︠911e6fad-558d-4b79-8f85-c6bd957ec821︠x derivative(x,x) ︡80b244a8-82fe-4dd5-8744-264dab7e7bcd︡︡{"stdout":"1\n","done":false}︡{"done":true} ︠a018a05d-7b54-4cfb-a213-5d71b184f843︠ x = 12 y = -12 ︡2245e20b-e5e0-48ef-8232-ca8324a7f60e︡︡{"done":true} ︠dad2d10e-87bd-4e8b-bcdb-fc8e0c02f49d︠ x/y ︡c97b1d01-c8d1-42b3-b78c-a13bf4a03b56︡︡{"stdout":"-1\n","done":false}︡{"done":true} ︠a8492742-46b4-4601-a68c-841f91d90cac︠ y/x ︡d5a7d42c-3831-419b-8eaa-7aa4a2a77b7b︡︡{"stdout":"-1\n","done":false}︡{"done":true} ︠12a3b794-0ef7-4aba-bc68-dd7ec0041f9b︠ x/y == y/x x^2 == y^2 x = +/- y ︡a058c16a-3322-429f-8138-4b90f82941f1︡︡{"stdout":"True\n","done":false}︡{"stdout":"True\n","done":false}︡{"done":false,"stderr":"Error in lines 3-3\nTraceback (most recent call last):\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py\", line 905, in execute\n exec compile(block+'\\n', '', 'single') in namespace, locals\n"}︡{"done":false,"stderr":" File \"\", line 1\n x = +/- y\n ^\nSyntaxError: invalid syntax\n"}︡{"done":true} ︠4400d904-02ce-4703-a116-fbf7f6050fbe︠ y = var('y') factor(2*y^2 + y - 1) ︡50e8d784-48f7-46a6-a106-f3ec9e98ad33︡︡{"stdout":"(2*y - 1)*(y + 1)","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠ac763a4d-faa2-4fc4-add5-4ea0aa48672e︠ t=var('t') plot(cos(t)+cos(t)*sin(t),t,-5,5) ︡06e8cdc0-cb15-43a8-9b0b-e7db6c47903e︡︡{"once":false,"done":false,"file":{"show":true,"uuid":"ab9fd4a2-8527-4652-ae4d-b25a76d2a742","filename":"/projects/c2665543-bc3d-430e-9234-2a0d9d08b797/.sage/temp/compute0-us/13990/tmp_ji9ulV.svg"}}︡{"html":"
","done":false}︡{"done":true} ︠bb41ae95-ddb7-4a95-836e-48d6841fb87f︠ n(3*sqrt(3)/4) ︡ef81a44a-8463-4666-a4f7-7db93d0a1971︡︡{"stdout":"1.29903810567666","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠a4b6ace6-0367-4c43-94bc-77c1ee8500df︠ t = var('t') f = cos(t)+cos(t)*sin(t) solve(derivative(f,t)==0,t) ︡087560d9-2909-487e-a879-0c605edf537e︡︡{"stdout":"[sin(t) == -1/2*sqrt(4*cos(t)^2 + 1) - 1/2, sin(t) == 1/2*sqrt(4*cos(t)^2 + 1) - 1/2]\n","done":false}︡{"done":true} ︠f891e02c-1844-4e15-9574-447567978f9c︠ x = var('x') y = var('y') integrate(integrate(e^(x^2),y,x,1),x,0,1) ︡45f38e86-0bec-4211-a8ed-4964e5481c67︡︡{"stdout":"-1/2*I*sqrt(pi)*erf(I) - 1/2*e + 1/2\n","done":false}︡{"done":true} ︠e8cba86c-403d-4f96-88f8-8776455b410a︠ integrate(e^(x^2),y,x,1) ︡1e649073-a564-4de3-bbf8-000d8e501061︡︡{"stdout":"-(x - 1)*e^(x^2)\n","done":false}︡{"done":true} ︠636d8c06-9187-4b1a-bb4b-ee25e2f1147e︠ integrate(1*e^(x^2),x,0,1) ︡17c57233-1594-4613-8707-cdf1db49f852︡︡{"stdout":"-1/2*I*sqrt(pi)*erf(I)\n","done":false}︡{"done":true} ︠ea99c8aa-9968-4deb-a2a6-0183c23d2698︠ integrate(integrate(e^(x^2),y,0,x),x,0,1) ︡ebbdcb21-dcb6-4808-b660-0b77ab01ec0c︡︡{"stdout":"1/2*e - 1/2","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠4d7badba-d67b-45ee-b442-6c89ea7f633b︠ plot(4-x^2,x,0,2) ︡b89150e8-748f-41fa-bc5f-234b2c3a0ddd︡︡{"once":false,"done":false,"file":{"show":true,"uuid":"1536bb13-a114-44e8-8689-ad70f5960586","filename":"/projects/c2665543-bc3d-430e-9234-2a0d9d08b797/.sage/temp/compute0-us/13990/tmp_iFXkpa.svg"}}︡{"html":"
