1.6: Difference between revisions

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{{DISPLAYTITLE:Algebraic and Analytic Methods - 1.6 Vectors and Vector-Valued Functions}}
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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex1 1.6#Ex1] || [[Item:Q171|<math>\mathbf{a} = (a_{1},a_{2},a_{3})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{a} = (a_{1},a_{2},a_{3})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a = (a[1], a[2], a[3])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a == (Subscript[a, 1], Subscript[a, 2], Subscript[a, 3])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex1 1.6#Ex1] || <math qid="Q171">\mathbf{a} = (a_{1},a_{2},a_{3})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{a} = (a_{1},a_{2},a_{3})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a = (a[1], a[2], a[3])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a == (Subscript[a, 1], Subscript[a, 2], Subscript[a, 3])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex2 1.6#Ex2] || [[Item:Q172|<math>\mathbf{b} = (b_{1},b_{2},b_{3})</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{b} = (b_{1},b_{2},b_{3})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b = (b[1], b[2], b[3])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b == (Subscript[b, 1], Subscript[b, 2], Subscript[b, 3])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex2 1.6#Ex2] || <math qid="Q172">\mathbf{b} = (b_{1},b_{2},b_{3})</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{b} = (b_{1},b_{2},b_{3})</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b = (b[1], b[2], b[3])</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">b == (Subscript[b, 1], Subscript[b, 2], Subscript[b, 3])</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6.E2 1.6.E2] || [[Item:Q173|<math>\mathbf{a}\cdot\mathbf{b} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{a}\cdot\mathbf{b} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a * b = a[1]*b[1]+ a[2]*b[2]+ a[3]*b[3]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a * b == Subscript[a, 1]*Subscript[b, 1]+ Subscript[a, 2]*Subscript[b, 2]+ Subscript[a, 3]*Subscript[b, 3]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6.E2 1.6.E2] || <math qid="Q173">\mathbf{a}\cdot\mathbf{b} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{a}\cdot\mathbf{b} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a * b = a[1]*b[1]+ a[2]*b[2]+ a[3]*b[3]</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a * b == Subscript[a, 1]*Subscript[b, 1]+ Subscript[a, 2]*Subscript[b, 2]+ Subscript[a, 3]*Subscript[b, 3]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6.E3 1.6.E3] || [[Item:Q174|<math>\|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Norm[a] == Sqrt[a * a]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6.E3 1.6.E3] || <math qid="Q174">\|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Error</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Norm[a] == Sqrt[a * a]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/1.6.E4 1.6.E4] || [[Item:Q175|<math>\cos@@{\theta} = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cos@@{\theta} = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cos[\[Theta]] == Divide[a * b,Norm[a]*Norm[b]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2694569811-.3969495503*I
| [https://dlmf.nist.gov/1.6.E4 1.6.E4] || <math qid="Q175">\cos@@{\theta} = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\cos@@{\theta} = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Cos[\[Theta]] == Divide[a * b,Norm[a]*Norm[b]]</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: -.2694569811-.3969495503*I
Test Values: {a = -1.5, b = -1.5, theta = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .227765517+.4690753764*I
Test Values: {a = -1.5, b = -1.5, theta = 1/2*3^(1/2)+1/2*I}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .227765517+.4690753764*I
Test Values: {a = -1.5, b = -1.5, theta = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .227765517+.4690753764*I
Test Values: {a = -1.5, b = -1.5, theta = -1/2+1/2*I*3^(1/2)}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .227765517+.4690753764*I
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Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br>... skip entries to safe data</div></div>
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6.E6 1.6.E6] || [[Item:Q179|<math>\mathbf{a} = a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{a} = a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a = a[1]*i + a[2]*j + a[3]*((0 , 0 , 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a == Subscript[a, 1]*i + Subscript[a, 2]*j + Subscript[a, 3]*((0 , 0 , 1))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6.E6 1.6.E6] || <math qid="Q179">\mathbf{a} = a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{a} = a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a = a[1]*i + a[2]*j + a[3]*((0 , 0 , 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a == Subscript[a, 1]*i + Subscript[a, 2]*j + Subscript[a, 3]*((0 , 0 , 1))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex6 1.6#Ex6] || [[Item:Q180|<math>\mathbf{i}\times\mathbf{j} = \mathbf{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{i}\times\mathbf{j} = \mathbf{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i * j = ((0 , 0 , 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i * j == ((0 , 0 , 1))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex6 1.6#Ex6] || <math qid="Q180">\mathbf{i}\times\mathbf{j} = \mathbf{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{i}\times\mathbf{j} = \mathbf{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i * j = ((0 , 0 , 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i * j == ((0 , 0 , 