<?xml version="1.0" encoding="UTF-8"?>
<quiz>
  <question type="multichoice">
    <name>
      <text>29.1.3.14 Conservative vector field or not</text>
    </name>
    <questiontext format="moodle_auto_format">
      <text><![CDATA[<p>Select the following vector fields which are conservative:</p>]]></text>
    </questiontext>
    <generalfeedback format="moodle_auto_format">
      <text><![CDATA[<p>We determine whether the vector fields are conservative by considering their curl:</p>

<p>(a) \(\underline{F} = (x − y)\underline{i} + (x + y)\underline{j}\)
\[\begin{aligned}
\underline{\nabla} \times \underline{F} &= \begin{vmatrix}
\underline{i} &\underline{j} & \underline{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
(x − y) &  (x + y) & 0
\end{vmatrix}\\
&= \left( \frac{\partial}{\partial y}(0) -  \frac{\partial}{\partial z}(x+y)\right)\underline{i}
- \left( \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(x-y)\right)\underline{j}
+ \left( \frac{\partial}{\partial x}(x+y) -  \frac{\partial}{\partial y}(x-y)\right)\underline{k}\\
&= 2\underline{k} \neq \underline{0}
\end{aligned}\]
So this vector field isn't conservative.</p>

<p>(b) \(\underline{F} = 3x^2y^2\underline{i} + (2x^3y − 1)\underline{j}\)
\[\begin{aligned}
\underline{\nabla} \times \underline{F} &= \begin{vmatrix}
\underline{i} &\underline{j} & \underline{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
(3x^2y^2) &  (2x^3y − 1) & 0
\end{vmatrix}\\
&= \left( \frac{\partial}{\partial y}(0) -  \frac{\partial}{\partial z}((2x^3y − 1))\right)\underline{i}
- \left( \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(3x^2y^2)\right)\underline{j}
+ \left( \frac{\partial}{\partial x}((2x^3y − 1)) -  \frac{\partial}{\partial y}(3x^2y^2)\right)\underline{k}\\
&= (6x^2y-6x^2y)\underline{k}\\
&= \underline{0}
\end{aligned}\]
So this vector field is conservative.</p>

<p>(c) \(\underline{F} = 2x\underline{i} + (xz − 2)\underline{j} + xy\underline{k}\)
\[\begin{aligned}
\underline{\nabla} \times \underline{F} &= \begin{vmatrix}
\underline{i} &\underline{j} & \underline{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
2x &  (xz − 2) & xy
\end{vmatrix}\\
&= \left( \frac{\partial}{\partial y}(xy) -  \frac{\partial}{\partial z}((xz − 2))\right)\underline{i}
- \left( \frac{\partial}{\partial x}(xy) - \frac{\partial}{\partial z}(2x)\right)\underline{j}
+ \left( \frac{\partial}{\partial x}((xz − 2)) -  \frac{\partial}{\partial y}(2x)\right)\underline{k}\\
&= (x-x)\underline{i} - (x - 0)\underline{j}) + (z-0)\underline{k}\\
&=-x\underline{j} + z\underline{k} \neq \underline{0}
\end{aligned}\]
So this vector field isn't conservative.</p>

<p>(d) \(\underline{F} = x^2z\underline{i} + y^2z\underline{j} + \dfrac{1}{3} (x^3 + y^3)\underline{k}\)
\[\begin{aligned}
\underline{\nabla} \times \underline{F} &= \begin{vmatrix}
\underline{i} &\underline{j} & \underline{k} \\
\frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\
x^2z &  y^2z & \frac{1}{3} (x^3 + y^3)
\end{vmatrix}\\
&= \left( \frac{\partial}{\partial y}(\frac{1}{3} (x^3 + y^3)) -  \frac{\partial}{\partial z}((y^2z))\right)\underline{i}
- \left( \frac{\partial}{\partial x}(\frac{1}{3} (x^3 + y^3)) - \frac{\partial}{\partial z}(x^2z)\right)\underline{j}
+ \left( \frac{\partial}{\partial x}((y^2z)) -  \frac{\partial}{\partial y}(x^2z)\right)\underline{k}\\
&= (y^2-y^2)\underline{i} - (x^2 - x^2)\underline{j}) + (0-0)\underline{k}\\\\
&= \underline{0}
\end{aligned}\]
So this vector field is conservative.</p>]]></text>
    </generalfeedback>
    <defaultgrade>1</defaultgrade>
    <penalty>0.3333333</penalty>
    <hidden>0</hidden>
    <idnumber/>
    <single>false</single>
    <shuffleanswers>true</shuffleanswers>
    <answernumbering>abc</answernumbering>
    <showstandardinstruction>0</showstandardinstruction>
    <correctfeedback format="html">
      <text>Your answer is correct.</text>
    </correctfeedback>
    <partiallycorrectfeedback format="html">
      <text>Your answer is partially correct.</text>
    </partiallycorrectfeedback>
    <incorrectfeedback format="html">
      <text>Your answer is incorrect.</text>
    </incorrectfeedback>
    <shownumcorrect/>
    <answer fraction="0" format="html">
      <text>\(\underline{F} = (x − y)\underline{i} + (x + y)\underline{j}\)</text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="50" format="html">
      <text>\(\underline{F} = 3x^2y^2\underline{i} + (2x^3y − 1)\underline{j}\)</text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="0" format="html">
      <text>\(\underline{F} = 2x\underline{i} + (xz − 2)\underline{j} + xy\underline{k}\)</text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
    <answer fraction="50" format="html">
      <text>\(\underline{F} = x^2z\underline{i} + y^2z\underline{j} + \dfrac{1}{3} (x^3 + y^3)\underline{k}\)</text>
      <feedback format="html">
        <text/>
      </feedback>
    </answer>
  </question>
</quiz>
