示例：带奇异值的等值线图

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示例：带奇异值的等值线图

1. 使用已知奇点定义 2 变量函数。

2. 在等值线图中F。

单击可复制此表达式

<region id="ID0E4V1IB" actualWidth="263.52" actualHeight="245.20000000000002" top="57.600000000000009" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-32" end="32">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>9</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="20" />
      <graph-size width="192" height="192" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

3. 将 Z 轴的范围设在 -25 到 25 之间。

单击可复制此表达式

<region id="ID0ENX1IB" actualWidth="263.52" actualHeight="245.20000000000002" top="326.40000000000003" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-25" end="25">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="21">
              <real xmlns="http://schemas.mathsoft.com/math50">-25</real>
            </startValue>
            <endValue resultRef="22">
              <real xmlns="http://schemas.mathsoft.com/math50">25</real>
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="23" />
      <graph-size width="192" height="192" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

4. 通过改变 Y 轴刻度线放大奇点。

单击可复制此表达式

<region id="ID0E4Y1IB" actualWidth="284.51999999999953" actualHeight="243.03333333333336" top="595.2" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.1" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="24">
              <real xmlns="http://schemas.mathsoft.com/math50">-0.1</real>
            </startValue>
            <endValue resultRef="25">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-25" end="25">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="26">
              <real xmlns="http://schemas.mathsoft.com/math50">-25</real>
            </startValue>
            <endValue resultRef="27">
              <real xmlns="http://schemas.mathsoft.com/math50">25</real>
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="28" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

红色和蓝色区域代表了 Z 轴的值超出 -25 和 +25 范围的区域。

5. 通过删除用户定义的全部三个刻度值来恢复默认 Z 轴的值。

单击可复制此表达式

<region id="ID0EH21IB" actualWidth="284.51999999999953" actualHeight="243.03333333333336" top="883.20000000000016" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.1" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="29">
              <real xmlns="http://schemas.mathsoft.com/math50">-0.1</real>
            </startValue>
            <endValue resultRef="30">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-3200" end="3200">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>9</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="31" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

6. 通过设置 Y 轴最小刻度值为零来查看绘图的上部分。

单击可复制此表达式

<region id="ID0EX31IB" actualWidth="284.51999999999953" actualHeight="243.03333333333336" top="1152.0000000000002" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="32">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <endValue resultRef="33">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-32" end="32">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="34" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

由于一奇点位于 y=0 处，因此等值线图无法计算数据。

7. 将 Y 轴的更低刻度值设成稍微比零小的值。

单击可复制此表达式

<region id="ID0EG61IB" actualWidth="284.51999999999953" actualHeight="243.03333333333336" top="1440.0000000000002" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.01" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="35">
              <real xmlns="http://schemas.mathsoft.com/math50">-0.01</real>
            </startValue>
            <endValue resultRef="36">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-7500" end="7500">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="37" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

等值线图可计算数据。

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