1))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex7 1.6#Ex7] || [[Item:Q181|<math>\mathbf{j}\times\mathbf{k} = \mathbf{i}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{j}\times\mathbf{k} = \mathbf{i}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j *((0 , 0 , 1)) = i</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j *((0 , 0 , 1)) == i</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex7 1.6#Ex7] || <math qid="Q181">\mathbf{j}\times\mathbf{k} = \mathbf{i}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{j}\times\mathbf{k} = \mathbf{i}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j *((0 , 0 , 1)) = i</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j *((0 , 0 , 1)) == i</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex8 1.6#Ex8] || [[Item:Q182|<math>\mathbf{k}\times\mathbf{i} = \mathbf{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{k}\times\mathbf{i} = \mathbf{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * i = j</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * i == j</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex8 1.6#Ex8] || <math qid="Q182">\mathbf{k}\times\mathbf{i} = \mathbf{j}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{k}\times\mathbf{i} = \mathbf{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * i = j</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * i == j</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex9 1.6#Ex9] || [[Item:Q183|<math>\mathbf{j}\times\mathbf{i} = -\mathbf{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{j}\times\mathbf{i} = -\mathbf{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j * i = -((0 , 0 , 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j * i == -((0 , 0 , 1))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex9 1.6#Ex9] || <math qid="Q183">\mathbf{j}\times\mathbf{i} = -\mathbf{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{j}\times\mathbf{i} = -\mathbf{k}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j * i = -((0 , 0 , 1))</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">j * i == -((0 , 0 , 1))</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex10 1.6#Ex10] || [[Item:Q184|<math>\mathbf{k}\times\mathbf{j} = -\mathbf{i}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{k}\times\mathbf{j} = -\mathbf{i}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * j = - i</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * j == - i</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex10 1.6#Ex10] || <math qid="Q184">\mathbf{k}\times\mathbf{j} = -\mathbf{i}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{k}\times\mathbf{j} = -\mathbf{i}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * j = - i</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">((0 , 0 , 1)) * j == - i</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6#Ex11 1.6#Ex11] || [[Item:Q185|<math>\mathbf{i}\times\mathbf{k} = -\mathbf{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{i}\times\mathbf{k} = -\mathbf{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i *((0 , 0 , 1)) = - j</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i *((0 , 0 , 1)) == - j</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6#Ex11 1.6#Ex11] || <math qid="Q185">\mathbf{i}\times\mathbf{k} = -\mathbf{j}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{i}\times\mathbf{k} = -\mathbf{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i *((0 , 0 , 1)) = - j</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">i *((0 , 0 , 1)) == - j</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
|- style="background: #dfe6e9;"
|- style="background: #dfe6e9;"
| [https://dlmf.nist.gov/1.6.E12 1.6.E12] || [[Item:Q189|<math>a_{j}b_{j} = \sum_{j=1}^{3}a_{j}b_{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{j}b_{j} = \sum_{j=1}^{3}a_{j}b_{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[j]*b[j] = sum(a[j]*b[j], j = 1..3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, j]*Subscript[b, j] == Sum[Subscript[a, j]*Subscript[b, j], {j, 1, 3}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6.E12 1.6.E12] || <math qid="Q189">a_{j}b_{j} = \sum_{j=1}^{3}a_{j}b_{j}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>a_{j}b_{j} = \sum_{j=1}^{3}a_{j}b_{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">a[j]*b[j] = sum(a[j]*b[j], j = 1..3)</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">Subscript[a, j]*Subscript[b, j] == Sum[Subscript[a, j]*Subscript[b, j], {j, 1, 3}, GenerateConditions->None]</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/1.6#Ex15 1.6#Ex15] || [[Item:Q194|<math>\LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[1, 2, 3] = LeviCivita[3, 1, 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 1, 2, 3] == Part[LeviCivitaTensor[3,List], 3, 1, 2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/1.6#Ex15 1.6#Ex15] || <math qid="Q194">\LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[1, 2, 3] = LeviCivita[3, 1, 2]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 1, 2, 3] == Part[LeviCivitaTensor[3,List], 3, 1, 2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/1.6#Ex15 1.6#Ex15] || [[Item:Q194|<math>\LeviCivitasym{3}{1}{2} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{3}{1}{2} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[3, 1, 2] = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 3, 1, 2] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/1.6#Ex15 1.6#Ex15] || <math qid="Q194">\LeviCivitasym{3}{1}{2} = 1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{3}{1}{2} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[3, 1, 2] = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 3, 1, 2] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/1.6#Ex16 1.6#Ex16] || [[Item:Q195|<math>\LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[2, 1, 