Algebraic and Analytic Methods - 1.5 Calculus of Two or More Variables

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.5.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv{f}{y} = D_{y}f}
\pderiv{f}{y} = D_{y}f		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(f, y) = D[y]*f

D[f, y] == Subscript[D, y]*f
Failure Failure
Failed [300 / 300]
Result: -.5000000004-.8660254040*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = 1/2*3^(1/2)+1/2*I}


Result: .8660254040-.5000000004*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = -1/2+1/2*I*3^(1/2)}


Result: -.8660254040+.5000000004*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = 1/2-1/2*I*3^(1/2)}


Result: .5000000004+.8660254040*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.5000000000000001, -0.8660254037844386]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5], Rule[Subscript[D, y], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8660254037844387, -0.49999999999999994]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5], Rule[Subscript[D, y], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle D_{y}f = f_{y}}
D_{y}f = f_{y}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
D[y]*f = f[y]

Subscript[D, y]*f == Subscript[f, y]
Failure Failure
Failed [300 / 300]
Result: -.3660254036+.3660254040*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = 1/2*3^(1/2)+1/2*I, f[y] = 1/2*3^(1/2)+1/2*I}


Result: 1.000000000+0.*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = 1/2*3^(1/2)+1/2*I, f[y] = -1/2+1/2*I*3^(1/2)}


Result: .4e-9+1.732050808*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = 1/2*3^(1/2)+1/2*I, f[y] = 1/2-1/2*I*3^(1/2)}


Result: 1.366025404+1.366025404*I
Test Values: {f = 1/2*3^(1/2)+1/2*I, y = -1.5, D[y] = 1/2*3^(1/2)+1/2*I, f[y] = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.3660254037844386, 0.36602540378443865]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5], Rule[Subscript[D, y], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, y], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.9999999999999999, -1.1102230246251565*^-16]
Test Values: {Rule[f, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[y, -1.5], Rule[Subscript[D, y], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[f, y], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5#Ex3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = r\cos@@{\phi}}
x = r\cos@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
x = r*cos(phi)

x == r*Cos[\[Phi]]
Failure Failure
Failed [180 / 180]
Result: 2.595814528-.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, x = 1.5}


Result: 1.595814528-.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, x = .5}


Result: 3.095814528-.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, x = 2}


Result: .404185472+.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = 1.5, x = 1.5}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.595814528585838, -0.5954243253435487]
Test Values: {Rule[r, -1.5], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.3416482752961656, 0.7036130644027555]
Test Values: {Rule[r, -1.5], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5#Ex4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = r\sin@@{\phi}}
y = r\sin@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
y = r*sin(phi)

y == r*Sin[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: -.211529498+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = -1.5}


Result: 2.788470502+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = 1.5}


Result: .788470502+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = -.5}


Result: 1.788470502+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.21152949854979308, 0.506394694834305]
Test Values: {Rule[r, -1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.506097038210817, 1.2879550752257174]
Test Values: {Rule[r, -1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv[2]{f}{x}+\pderiv[2]{f}{y} = \pderiv[2]{f}{r}+\frac{1}{r}\pderiv{f}{r}+\frac{1}{r^{2}}\pderiv[2]{f}{\phi}}
\pderiv[2]{f}{x}+\pderiv[2]{f}{y} = \pderiv[2]{f}{r}+\frac{1}{r}\pderiv{f}{r}+\frac{1}{r^{2}}\pderiv[2]{f}{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(f, [x$(2)])+ diff(f, [y$(2)]) = diff(f, [r$(2)])+(1)/(r)*diff(f, r)+(1)/((r)^(2))*diff(f, [phi$(2)])

D[f, {x, 2}]+ D[f, {y, 2}] == D[f, {r, 2}]+Divide[1,r]*D[f, r]+Divide[1,(r)^(2)]*D[f, {\[Phi], 2}]
Successful Successful - Successful [Tested: 300]
1.5#Ex5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = r\cos@@{\phi}}
x = r\cos@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
x = r*cos(phi)

x == r*Cos[\[Phi]]
Failure Failure
Failed [180 / 180]
Result: 2.595814528-.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, x = 1.5}


Result: 1.595814528-.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, x = .5}


Result: 3.095814528-.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, x = 2}


Result: .404185472+.5954243254*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = 1.5, x = 1.5}