","done":false}︡{"done":true} ︠3b93bcda-9ea9-472f-970d-0f7d10b69d9b︠ integrate(integrate(x^2,y,0,4-x^2),x,0,2) ︡0de951c0-b763-4b02-a660-7354e25e5105︡︡{"stdout":"64/15","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠1202a5dd-e221-4890-bebc-dca564a03a97︠ x = var('x') ︡d51b4fe9-a754-4f0c-a53f-ddf88e10694b︡︡{"done":true} ︠b9c2b60d-7bcf-42a9-94e3-bbb6a6bb648c︠ y = var('y') z = var('z') rho = sqrt(x^2+y^2+z^2) expand(rho^4) %typeset_mode True ︡e1ce7130-0bc7-4b97-9da9-b2c38365b049︡︡{"html":"
$\\displaystyle x^{4} + 2 \\, x^{2} y^{2} + y^{4} + 2 \\, x^{2} z^{2} + 2 \\, y^{2} z^{2} + z^{4}$
","done":false}︡{"done":true} ︠bff84c47-2bbd-42e6-9e8e-e66ca806ced5︠ ︠5b89100a-2789-49aa-b459-8c8201f79192︠ integrate(integrate(cos(x*y),y,0,sin(x)),x,0,1) integrate(integrate(cos(x*y),x,arcsin(y),1),y,0,sin(1)) ︡542ace23-d87c-4f96-b387-1d6e61604c27︡︡{"done":false,"stderr":"Error in lines 1-1\nTraceback (most recent call last):\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/smc_sagews/sage_server.py\", line 905, in execute\n exec compile(block+'\\n', '', 'single') in namespace, locals\n File \"\", line 1, in \n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/misc/functional.py\", line 663, in integral\n return x.integral(*args, **kwds)\n File \"sage/symbolic/expression.pyx\", line 11269, in sage.symbolic.expression.Expression.integral (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/expression.cpp:59975)\n return integral(self, *args, **kwds)\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py\", line 761, in integrate\n return definite_integral(expression, v, a, b, hold=hold)\n File \"sage/symbolic/function.pyx\", line 994, in sage.symbolic.function.BuiltinFunction.__call__ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/function.cpp:11377)\n res = super(BuiltinFunction, self).__call__(\n File \"sage/symbolic/function.pyx\", line 502, in sage.symbolic.function.Function.__call__ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/function.cpp:7144)\n res = g_function_evalv(self._serial, vec, hold)\n File \"sage/symbolic/function.pyx\", line 1065, in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (/projects/sage/sage-6.9/src/build/cythonized/sage/symbolic/function.cpp:12106)\n return self._eval0_(*args)\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/symbolic/integration/integral.py\", line 176, in _eval_\n return integrator(*args)\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/symbolic/integration/external.py\", line 23, in maxima_integrator\n result = maxima.sr_integral(expression, v, a, b)\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py\", line 784, in sr_integral\n self._missing_assumption(s)\n File \"/projects/sage/sage-6.9/local/lib/python2.7/site-packages/sage/interfaces/maxima_lib.py\", line 993, in _missing_assumption\n raise ValueError(outstr)\nValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(sin(x)>0)', see `assume?