3] = LeviCivita[3, 2, 1]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 2, 1, 3] == Part[LeviCivitaTensor[3,List], 3, 2, 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/1.6#Ex16 1.6#Ex16] || <math qid="Q195">\LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[2, 1, 3] = LeviCivita[3, 2, 1]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 2, 1, 3] == Part[LeviCivitaTensor[3,List], 3, 2, 1]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/1.6#Ex16 1.6#Ex16] || [[Item:Q195|<math>\LeviCivitasym{3}{2}{1} = -1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{3}{2}{1} = -1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[3, 2, 1] = - 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 3, 2, 1] == - 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/1.6#Ex16 1.6#Ex16] || <math qid="Q195">\LeviCivitasym{3}{2}{1} = -1</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{3}{2}{1} = -1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[3, 2, 1] = - 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 3, 2, 1] == - 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/1.6#Ex17 1.6#Ex17] || [[Item:Q196|<math>\LeviCivitasym{2}{2}{1} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{2}{2}{1} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[2, 2, 1] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 2, 2, 1] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
| [https://dlmf.nist.gov/1.6#Ex17 1.6#Ex17] || <math qid="Q196">\LeviCivitasym{2}{2}{1} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{2}{2}{1} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[2, 2, 1] = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], 2, 2, 1] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/1.6.E16 1.6.E16] || [[Item:Q197|<math>\LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[j, k, ell]*LeviCivita[ell, m, n] = KroneckerDelta[j, m]*KroneckerDelta[k, n]- KroneckerDelta[j, n]*KroneckerDelta[k, m]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Part[LeviCivitaTensor[3,List], \[ScriptL], m, n] == KroneckerDelta[j, m]*KroneckerDelta[k, n]- KroneckerDelta[j, n]*KroneckerDelta[k, m]</syntaxhighlight> || Failure || Failure || Error || Error
| [https://dlmf.nist.gov/1.6.E16 1.6.E16] || <math qid="Q197">\LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>LeviCivita[j, k, ell]*LeviCivita[ell, m, n] = KroneckerDelta[j, m]*KroneckerDelta[k, n]- KroneckerDelta[j, n]*KroneckerDelta[k, m]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Part[LeviCivitaTensor[3,List], \[ScriptL], m, n] == KroneckerDelta[j, m]*KroneckerDelta[k, n]- KroneckerDelta[j, n]*KroneckerDelta[k, m]</syntaxhighlight> || Failure || Failure || Error || Error
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| [https://dlmf.nist.gov/1.6.E17 1.6.E17] || [[Item:Q198|<math>\mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>e[j] * e[k] = LeviCivita[j, k, ell]*e[ell]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[e, j] * Subscript[e, k] == Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Subscript[e, \[ScriptL]]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/1.6.E17 1.6.E17] || <math qid="Q198">\mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>e[j] * e[k] = LeviCivita[j, k, ell]*e[ell]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[e, j] * Subscript[e, k] == Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Subscript[e, \[ScriptL]]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/1.6.E18 1.6.E18] || [[Item:Q199|<math>a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}a_{j}b_{k}\mathbf{e}_{\ell}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}a_{j}b_{k}\mathbf{e}_{\ell}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>a[j]*e[j] * b[k]*e[k] = LeviCivita[j, k, ell]*a[j]*b[k]*e[ell]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[a, j]*Subscript[e, j] * Subscript[b, k]*Subscript[e, k] == Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Subscript[a, j]*Subscript[b, k]*Subscript[e, \[ScriptL]]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/1.6.E18 1.6.E18] || <math qid="Q199">a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}a_{j}b_{k}\mathbf{e}_{\ell}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}a_{j}b_{k}\mathbf{e}_{\ell}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>a[j]*e[j] * b[k]*e[k] = LeviCivita[j, k, ell]*a[j]*b[k]*e[ell]</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[a, j]*Subscript[e, j] * Subscript[b, k]*Subscript[e, k] == Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Subscript[a, j]*Subscript[b, k]*Subscript[e, \[ScriptL]]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/1.6.E43 1.6.E43] || [[Item:Q224|<math>\mathbf{F}(x,y) = F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{F}(x,y) = F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">F(x , y) = F[1](x , y)* i + F[2](x , y)* j</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">F[x , y] == Subscript[F, 1][x , y]* i + Subscript[F, 2][x , y]* j</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
| [https://dlmf.nist.gov/1.6.E43 1.6.E43] || <math qid="Q224">\mathbf{F}(x,y) = F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}</math><br><syntaxhighlight lang="tex" style="font-size: 75%; background: inherit;" inline>\mathbf{F}(x,y) = F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}</syntaxhighlight> || <math></math> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">F(x , y) = F[1](x , y)* i + F[2](x , y)* j</pre></div> || <div class="mw-highlight mw-highlight-lang-mathematica mw-content-ltr" dir="ltr"><pre style="background: inherit;">F[x , y] == Subscript[F, 1][x , y]* i + Subscript[F, 2][x , y]* j</pre></div> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| [https://dlmf.nist.gov/1.6.E46 1.6.E46] || [[Item:Q227|<math>\mathbf{T}_{u} = \pderiv{x}{u}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{u}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{u}(u_{0},v_{0})\mathbf{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mathbf{T}_{u} = \pderiv{x}{u}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{u}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{u}(u_{0},v_{0})\mathbf{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>T[u] = diff(x, u)*(u[0], v[0])*i + diff(y, u)*(u[0], v[0])*j + diff(x + y*I, u)*(u[0], v[0])*((0 , 0 , 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[T, u] == D[x, u]*(Subscript[u, 0], Subscript[v, 0])*i + D[y, u]*(Subscript[u, 0], Subscript[v, 0])*j + D[x + y*I, u]*(Subscript[u, 0], Subscript[v, 0])*((0 , 0 , 1))</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
