f:=x^2*y^3;

NiM+SSJmRzYiKiZJInhHRiUiIiNJInlHRiUiIiQ=

diff(f,x);

NiMsJComSSJ4RzYiIiIiSSJ5R0YmIiIkIiIj

diff(f,y);

NiMsJComSSJ4RzYiIiIjSSJ5R0YmRiciIiQ=

V:=<y*z,z*x,x*y>;

NiM+SSJWRzYiLUknUlRBQkxFR0YlNiUiJ08/di1JJ01BVFJJWEdGJTYjNyU3IyomSSJ5R0YlIiIiSSJ6R0YlRjE3IyomRjJGMUkieEdGJUYxNyMqJkY1RjFGMEYxJkknVmVjdG9yRzYkSSpwcm90ZWN0ZWRHRjtJKF9zeXNsaWJHRiU2I0knY29sdW1uR0Yl

VectorField(V);

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with(VectorCalculus);

Warning, the assigned names <,> and <|> now have a global binding

Warning, these protected names have been redefined and unprotected: *, +, ., D, Vector, diff, int, limit, series

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

VF:=VectorField(V,'cartesian'[x,y,z]);

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curlVF:=Curl(VF);

NiM+SSdjdXJsVkZHNiItSSdSVEFCTEVHRiU2JSIoZzUxIi1JJ01BVFJJWEdGJTYlNyU3IyIiIUYuRi43JVEieEYlUSJ5RiVRInpGJVEmZmllbGRGJSZJJ1ZlY3Rvckc2JEkqcHJvdGVjdGVkR0Y4SShfc3lzbGliR0YlNiNJJ2NvbHVtbkdGJQ==

Gradient(x*y*z,'cartesian'[x,y,z]);

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SetCoordinates('spherical'[r,theta,phi]);

NiMmSSpzcGhlcmljYWxHNiI2JUkickdGJUkmdGhldGFHRiVJJHBoaUdGJQ==

WF:=Gradient(r*cos(theta));

NiM+SSNXRkc2Ii1JJ1JUQUJMRUdGJTYlIilDJWUiPi1JJ01BVFJJWEdGJTYlNyU3Iy1JJGNvc0c2JEkqcHJvdGVjdGVkR0YySShfc3lzbGliR0YlNiNJJnRoZXRhR0YlNyMsJC1JJHNpbkdGMUY0ISIiNyMiIiE3JVEickYlUSZ0aGV0YUYlUSRwaGlGJVEmZmllbGRGJSZJJ1ZlY3RvckdGMTYjSSdjb2x1bW5HRiU=

curlWF:=Curl(WF);

NiM+SSdjdXJsV0ZHNiItSSdSVEFCTEVHRiU2JSIpLyZlIj4tSSdNQVRSSVhHRiU2JTclNyMiIiFGLkYuNyVRInJGJVEmdGhldGFGJVEkcGhpRiVRJmZpZWxkRiUmSSdWZWN0b3JHNiRJKnByb3RlY3RlZEdGOEkoX3N5c2xpYkdGJTYjSSdjb2x1bW5HRiU=

dZeroForm:=proc(f) local a,b; description "d of 0-form f (function), returns 1-form";<diff(f,r),diff(f,theta),diff(f,phi)>;end proc;

NiM+SSpkWmVyb0Zvcm1HNiJmKjYjSSJmR0YlNiRJImFHRiVJImJHRiVGJUYlLV9JL1ZlY3RvckNhbGN1bHVzRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiVJJDwsPkdGMTYlLV9GLkklZGlmZkdGMDYkOSRJInJHRiUtRjU2JEY4SSZ0aGV0YUdGJS1GNTYkRjhJJHBoaUdGJTYjUUlkfm9mfjAtZm9ybX5mfihmdW5jdGlvbiksfnJldHVybnN+MS1mb3JtRiVGJUYl

dOneForm:=proc(U) local a,b; description "d of 1-form U, returns 2-form"; <diff(U[3],theta)-diff(U[2],phi),diff(U[1],phi)-diff(U[3],r),diff(U[2],r)-diff(U[1],theta)>;end proc;