... skip entries to safe data
Failed [180 / 180]
Result: Complex[2.595814528585838, -0.5954243253435487]
Test Values: {Rule[r, -1.5], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[3.3416482752961656, 0.7036130644027555]
Test Values: {Rule[r, -1.5], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5#Ex6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = r\sin@@{\phi}}
y = r\sin@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
y = r*sin(phi)

y == r*Sin[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: -.211529498+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = -1.5}


Result: 2.788470502+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = 1.5}


Result: .788470502+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = -.5}


Result: 1.788470502+.5063946946*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, r = -1.5, y = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-0.21152949854979308, 0.506394694834305]
Test Values: {Rule[r, -1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-2.506097038210817, 1.2879550752257174]
Test Values: {Rule[r, -1.5], Rule[y, -1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5#Ex7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z = z}
z = z		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
z = z
 
z == z
 Skipped - no semantic math Skipped - no semantic math - -
1.5.E15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv[2]{f}{x}+\pderiv[2]{f}{y}+\pderiv[2]{f}{z} = \pderiv[2]{f}{r}+\frac{1}{r}\pderiv{f}{r}+\frac{1}{r^{2}}\pderiv[2]{f}{\phi}+\pderiv[2]{f}{z}}
\pderiv[2]{f}{x}+\pderiv[2]{f}{y}+\pderiv[2]{f}{z} = \pderiv[2]{f}{r}+\frac{1}{r}\pderiv{f}{r}+\frac{1}{r^{2}}\pderiv[2]{f}{\phi}+\pderiv[2]{f}{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(f, [x$(2)])+ diff(f, [y$(2)])+ subs( temp=(x + y*I), diff( f, temp$(2) ) ) = diff(f, [r$(2)])+(1)/(r)*diff(f, r)+(1)/((r)^(2))*diff(f, [phi$(2)])+ subs( temp=(x + y*I), diff( f, temp$(2) ) )

D[f, {x, 2}]+ D[f, {y, 2}]+ (D[f, {temp, 2}]/.temp-> (x + y*I)) == D[f, {r, 2}]+Divide[1,r]*D[f, r]+Divide[1,(r)^(2)]*D[f, {\[Phi], 2}]+ (D[f, {temp, 2}]/.temp-> (x + y*I))
Successful Successful - Successful [Tested: 300]
1.5#Ex8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x = \rho\sin@@{\theta}\cos@@{\phi}}
x = \rho\sin@@{\theta}\cos@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
x = rho*sin(theta)*cos(phi)

x == \[Rho]*Sin[\[Theta]]*Cos[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: .7933233780-.2990618885*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = 1.5}


Result: -.2066766220-.2990618885*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = .5}


Result: 1.293323378-.2990618885*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, x = 2}


Result: 2.075937432-.6992275886*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = -1/2+1/2*I*3^(1/2), x = 1.5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.7933233783531031, -0.29906188871772155]
Test Values: {Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.1325168401075085, -1.15603698835011]
Test Values: {Rule[x, 1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5#Ex9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = \rho\sin@@{\theta}\sin@@{\phi}}
y = \rho\sin@@{\theta}\sin@@{\phi}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
y = rho*sin(theta)*sin(phi)

y == \[Rho]*Sin[\[Theta]]*Sin[\[Phi]]
Failure Failure
Failed [300 / 300]
Result: -1.750303606-.8142131232*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = -1.5}


Result: 1.249696394-.8142131232*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = 1.5}


Result: -.7503036064-.8142131232*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = -.5}


Result: .2496963936-.8142131232*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, y = .5}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[-1.750303605777159, -0.8142131229438421]
Test Values: {Rule[y, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.49444885508381575, -0.009630141394747549]
Test Values: {Rule[y, -1.5], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϕ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5#Ex10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z = \rho\cos@@{\theta}}
z = \rho\cos@@{\theta}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
z = rho*cos(theta)

z == \[Rho]*Cos[\[Theta]]
Failure Failure
Failed [300 / 300]
Result: .348818158e-1+.4784968852*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = 1/2*3^(1/2)+1/2*I}


Result: -1.331143588+.8445222892*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = -1/2+1/2*I*3^(1/2)}


Result: -.3311435882-.8875285188*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = 1/2-1/2*I*3^(1/2)}


