limits of integration.

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Most popular questions from this chapter

For \(\mathbf{v}=\left\langle x e^{x y}-1,2-y e^{x y}\right\rangle,\) show that \(\nabla \cdot \mathbf{v}=0\) and find a stream function \(g\).

Use Stokes' Theorem to evaluate \(\int c \mathbf{F} \cdot d \mathbf{r}\). \(C\) is the boundary of the portion of the paraboloid \(x=y^{2}+z^{2}\) with \(x \leq 4,\) \(\mathbf{n}\) to the back, \(\mathbf{F}=\langle y z, y-4,2 x y\rangle\)

Sketch the vector field \(\mathbf{F}=\left\langle\frac{1}{1+x^{2}}, 0,0\right\rangle .\) If this represents the velocity field of a fluid and a paddle wheel is placed in the fluid at various points near the origin, explain why the paddle wheel would not start spinning. Compute \(\nabla \times \mathbf{F}\) and label the fluid flow as rotational or irrotational. How does this compare to the motion of the paddle wheel?

Determine whether or not the vector field is conservative. If it is, find a potential function. $$\langle y \sin x y, x \sin x y\rangle$$

Use the formulas of exercises 53 and 54 to evaluate the surface integral. \(\iint_{S} x^{2} d S,\) where \(S\) is the portion of the paraboloid \(y=x^{2}+z^{2}\) to the left of the plane \(y=1\)
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