NiM+SSlkT25lRm9ybUc2ImYqNiNJIlVHRiU2JEkiYUdGJUkiYkdGJUYlRiUtX0kvVmVjdG9yQ2FsY3VsdXNHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJUkkPCw+R0YxNiUtX0YuSSIrR0YwNiQtX0YuSSVkaWZmR0YwNiQmOSQ2IyIiJEkmdGhldGFHRiUtX0YuSSIqR0YwNiQtRjk2JCZGPTYjIiIjSSRwaGlHRiUhIiItRjU2JC1GOTYkJkY9NiMiIiJGSi1GQjYkLUY5NiRGPEkickdGJUZLLUY1NiQtRjk2JEZHRlctRkI2JC1GOTYkRlBGQEZLNiNRP2R+b2Z+fjEtZm9ybX5VLH5yZXR1cm5zfjItZm9ybUYlRiVGJQ==

dTwoForm:=proc(U) local a,b; description "d of 2-form U, returns 3-form";diff(U[1],r)+diff(U[2],theta)+diff(U[3],phi);end proc;

NiM+SSlkVHdvRm9ybUc2ImYqNiNJIlVHRiU2JEkiYUdGJUkiYkdGJUYlRiUtX0kvVmVjdG9yQ2FsY3VsdXNHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJUkiK0dGMDYkLUYtNiQtX0YuSSVkaWZmR0YwNiQmOSQ2IyIiIkkickdGJS1GNzYkJkY7NiMiIiNJJnRoZXRhR0YlLUY3NiQmRjs2IyIiJEkkcGhpR0YlNiNRPmR+b2Z+Mi1mb3JtflUsfnJldHVybnN+My1mb3JtRiVGJUYl

StarZeroForm:=proc(f) local a,b; description "Hodge * of 0-form f, returns 3-form";f*r^2*sin(theta);end proc;

NiM+SS1TdGFyWmVyb0Zvcm1HNiJmKjYjSSJmR0YlNiRJImFHRiVJImJHRiVGJUYlLV9JL1ZlY3RvckNhbGN1bHVzRzYkSSpwcm90ZWN0ZWRHRjBJKF9zeXNsaWJHRiVJIipHRjA2JC1GLTYkOSQqJEkickdGJSIiIy1JJHNpbkdGLzYjSSZ0aGV0YUdGJTYjUURIb2RnZX4qfm9mfjAtZm9ybX5mLH5yZXR1cm5zfjMtZm9ybUYlRiVGJQ==

StarOneForm:=proc(U) local a,b; description "Hodge * of 1-form U, returns 2-form";<r^2*sin(theta)*U[1],sin(theta)*U[2],U[3]/sin(theta)>;end proc;

NiM+SSxTdGFyT25lRm9ybUc2ImYqNiNJIlVHRiU2JEkiYUdGJUkiYkdGJUYlRiUtX0kvVmVjdG9yQ2FsY3VsdXNHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJUkkPCw+R0YxNiUtX0YuSSIqR0YwNiQtRjU2JCokSSJyR0YlIiIjLUkkc2luR0YvNiNJJnRoZXRhR0YlJjkkNiMiIiItRjU2JEY9JkZCNiNGPC1GNTYkJkZCNiMiIiQqJEY9ISIiNiNRREhvZGdlfip+b2Z+MS1mb3JtflUsfnJldHVybnN+Mi1mb3JtRiVGJUYl

StarTwoForm:=proc(U) local a,b; description "Hodge * of 2-form U, returns 1-form";<-U[1]/(r^2*sin(theta)),-U[2]/sin(theta),-sin(theta)*U[3]>;end proc;

NiM+SSxTdGFyVHdvRm9ybUc2ImYqNiNJIlVHRiU2JEkiYUdGJUkiYkdGJUYlRiUtX0kvVmVjdG9yQ2FsY3VsdXNHNiRJKnByb3RlY3RlZEdGMEkoX3N5c2xpYkdGJUkkPCw+R0YxNiUtX0YuSSIqR0YwNiQtRjU2JCY5JDYjIiIiKiQtRjU2JCokSSJyR0YlIiIjLUkkc2luR0YvNiNJJnRoZXRhR0YlISIiRkgtRjU2JC1GNTYkJkY7NiNGQyokRkRGSEZILUY1NiQtRjU2JEZEJkY7NiMiIiRGSDYjUURIb2RnZX4qfm9mfjItZm9ybX5VLH5yZXR1cm5zfjEtZm9ybUYlRiVGJQ==

