Example: Piecewise Functions

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Example: Piecewise Functions

Using the Kronecker delta (δ) and the Heaviside Step (Φ) Functions

1. Define the following piecewise function.

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<region id="ID0E2BP5" top="153.60000000000002" left="57.600000000000009" height="82.104000000000013" width="181.26333333333335" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="26">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <function>
        <id labels="VARIABLE" xml:space="preserve">f</id>
        <boundVars>
          <id labels="VARIABLE" xml:space="preserve">x</id>
        </boundVars>
      </function>
      <program>
        <if>
          <test>
            <apply>
              <lessThan />
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
              <real>0</real>
            </apply>
          </test>
          <then>
            <program>
              <return>
                <apply>
                  <neg />
                  <parens>
                    <apply>
                      <pow />
                      <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
                      <real>2</real>
                    </apply>
                  </parens>
                </apply>
              </return>
            </program>
          </then>
        </if>
        <apply>
          <pow />
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
          <real>2</real>
        </apply>
      </program>
    </define>
  </math>
</region>

2. Use the sign function to see whether various values returned by the function f are positive, negative, or equal to 0.

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3. Use the kronecker's delta (δ) function to see whether the function returns the same value for two results.

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4. Use the Heaviside step (Φ) function to see the sign of the returned value of the matrix.

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Using the Antisymmetric Tensor (ε) Function

1. Use the antisymmetric tensor (ε) function.

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If the function's arguments are an even permutation of (0,1,2), then the function returns 1. For odd permutations, the function returns -1.

2. Call the function again with at least two identical parameters.

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In this case, the function returns 0.