| [https://dlmf.nist.gov/1.6.E46 1.6.E46] || <math qid="Q227">\mathbf{T}_{u} = \pderiv{x}{u}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{u}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{u}(u_{0},v_{0})\mathbf{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mathbf{T}_{u} = \pderiv{x}{u}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{u}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{u}(u_{0},v_{0})\mathbf{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>T[u] = diff(x, u)*(u[0], v[0])*i + diff(y, u)*(u[0], v[0])*j + diff(x + y*I, u)*(u[0], v[0])*((0 , 0 , 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[T, u] == D[x, u]*(Subscript[u, 0], Subscript[v, 0])*i + D[y, u]*(Subscript[u, 0], Subscript[v, 0])*j + D[x + y*I, u]*(Subscript[u, 0], Subscript[v, 0])*((0 , 0 , 1))</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Line 68: Line 70:
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || -
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || -
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| [https://dlmf.nist.gov/1.6.E47 1.6.E47] || [[Item:Q228|<math>\mathbf{T}_{v} = \pderiv{x}{v}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{v}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{v}(u_{0},v_{0})\mathbf{k}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mathbf{T}_{v} = \pderiv{x}{v}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{v}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{v}(u_{0},v_{0})\mathbf{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>T[v] = diff(x, v)*(u[0], v[0])*i + diff(y, v)*(u[0], v[0])*j + diff(x + y*I, v)*(u[0], v[0])*((0 , 0 , 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[T, v] == D[x, v]*(Subscript[u, 0], Subscript[v, 0])*i + D[y, v]*(Subscript[u, 0], Subscript[v, 0])*j + D[x + y*I, v]*(Subscript[u, 0], Subscript[v, 0])*((0 , 0 , 1))</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
| [https://dlmf.nist.gov/1.6.E47 1.6.E47] || <math qid="Q228">\mathbf{T}_{v} = \pderiv{x}{v}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{v}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{v}(u_{0},v_{0})\mathbf{k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\mathbf{T}_{v} = \pderiv{x}{v}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{v}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{v}(u_{0},v_{0})\mathbf{k}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>T[v] = diff(x, v)*(u[0], v[0])*i + diff(y, v)*(u[0], v[0])*j + diff(x + y*I, v)*(u[0], v[0])*((0 , 0 , 1))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Subscript[T, v] == D[x, v]*(Subscript[u, 0], Subscript[v, 0])*i + D[y, v]*(Subscript[u, 0], Subscript[v, 0])*j + D[x + y*I, v]*(Subscript[u, 0], Subscript[v, 0])*((0 , 0 , 1))</syntaxhighlight> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: .8660254040+.5000000000*I
Line 74: Line 76:
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}</syntaxhighlight><br>... skip entries to safe data</div></div> || Error
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| [https://dlmf.nist.gov/1.6.E49 1.6.E49] || [[Item:Q230|<math>\|\mathbf{T}_{u}\times\mathbf{T}_{v}\| = \sqrt{\left(\pderiv{(x,y)}{(u,v)}\right)^{2}+\left(\pderiv{(y,z)}{(u,v)}\right)^{2}+\left(\pderiv{(x,z)}{(u,v)}\right)^{2}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\|\mathbf{T}_{u}\times\mathbf{T}_{v}\| = \sqrt{\left(\pderiv{(x,y)}{(u,v)}\right)^{2}+\left(\pderiv{(y,z)}{(u,v)}\right)^{2}+\left(\pderiv{(x,z)}{(u,v)}\right)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Norm[Subscript[T, u] * Subscript[T, v]] == Sqrt[(((D[(x , y), {temp, 1}]/.temp-> (u , v))))^(2)+(((D[(y ,(x + y*I)), {temp, 1}]/.temp-> (u , v))))^(2)+(((D[(x ,(x + y*I)), {temp, 1}]/.temp-> (u , v))))^(2)]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/1.6.E49 1.6.E49] || <math qid="Q230">\|\mathbf{T}_{u}\times\mathbf{T}_{v}\| = \sqrt{\left(\pderiv{(x,y)}{(u,v)}\right)^{2}+\left(\pderiv{(y,z)}{(u,v)}\right)^{2}+\left(\pderiv{(x,z)}{(u,v)}\right)^{2}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\|\mathbf{T}_{u}\times\mathbf{T}_{v}\| = \sqrt{\left(\pderiv{(x,y)}{(u,v)}\right)^{2}+\left(\pderiv{(y,z)}{(u,v)}\right)^{2}+\left(\pderiv{(x,z)}{(u,v)}\right)^{2}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Norm[Subscript[T, u] * Subscript[T, v]] == Sqrt[(((D[(x , y), {temp, 1}]/.temp-> (u , v))))^(2)+(((D[(y ,(x + y*I)), {temp, 1}]/.temp-> (u , v))))^(2)+(((D[(x ,(x + y*I)), {temp, 1}]/.temp-> (u , v))))^(2)]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/1.6.E50 1.6.E50] || [[Item:Q231|<math>\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Norm[Subscript[T, \[Theta]] * Subscript[T, \[Phi]]] == \[Rho]^(2)* Abs[Sin[\[Theta]]]</syntaxhighlight> || Translation Error || Translation Error || - || -
| [https://dlmf.nist.gov/1.6.E50 1.6.E50] || <math qid="Q231">\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>Norm[Subscript[T, \[Theta]] * Subscript[T, \[Phi]]] == \[Rho]^(2)* Abs[Sin[\[Theta]]]</syntaxhighlight> || Translation Error || Translation Error || - || -
|}
|}
</div>
</div>