` for more details)\nIs sin(x) positive, negative or zero?\n"}︡{"done":true} ︠6f428f92-0c1c-4338-8b97-798dd3ba9b33︠ theta = var('theta') r = var('r') integrate(integrate(integrate(r,z,0,9-r^2),r,0,3),theta,0,2*pi) ︡bf3921ab-742c-4b7f-ab7d-194946ea969e︡︡{"html":"
$\\displaystyle \\frac{81}{2} \\, \\pi$
","done":false}︡{"done":true} ︠6c301ef6-4d10-4772-bff6-ae4cf14802ea︠ x = var('x') y = var('y') integrate(integrate(2*sin(x^2),y,0,x),x,0,pi) ︡b7ceb129-8edc-4b01-a288-2760942124b3︡︡{"html":"
$\\displaystyle -\\cos\\left(\\pi^{2}\\right) + 1$
","done":false}︡{"done":true} ︠c1bf9e26-6f9c-476b-96ae-0d838794834d︠ integrate(2*sin(x^2),y,0,x) ︡42fab445-7a07-4141-bf19-fc0b512183a8︡︡{"html":"
$\\displaystyle 2 \\, x \\sin\\left(x^{2}\\right)$
","done":false}︡{"done":true} ︠9d3c81e4-d12c-413a-b1d0-07839008786b︠ x = var('x') y = var('y') z = var('z') integrate(integrate(integrate(x,z,0,1-x-y),y,0,1-x),x,0,1) ︡732e031f-a22f-405c-b561-542fd7ec6b09︡︡{"html":"
$\\displaystyle \\frac{1}{24}$
","done":false}︡{"done":true} ︠b20b2a58-22a2-4840-b26e-f62c891c4067︠ r = var('r') theta = var('theta') x = r*cos(theta) y = r*sin(theta) integrate(integrate((x+y)*r,r,0,1),theta,0,pi/2) ︡1eaae19b-113a-4619-888f-d9beb36442f2︡︡{"html":"
$\\displaystyle \\frac{2}{3}$
","done":false}︡{"done":true} ︠3dde0a8e-8d9e-4c96-b334-eda3d0231bfa︠ integrate(integrate((x)*r,r,0,1),theta,0,pi/2) ︡849c972f-2326-4e4b-85b2-7a7bcbdea38d︡︡{"html":"
$\\displaystyle \\frac{1}{3}$
","done":false}︡{"done":true} ︠3f077ea7-6c21-453b-a1a4-2aced7389169︠ x = var('x') y = var('y') expand((x+I*y)^8) ︡9a232634-38b8-4437-b686-dff93724f078︡︡{"html":"
$\\displaystyle x^{8} + 8 i \\, x^{7} y - 28 \\, x^{6} y^{2} - 56 i \\, x^{5} y^{3} + 70 \\, x^{4} y^{4} + 56 i \\, x^{3} y^{5} - 28 \\, x^{2} y^{6} - 8 i \\, x y^{7} + y^{8}$
","done":false}︡{"done":true} ︠f571a4f0-cf9c-4e09-95ae-26ce6485fee8︠ x = r*cos(theta) y = r*sin(theta) expand((x+I*y)^8) ︡1105e7f2-bb36-4745-b246-6d899aa8ba56︡︡{"html":"
$\\displaystyle r^{8} \\cos\\left(\\theta\\right)^{8} + 8 i \\, r^{8} \\cos\\left(\\theta\\right)^{7} \\sin\\left(\\theta\\right) - 28 \\, r^{8} \\cos\\left(\\theta\\right)^{6} \\sin\\left(\\theta\\right)^{2} - 56 i \\, r^{8} \\cos\\left(\\theta\\right)^{5} \\sin\\left(\\theta\\right)^{3} + 70 \\, r^{8} \\cos\\left(\\theta\\right)^{4} \\sin\\left(\\theta\\right)^{4} + 56 i \\, r^{8} \\cos\\left(\\theta\\right)^{3} \\sin\\left(\\theta\\right)^{5} - 28 \\, r^{8} \\cos\\left(\\theta\\right)^{2} \\sin\\left(\\theta\\right)^{6} - 8 i \\, r^{8} \\cos\\left(\\theta\\right) \\sin\\left(\\theta\\right)^{7} + r^{8} \\sin\\left(\\theta\\right)^{8}$
