Algebraic and Analytic Methods - 1.6 Vectors and Vector-Valued Functions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.6#Ex1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{a} = (a_{1},a_{2},a_{3})}
\mathbf{a} = (a_{1},a_{2},a_{3})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{b} = (b_{1},b_{2},b_{3})}
\mathbf{b} = (b_{1},b_{2},b_{3})		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}}
\|\mathbf{a}\| = \sqrt{\mathbf{a}\cdot\mathbf{a}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error
 
Norm[a] == Sqrt[a * a]
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}\|}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{i}\times\mathbf{j} = \mathbf{k}}
\mathbf{i}\times\mathbf{j} = \mathbf{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
i * j = ((0 , 0 , 1))
 
i * j == ((0 , 0 , 1))
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{j}\times\mathbf{k} = \mathbf{i}}
\mathbf{j}\times\mathbf{k} = \mathbf{i}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
j *((0 , 0 , 1)) = i
 
j *((0 , 0 , 1)) == i
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{k}\times\mathbf{i} = \mathbf{j}}
\mathbf{k}\times\mathbf{i} = \mathbf{j}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((0 , 0 , 1)) * i = j
 
((0 , 0 , 1)) * i == j
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{j}\times\mathbf{i} = -\mathbf{k}}
\mathbf{j}\times\mathbf{i} = -\mathbf{k}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
j * i = -((0 , 0 , 1))
 
j * i == -((0 , 0 , 1))
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{k}\times\mathbf{j} = -\mathbf{i}}
\mathbf{k}\times\mathbf{j} = -\mathbf{i}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
((0 , 0 , 1)) * j = - i
 
((0 , 0 , 1)) * j == - i
 Skipped - no semantic math Skipped - no semantic math - -
1.6#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{i}\times\mathbf{k} = -\mathbf{j}}
\mathbf{i}\times\mathbf{k} = -\mathbf{j}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
i *((0 , 0 , 1)) = - j
 
i *((0 , 0 , 1)) == - j
 Skipped - no semantic math Skipped - no semantic math - -
1.6.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}}
\LeviCivitasym{1}{2}{3} = \LeviCivitasym{3}{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LeviCivitasym{3}{1}{2} = 1}
\LeviCivitasym{3}{1}{2} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
LeviCivita[3, 1, 2] = 1

Part[LeviCivitaTensor[3,List], 3, 1, 2] == 1
Successful Successful - Successful [Tested: 1]
1.6#Ex16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}}
\LeviCivitasym{2}{1}{3} = \LeviCivitasym{3}{2}{1}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LeviCivitasym{3}{2}{1} = -1}
\LeviCivitasym{3}{2}{1} = -1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
LeviCivita[3, 2, 1] = - 1

Part[LeviCivitaTensor[3,List], 3, 2, 1] == - 1
Successful Successful - Successful [Tested: 1]
1.6#Ex17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LeviCivitasym{2}{2}{1} = 0}
\LeviCivitasym{2}{2}{1} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
LeviCivita[2, 2, 1] = 0

Part[LeviCivitaTensor[3,List], 2, 2, 1] == 0
Successful Successful - Successful [Tested: 1]
1.6.E16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}}
\LeviCivitasym{j}{k}{\ell}\LeviCivitasym{\ell}{m}{n} = \Kroneckerdelta{j}{m}\Kroneckerdelta{k}{n}-\Kroneckerdelta{j}{n}\Kroneckerdelta{k}{m}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}}
\mathbf{e}_{j}\times\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\ell}\mathbf{e}_{\ell}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a_{j}\mathbf{e}_{j}\times b_{k}\mathbf{e}_{k} = \LeviCivitasym{j}{k}{\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}
\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}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \|\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}}}
\|\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}}
\|\mathbf{T}_{\theta}\times\mathbf{T}_{\phi}\| = \rho^{2}\abs{\sin@@{\theta}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
Error

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