Latest revision as of 10:59, 28 June 2021


DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.6#Ex1 𝐚 = ( a 1 , a 2 , a 3 ) 𝐚 subscript 𝑎 1 subscript 𝑎 2 subscript 𝑎 3 {\displaystyle{\displaystyle\mathbf{a}=(a_{1},a_{2},a_{3})}}
\mathbf{a} = (a_{1},a_{2},a_{3})		 

a = (a[1], a[2], a[3])
 
a == (Subscript[a, 1], Subscript[a, 2], Subscript[a, 3])
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex2 𝐛 = ( b 1 , b 2 , b 3 ) 𝐛 subscript 𝑏 1 subscript 𝑏 2 subscript 𝑏 3 {\displaystyle{\displaystyle\mathbf{b}=(b_{1},b_{2},b_{3})}}
\mathbf{b} = (b_{1},b_{2},b_{3})		 

b = (b[1], b[2], b[3])
 
b == (Subscript[b, 1], Subscript[b, 2], Subscript[b, 3])
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E2 𝐚 ⋅ 𝐛 = a 1 ⁢ b 1 + a 2 ⁢ b 2 + a 3 ⁢ b 3 ⋅ 𝐚 𝐛 subscript 𝑎 1 subscript 𝑏 1 subscript 𝑎 2 subscript 𝑏 2 subscript 𝑎 3 subscript 𝑏 3 {\displaystyle{\displaystyle\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_% {3}b_{3}}}
\mathbf{a}\cdot\mathbf{b} = a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}		 

a * b = a[1]*b[1]+ a[2]*b[2]+ a[3]*b[3]
 
a * b == Subscript[a, 1]*Subscript[b, 1]+ Subscript[a, 2]*Subscript[b, 2]+ Subscript[a, 3]*Subscript[b, 3]
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E3 ∥ 𝐚 ∥ = 𝐚 ⋅ 𝐚 norm 𝐚 ⋅ 𝐚 𝐚 {\displaystyle{\displaystyle\|\mathbf{a}\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}}}
\|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}		 

Error
 
Norm[a] == Sqrt[a * a]
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E4 cos ⁡ θ = 𝐚 ⋅ 𝐛 ∥ 𝐚 ∥ ⁢ ∥ 𝐛 ∥ 𝜃 ⋅ 𝐚 𝐛 norm 𝐚 norm 𝐛 {\displaystyle{\displaystyle\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|% \mathbf{a}\|\;\|\mathbf{b}\|}}}
\cos@@{\theta} = \frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|}		 

Error

Cos[\[Theta]] == Divide[a * b,Norm[a]*Norm[b]]
Failure Failure
Failed [300 / 300]
Result: -.2694569811-.3969495503*I
Test Values: {a = -1.5, b = -1.5, theta = 1/2*3^(1/2)+1/2*I}


Result: .227765517+.4690753764*I
Test Values: {a = -1.5, b = -1.5, theta = -1/2+1/2*I*3^(1/2)}


Result: .227765517+.4690753764*I
Test Values: {a = -1.5, b = -1.5, theta = 1/2-1/2*I*3^(1/2)}


Result: -.2694569811-.3969495503*I
Test Values: {a = -1.5, b = -1.5, theta = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.2694569809427748, -0.3969495502290325]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.2277655168641104, 0.46907537626850365]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.6.E6 𝐚 = a 1 ⁢ 𝐢 + a 2 ⁢ 𝐣 + a 3 ⁢ 𝐤 𝐚 subscript 𝑎 1 𝐢 subscript 𝑎 2 𝐣 subscript 𝑎 3 𝐤 {\displaystyle{\displaystyle\mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}% \mathbf{k}}}
\mathbf{a} = a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}		 

a = a[1]*i + a[2]*j + a[3]*((0 , 0 , 1))
 
a == Subscript[a, 1]*i + Subscript[a, 2]*j + Subscript[a, 3]*((0 , 0 , 1))
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex6 𝐢 × 𝐣 = 𝐤 𝐢 𝐣 𝐤 {\displaystyle{\displaystyle\mathbf{i}\times\mathbf{j}=\mathbf{k}}}
\mathbf{i}\times\mathbf{j} = \mathbf{k}		 

i * j = ((0 , 0 , 1))
 
i * j == ((0 , 0 , 1))
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex7 𝐣 × 𝐤 = 𝐢 𝐣 𝐤 𝐢 {\displaystyle{\displaystyle\mathbf{j}\times\mathbf{k}=\mathbf{i}}}
\mathbf{j}\times\mathbf{k} = \mathbf{i}		 

j *((0 , 0 , 1)) = i
 
j *((0 , 0 , 1)) == i
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex8 𝐤 × 𝐢 = 𝐣 𝐤 𝐢 𝐣 {\displaystyle{\displaystyle\mathbf{k}\times\mathbf{i}=\mathbf{j}}}
\mathbf{k}\times\mathbf{i} = \mathbf{j}		 

