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Chapter 14: Q 9 (page 1154)

Determine whether the vector fields that follow are conservative. If the field is conservative, find a potential function for it.
$G (x, y, z) = \cos y \sin z i - x \sin y \sin z j - x \cos y \cos z k$

Short Answer

Expert verified

The vector field is non conservative

Step by step solution

01

Step 1:Given information

The given expression is $G (x, y, z) = \cos y \sin z i - x \sin y \sin z j - x \cos y \cos z k$

02

Step 2:Simplification

Consider the vector field,

$G (x, y, z) = \cos y \sin z i - x \sin y \sin z j - x \cos y \cos z k$

$鈬� c u r l \overset{鈫�}{G = 0}$ ,then it is conservative

now,

$c u r l \overset{鈫�}{G} = |\begin{matrix} i & j & k \\ \frac{鈭�}{鈭� x} & \frac{鈭�}{鈭� y} & \frac{鈭�}{鈭� z} \\ (\cos y \sin z) & (- x \sin y \sin z) & - x \cos y \cos z \end{matrix}|$

$= i (\frac{鈭� (- x \cos y \cos z)}{鈭� y} - \frac{鈭� (- x \sin y \sin z)}{鈭� z}) - j (\frac{鈭� (- x \cos y \cos z)}{鈭� x} - \frac{鈭� (\cos y \sin z)}{鈭� z})$

$+ k (\frac{鈭� (- x \sin y \sin z)}{鈭� x} - \frac{鈭� (\cos y \sin z)}{鈭� y})$

$= i (x \sin y \cos z + x \sin y \cos z) - j (- \cos y \cos z - \cos y \cos z)$ $+ k (- \sin y \sin z + \sin z \sin y)$

$鈮� 0$

Since $curl G 鈮� 0$

Therefore,The vector field is non conservative

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Most popular questions from this chapter

What are the outputs of a vector field in the Cartesian plane?

Find ${鈭�}_{S} 鈥� 1 d S$ , where S is the portion of the surface with equation $x = e^{y z} 鈭� e^{鈭� y z}$ that lies on the positive side of the circle of radius 3 and centered at the origin in the yz-plane.

Use what you know about averages to propose a formula for the average rate of flux of a vector field F(x, y ,z) through a smooth surface S in the direction of n.

Find the work done by the vector field

$F (x, y) = \frac{x e \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} i + \frac{y e \sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2}}} j$

in moving an object around the unit circle, starting and ending at $(1, 0)$ .

$Compute the divergence of the vector fields in Exercises 17 鈥� 22 . F (x, y, z) = {xe}^{yz} i + {ye}^{xz} j + {ze}^{xy} k$

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