Result: -1.697168992-.5215031148*I
Test Values: {rho = 1/2*3^(1/2)+1/2*I, theta = 1/2*3^(1/2)+1/2*I, z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [300 / 300]
Result: Complex[0.03488181560898618, 0.4784968849905366]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.8875285187939019, -0.3311435881754526]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[θ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ρ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
1.5.E17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv[2]{f}{x}+\pderiv[2]{f}{y}+\pderiv[2]{f}{z} = {\frac{1}{\rho^{2}}\pderiv{}{\rho}\left(\rho^{2}\pderiv{f}{\rho}\right)+\frac{1}{\rho^{2}\sin^{2}@@{\theta}}\pderiv[2]{f}{\phi}}+\frac{1}{\rho^{2}\sin@@{\theta}}\pderiv{}{\theta}\left(\sin@@{\theta}\pderiv{f}{\theta}\right)}
\pderiv[2]{f}{x}+\pderiv[2]{f}{y}+\pderiv[2]{f}{z} = {\frac{1}{\rho^{2}}\pderiv{}{\rho}\left(\rho^{2}\pderiv{f}{\rho}\right)+\frac{1}{\rho^{2}\sin^{2}@@{\theta}}\pderiv[2]{f}{\phi}}+\frac{1}{\rho^{2}\sin@@{\theta}}\pderiv{}{\theta}\left(\sin@@{\theta}\pderiv{f}{\theta}\right)		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(f, [x$(2)])+ diff(f, [y$(2)])+ subs( temp=(x + y*I), diff( f, temp$(2) ) ) = (1)/((rho)^(2))*diff(((rho)^(2)* diff(f, rho))+(1)/((rho)^(2)* (sin(theta))^(2))*diff(f, [phi$(2)]), rho)+(1)/((rho)^(2)* sin(theta))*diff(sin(theta)*diff(f, theta), theta)

D[f, {x, 2}]+ D[f, {y, 2}]+ (D[f, {temp, 2}]/.temp-> (x + y*I)) == Divide[1,\[Rho]^(2)]*D[(\[Rho]^(2)* D[f, \[Rho]])+Divide[1,\[Rho]^(2)* (Sin[\[Theta]])^(2)]*D[f, {\[Phi], 2}], \[Rho]]+Divide[1,\[Rho]^(2)* Sin[\[Theta]]]*D[Sin[\[Theta]]*D[f, \[Theta]], \[Theta]]
Successful Successful - Successful [Tested: 300]
1.5.E19 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \pderiv{f}{x} = \pderiv{f}{y}}
\pderiv{f}{x} = \pderiv{f}{y}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
diff(f, x) = diff(f, y)

D[f, x] == D[f, y]
Successful Successful - Successful [Tested: 180]
1.5.E23 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \abs{\int_{c_{1}}^{d}(\ipderiv{f}{x})\diff{y}} < \epsilon}
\abs{\int_{c_{1}}^{d}(\ipderiv{f}{x})\diff{y}} < \epsilon		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
abs(int(diff(f, x), y = c[1]..d)) < epsilon

Abs[Integrate[D[f, x], {y, Subscript[c, 1], d}, GenerateConditions->None]] < \[Epsilon]
Failure Failure Successful [Tested: 300] Successful [Tested: 300]
1.5.E25 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a = x_{0}}
a = x_{0}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a = x[0]
 
a == Subscript[x, 0]
 Skipped - no semantic math Skipped - no semantic math - -
1.5.E26 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c = y_{0}}
c = y_{0}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c = y[0]
 
c == Subscript[y, 0]
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a \leq x}
a \leq x		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a <= x
 
a <= x
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi_{1}(x) \leq y}
\phi_{1}(x) \leq y		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
phi[1](x) <= y
 
Subscript[\[Phi], 1][x] <= y
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle c \leq y}
c \leq y		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
c <= y
 
c <= y
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \psi_{1}(y) \leq x}
\psi_{1}(y) \leq x		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
psi[1](y) <= x
 
Subscript[\[Psi], 1][y] <= x
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle a \leq x}
a \leq x		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
a <= x
 
a <= x
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex16 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \phi_{1}(x) \leq y}
\phi_{1}(x) \leq y		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
phi[1](x) <= y
 
Subscript[\[Phi], 1][x] <= y
 Skipped - no semantic math Skipped - no semantic math - -
1.5#Ex17 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \psi_{1}(x,y) \leq z}
\psi_{1}(x,y) \leq z		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
psi[1](x , y) <= (x + y*I)
 
Subscript[\[Psi], 1][x , y] <= (x + y*I)
 Skipped - no semantic math Skipped - no semantic math - -
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