((0 , 0 , 1)) * i = j
 
((0 , 0 , 1)) * i == j
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex9 𝐣 × 𝐢 = - 𝐤 𝐣 𝐢 𝐤 {\displaystyle{\displaystyle\mathbf{j}\times\mathbf{i}=-\mathbf{k}}}
\mathbf{j}\times\mathbf{i} = -\mathbf{k}		 

j * i = -((0 , 0 , 1))
 
j * i == -((0 , 0 , 1))
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex10 𝐤 × 𝐣 = - 𝐢 𝐤 𝐣 𝐢 {\displaystyle{\displaystyle\mathbf{k}\times\mathbf{j}=-\mathbf{i}}}
\mathbf{k}\times\mathbf{j} = -\mathbf{i}		 

((0 , 0 , 1)) * j = - i
 
((0 , 0 , 1)) * j == - i
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex11 𝐢 × 𝐤 = - 𝐣 𝐢 𝐤 𝐣 {\displaystyle{\displaystyle\mathbf{i}\times\mathbf{k}=-\mathbf{j}}}
\mathbf{i}\times\mathbf{k} = -\mathbf{j}		 

i *((0 , 0 , 1)) = - j
 
i *((0 , 0 , 1)) == - j
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E12 a j ⁢ b j = ∑ j = 1 3 a j ⁢ b j subscript 𝑎 𝑗 subscript 𝑏 𝑗 superscript subscript 𝑗 1 3 subscript 𝑎 𝑗 subscript 𝑏 𝑗 {\displaystyle{\displaystyle a_{j}b_{j}=\sum_{j=1}^{3}a_{j}b_{j}}}
a_{j}b_{j} = \sum_{j=1}^{3}a_{j}b_{j}		 

a[j]*b[j] = sum(a[j]*b[j], j = 1..3)
 
Subscript[a, j]*Subscript[b, j] == Sum[Subscript[a, j]*Subscript[b, j], {j, 1, 3}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex15 ϵ 1 ⁣ 2 ⁣ 3 = ϵ 3 ⁣ 1 ⁣ 2 Levi-Civita 1 2 3 Levi-Civita 3 1 2 {\displaystyle{\displaystyle\epsilon_{123}=\epsilon_{312}}}
\LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}		 

LeviCivita[1, 2, 3] = LeviCivita[3, 1, 2]

Part[LeviCivitaTensor[3,List], 1, 2, 3] == Part[LeviCivitaTensor[3,List], 3, 1, 2]
Successful Successful - Successful [Tested: 1]
1.6#Ex15 ϵ 3 ⁣ 1 ⁣ 2 = 1 Levi-Civita 3 1 2 1 {\displaystyle{\displaystyle\epsilon_{312}=1}}
\LeviCivitasym{3}{1}{2} = 1		 

LeviCivita[3, 1, 2] = 1

Part[LeviCivitaTensor[3,List], 3, 1, 2] == 1
Successful Successful - Successful [Tested: 1]
1.6#Ex16 ϵ 2 ⁣ 1 ⁣ 3 = ϵ 3 ⁣ 2 ⁣ 1 Levi-Civita 2 1 3 Levi-Civita 3 2 1 {\displaystyle{\displaystyle\epsilon_{213}=\epsilon_{321}}}
\LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}		 

LeviCivita[2, 1, 3] = LeviCivita[3, 2, 1]

Part[LeviCivitaTensor[3,List], 2, 1, 3] == Part[LeviCivitaTensor[3,List], 3, 2, 1]
Successful Successful - Successful [Tested: 1]
1.6#Ex16 ϵ 3 ⁣ 2 ⁣ 1 = - 1 Levi-Civita 3 2 1 1 {\displaystyle{\displaystyle\epsilon_{321}=-1}}
\LeviCivitasym{3}{2}{1} = -1		 

LeviCivita[3, 2, 1] = - 1

Part[LeviCivitaTensor[3,List], 3, 2, 1] == - 1
Successful Successful - Successful [Tested: 1]
1.6#Ex17 ϵ 2 ⁣ 2 ⁣ 1 = 0 Levi-Civita 2 2 1 0 {\displaystyle{\displaystyle\epsilon_{221}=0}}
\LeviCivitasym{2}{2}{1} = 0		 

LeviCivita[2, 2, 1] = 0

Part[LeviCivitaTensor[3,List], 2, 2, 1] == 0
Successful Successful - Successful [Tested: 1]
1.6.E16 ϵ j ⁣ k ⁣ ℓ ⁢ ϵ ℓ ⁣ m ⁣ n = δ j , m ⁢ δ k , n - δ j , n ⁢ δ k , m Levi-Civita 𝑗 𝑘 ℓ Levi-Civita ℓ 𝑚 𝑛 Kronecker 𝑗 𝑚 Kronecker 𝑘 𝑛 Kronecker 𝑗 𝑛 Kronecker 𝑘 𝑚 {\displaystyle{\displaystyle\epsilon_{jk\ell}\epsilon_{\ell mn}=\delta_{j,m}% \delta_{k,n}-\delta_{j,n}\delta_{k,m}}}
\LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}		 