","done":false}︡{"done":true} ︠e97d2eb8-62b0-48cd-a4bb-029c6d71841c︠ (x+I*y)^8==(r*e^(I*theta))^8 ︡52b022de-34f0-47a3-82db-9c935022ef5c︡︡{"html":"
$\\displaystyle {\\left(r \\cos\\left(\\theta\\right) + i \\, r \\sin\\left(\\theta\\right)\\right)}^{8} = r^{8} e^{\\left(8 i \\, \\theta\\right)}$
","done":false}︡{"done":true} ︠a667be3b-56e8-4d3c-b0c6-6b45804247eb︠ integrate(integrate(r^8 * e^(8*I*theta)*r, r,0,1),theta,0,2*pi) ︡43ba718c-2709-422e-bd8a-b02af6ec5518︡︡{"html":"
$\\displaystyle 0$
","done":false}︡{"done":true} ︠2cb54487-03ea-4bfe-938b-be991568d050︠ integrate(e^(8*I*theta),theta,0,2*pi) ︡82205c39-ad1b-4555-8975-ad01f8ee09ed︡︡{"html":"
$\\displaystyle 0$
","done":false}︡{"done":true} ︠b9eb506f-13d4-4eff-b350-eb8aae3e8967︠ ︠557a0a2b-640a-4d24-8fce-90ad569e8e94︠ ︠e2ba7585-d478-405e-8a53-36832eebc13di︠ x = r*cos(theta) y = r*sin(theta) expand((x+I*y)^4) ︡bc0bb845-7ac8-4832-aeeb-5eee1a1c1d90︡︡{"html":"
$\\displaystyle r^{4} \\cos\\left(\\theta\\right)^{4} + 4 i \\, r^{4} \\cos\\left(\\theta\\right)^{3} \\sin\\left(\\theta\\right) - 6 \\, r^{4} \\cos\\left(\\theta\\right)^{2} \\sin\\left(\\theta\\right)^{2} - 4 i \\, r^{4} \\cos\\left(\\theta\\right) \\sin\\left(\\theta\\right)^{3} + r^{4} \\sin\\left(\\theta\\right)^{4}$
","done":false}︡{"done":true} %default_mode mode_name ︠1874e736-4a62-4bbd-b693-77de93afbb85︠ ︠11836161-5ad2-4d28-859b-85883465fab6︠ ︠8cc492c9-41e1-4a28-aa45-76403bbb7bde︠ ︠fb8bcefb-794f-443d-9441-a3a2a41c6417︠ cos(2*theta) == cos(theta)^2 - sin(theta)^2 ︡710ecf1c-281f-41fb-ade8-a0c043b13427︡︡{"html":"
$\\displaystyle \\cos\\left(2 \\, \\theta\\right) = \\cos\\left(\\theta\\right)^{2} - \\sin\\left(\\theta\\right)^{2}$
","done":false}︡{"done":true} ︠fef671d4-9ab7-436b-b10f-93c9993b2b9cs︠ K. = SR['x'] Q. = QuaternionAlgebra(Frac(K),-1,-1) ︡050aeb4c-d96c-44fe-a81b-6f083b966339︡︡{"done":true} ︠16ef0b02-8318-4444-b356-41e8468b077ds︠ Q ︡21e305b5-152e-4c99-b218-9fdde81d84d6︡︡{"stdout":"Quaternion Algebra (-1, -1) with base ring Fraction Field of Univariate Polynomial Ring in x over Rational Field\n","done":false}︡{"done":true} ︠5910d0b8-dcd6-4bce-8abc-c6117046fba5s︠ AR = (1/sqrt(2) + i/sqrt(2)) AY = j ︡8b3fa036-fe5c-4bbe-b213-00d5845dec18︡︡{"done":true} ︠0a739bd5-d1f2-4359-ae6a-a2d16c848e9es︠ AR * AR * AR * AR ︡e6521e45-6a82-4317-8fbc-4bbf116743ea︡︡{"stdout":"-1\n","done":false}︡{"done":true} ︠6759ab4e-a086-41d0-b98c-026045127bb3︠ (AR*AR*AR)*AY*A ︡5b2b4381-3e2e-48c3-bfbb-c4a23b746296︡︡{"stdout":"k","done":false}︡{"stdout":"\n","done":false}︡{"done":true} ︠bf62c6ea-bc09-471c-bb27-33f51957ffe8s︠ AX = k AX*AY*AX*AY ︡58a8a8f6-c30b-45ac-a5cd-528dd6e59dbe︡︡{"stdout":"-1\n","done":false}︡{"done":true} ︠53033911-346d-40b5-bd25-77cd2763b427︠