LeviCivita[j, k, ell]*LeviCivita[ell, m, n] = KroneckerDelta[j, m]*KroneckerDelta[k, n]- KroneckerDelta[j, n]*KroneckerDelta[k, m]

Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Part[LeviCivitaTensor[3,List], \[ScriptL], m, n] == KroneckerDelta[j, m]*KroneckerDelta[k, n]- KroneckerDelta[j, n]*KroneckerDelta[k, m]
Failure Failure Error Error
1.6.E17 𝐞 j × 𝐞 k = ϵ j ⁣ k ⁣ ℓ ⁢ 𝐞 ℓ subscript 𝐞 𝑗 subscript 𝐞 𝑘 Levi-Civita 𝑗 𝑘 ℓ subscript 𝐞 ℓ {\displaystyle{\displaystyle\mathbf{e}_{j}\times\mathbf{e}_{k}=\epsilon_{jk% \ell}\mathbf{e}_{\ell}}}
\mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}		 

e[j] * e[k] = LeviCivita[j, k, ell]*e[ell]

Subscript[e, j] * Subscript[e, k] == Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Subscript[e, \[ScriptL]]
Translation Error Translation Error - -
1.6.E18 a j ⁢ 𝐞 j × b k ⁢ 𝐞 k = ϵ j ⁣ k ⁣ ℓ ⁢ a j ⁢ b k ⁢ 𝐞 ℓ subscript 𝑎 𝑗 subscript 𝐞 𝑗 subscript 𝑏 𝑘 subscript 𝐞 𝑘 Levi-Civita 𝑗 𝑘 ℓ subscript 𝑎 𝑗 subscript 𝑏 𝑘 subscript 𝐞 ℓ {\displaystyle{\displaystyle a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k}=% \epsilon_{jk\ell}a_{j}b_{k}\mathbf{e}_{\ell}}}
a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}a_{j}b_{k}\mathbf{e}_{\ell}		 

a[j]*e[j] * b[k]*e[k] = LeviCivita[j, k, ell]*a[j]*b[k]*e[ell]

Subscript[a, j]*Subscript[e, j] * Subscript[b, k]*Subscript[e, k] == Part[LeviCivitaTensor[3,List], j, k, \[ScriptL]]*Subscript[a, j]*Subscript[b, k]*Subscript[e, \[ScriptL]]
Translation Error Translation Error - -
1.6.E43 𝐅 ⁢ ( x , y ) = F 1 ⁢ ( x , y ) ⁢ 𝐢 + F 2 ⁢ ( x , y ) ⁢ 𝐣 𝐅 𝑥 𝑦 subscript 𝐹 1 𝑥 𝑦 𝐢 subscript 𝐹 2 𝑥 𝑦 𝐣 {\displaystyle{\displaystyle\mathbf{F}(x,y)=F_{1}(x,y)\mathbf{i}+F_{2}(x,y)% \mathbf{j}}}
\mathbf{F}(x,y) = F_{1}(x,y)\mathbf{i}+F_{2}(x,y)\mathbf{j}		 

F(x , y) = F[1](x , y)* i + F[2](x , y)* j
 
F[x , y] == Subscript[F, 1][x , y]* i + Subscript[F, 2][x , y]* j
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E46 𝐓 u = ∂ ⁡ x ∂ ⁡ u ⁢ ( u 0 , v 0 ) ⁢ 𝐢 + ∂ ⁡ y ∂ ⁡ u ⁢ ( u 0 , v 0 ) ⁢ 𝐣 + ∂ ⁡ z ∂ ⁡ u ⁢ ( u 0 , v 0 ) ⁢ 𝐤 subscript 𝐓 𝑢 partial-derivative 𝑥 𝑢 subscript 𝑢 0 subscript 𝑣 0 𝐢 partial-derivative 𝑦 𝑢 subscript 𝑢 0 subscript 𝑣 0 𝐣 partial-derivative 𝑧 𝑢 subscript 𝑢 0 subscript 𝑣 0 𝐤 {\displaystyle{\displaystyle\mathbf{T}_{u}=\frac{\partial x}{\partial u}(u_{0}% ,v_{0})\mathbf{i}+\frac{\partial y}{\partial u}(u_{0},v_{0})\mathbf{j}+\frac{% \partial z}{\partial u}(u_{0},v_{0})\mathbf{k}}}
\mathbf{T}_{u} = \pderiv{x}{u}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{u}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{u}(u_{0},v_{0})\mathbf{k}		 

T[u] = diff(x, u)*(u[0], v[0])*i + diff(y, u)*(u[0], v[0])*j + diff(x + y*I, u)*(u[0], v[0])*((0 , 0 , 1))

Subscript[T, u] == D[x, u]*(Subscript[u, 0], Subscript[v, 0])*i + D[y, u]*(Subscript[u, 0], Subscript[v, 0])*j + D[x + y*I, u]*(Subscript[u, 0], Subscript[v, 0])*((0 , 0 , 1))
Failure Failure
Failed [300 / 300]
Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}


Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}


Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 3}


Result: .8660254040+.5000000000*I
Test Values: {u = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[u] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}

... skip entries to safe data
 -
1.6.E47 𝐓 v = ∂ ⁡ x ∂ ⁡ v ⁢ ( u 0 , v 0 ) ⁢ 𝐢 + ∂ ⁡ y ∂ ⁡ v ⁢ ( u 0 , v 0 ) ⁢ 𝐣 + ∂ ⁡ z ∂ ⁡ v ⁢ ( u 0 , v 0 ) ⁢ 𝐤 subscript 𝐓 𝑣 partial-derivative 𝑥 𝑣 subscript 𝑢 0 subscript 𝑣 0 𝐢 partial-derivative 𝑦 𝑣 subscript 𝑢 0 subscript 𝑣 0 𝐣 partial-derivative 𝑧 𝑣 subscript 𝑢 0 subscript 𝑣 0 𝐤 {\displaystyle{\displaystyle\mathbf{T}_{v}=\frac{\partial x}{\partial v}(u_{0}% ,v_{0})\mathbf{i}+\frac{\partial y}{\partial v}(u_{0},v_{0})\mathbf{j}+\frac{% \partial z}{\partial v}(u_{0},v_{0})\mathbf{k}}}
\mathbf{T}_{v} = \pderiv{x}{v}(u_{0},v_{0})\mathbf{i}+\pderiv{y}{v}(u_{0},v_{0})\mathbf{j}+\pderiv{z}{v}(u_{0},v_{0})\mathbf{k}		 

T[v] = diff(x, v)*(u[0], v[0])*i + diff(y, v)*(u[0], v[0])*j + diff(x + y*I, v)*(u[0], v[0])*((0 , 0 , 1))

Subscript[T, v] == D[x, v]*(Subscript[u, 0], Subscript[v, 0])*i + D[y, v]*(Subscript[u, 0], Subscript[v, 0])*j + D[x + y*I, v]*(Subscript[u, 0], Subscript[v, 0])*((0 , 0 , 1))
Failure Failure
Failed [300 / 300]
Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 1}


Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 2}


Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 1, k = 3}


Result: .8660254040+.5000000000*I
Test Values: {v = 1/2*3^(1/2)+1/2*I, x = 1.5, y = -1.5, T[v] = 1/2*3^(1/2)+1/2*I, u[0] = 1/2*3^(1/2)+1/2*I, v[0] = 1/2*3^(1/2)+1/2*I, j = 2, k = 1}

... skip entries to safe data
 Error
1.6.E49 ∥ 𝐓 u × 𝐓 v ∥ = ( ∂ ⁡ ( x , y ) ∂ ⁡ ( u , v ) ) 2 + ( ∂ ⁡ ( y , z ) ∂ ⁡ ( u , v ) ) 2 + ( ∂ ⁡ ( x , z ) ∂ ⁡ ( u , v ) ) 2 norm subscript 𝐓 𝑢 subscript 𝐓 𝑣 superscript partial-derivative 𝑥 𝑦 𝑢 𝑣 2 superscript partial-derivative 𝑦 𝑧 𝑢 𝑣 2 superscript partial-derivative 𝑥 𝑧 𝑢 𝑣 2 {\displaystyle{\displaystyle\|\mathbf{T}_{u}\times\mathbf{T}_{v}\|=\sqrt{\left% (\frac{\partial(x,y)}{\partial(u,v)}\right)^{2}+\left(\frac{\partial(y,z)}{% \partial(u,v)}\right)^{2}+\left(\frac{\partial(x,z)}{\partial(u,v)}\right)^{2}% }}}
\|\mathbf{T}_{u}\times\mathbf{T}_{v}\| = \sqrt{\left(\pderiv{(x,y)}{(u,v)}\right)^{2}+\left(\pderiv{(y,z)}{(u,v)}\right)^{2}+\left(\pderiv{(x,z)}{(u,v)}\right)^{2}}		 

Error

Norm[Subscript[T, u] * Subscript[T, v]] == Sqrt[(((D[(x , y), {temp, 1}]/.temp-> (u , v))))^(2)+(((D[(y ,(x + y*I)), {temp, 1}]/.temp-> (u , v))))^(2)+(((D[(x ,(x + y*I)), {temp, 1}]/.temp-> (u , v))))^(2)]
Translation Error Translation Error - -
1.6.E50 ∥ 𝐓 θ × 𝐓 ϕ ∥ = ρ 2 ⁢ | sin ⁡ θ | norm subscript 𝐓 𝜃 subscript 𝐓 italic-ϕ superscript 𝜌 2 𝜃 {\displaystyle{\displaystyle\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\|=% \rho^{2}\left|\sin\theta\right|}}
\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}		 

Error

Norm[Subscript[T, \[Theta]] * Subscript[T, \[Phi]]] == \[Rho]^(2)* Abs[Sin[\[Theta]]]
Translation Error Translation Error - -
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