StarThreeForm:=proc(U) local a,b; description "Hodge * of 3-form U, returns 0-form";-U/(r^2*sin(theta));end proc;

NiM+SS5TdGFyVGhyZWVGb3JtRzYiZio2I0kiVUdGJTYkSSJhR0YlSSJiR0YlRiVGJS1fSS9WZWN0b3JDYWxjdWx1c0c2JEkqcHJvdGVjdGVkR0YwSShfc3lzbGliR0YlSSIqR0YwNiQtRi02JDkkKiQtRi02JCokSSJyR0YlIiIjLUkkc2luR0YvNiNJJnRoZXRhR0YlISIiRkE2I1FESG9kZ2V+Kn5vZn4zLWZvcm1+VSx+cmV0dXJuc34wLWZvcm1GJUYlRiU=

dOneForm(<0,r^2,r^2*sin(theta)>);

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dTwoForm(%);

NiMiIiE=

dZeroForm(r^2*sin(theta));

NiMtSSdSVEFCTEVHNiI2JSIpP3kmKj4tSSdNQVRSSVhHRiU2JDclNyMsJComSSJyR0YlIiIiLUkkc2luRzYkSSpwcm90ZWN0ZWRHRjRJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiVGMCIiIzcjKiZGL0Y4LUkkY29zR0YzRjZGMDcjIiIhNyVRInJGJVEmdGhldGFGJVEkcGhpRiUmSSdWZWN0b3JHRjM2I0knY29sdW1uR0Yl

dOneForm(%);

NiMtSSdSVEFCTEVHNiI2JSIpK3omKj4tSSdNQVRSSVhHRiU2JDclNyMiIiFGLEYsNyVRInJGJVEmdGhldGFGJVEkcGhpRiUmSSdWZWN0b3JHNiRJKnByb3RlY3RlZEdGNUkoX3N5c2xpYkdGJTYjSSdjb2x1bW5HRiU=

dP:=dZeroForm((3*cos(theta)^2-1)/r^3);

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StarOneForm(%);

NiMtSSdSVEFCTEVHNiI2JSIpZyFlKj4tSSdNQVRSSVhHRiU2JDclNyMsJCooSSJyR0YlISIjLUkkc2luRzYkSSpwcm90ZWN0ZWRHRjRJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiUiIiIsJiokLUkkY29zR0YzRjYiIiMiIiQhIiJGOEY4ISIkNyMsJCooRjFGPUY7RjhGL0ZAISInNyMiIiE3JVEickYlUSZ0aGV0YUYlUSRwaGlGJSZJJ1ZlY3RvckdGMzYjSSdjb2x1bW5HRiU=

dTwoForm(%);

NiMsKCooSSJyRzYiISIkLUkkc2luRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2I0kmdGhldGFHRiYiIiIsJiokLUkkY29zR0YqRi0iIiMiIiQhIiJGL0YvIiInKihGKEYvRjJGNEYlRichIzcqJkYoRjVGJUYnRjc=

simplify(%);

NiMiIiE=

StarThreeForm(%);

NiMiIiE=

dQ:=dZeroForm(r^2*(3*cos(theta)^2-1));

NiM+SSNkUUc2Ii1JJ1JUQUJMRUdGJTYlIihfPDkiLUknTUFUUklYR0YlNiQ3JTcjLCQqJkkickdGJSIiIiwmKiQtSSRjb3NHNiRJKnByb3RlY3RlZEdGOEkoX3N5c2xpYkdGJTYjSSZ0aGV0YUdGJSIiIyIiJCEiIkYyRjJGPDcjLCQqKEYxRjxGNUYyLUkkc2luR0Y3RjpGMiEiJzcjIiIhNyVRInJGJVEmdGhldGFGJVEkcGhpRiUmSSdWZWN0b3JHRjc2I0knY29sdW1uR0Yl

StarOneForm(%);

NiMtSSdSVEFCTEVHNiI2JSIpP2NUPy1JJ01BVFJJWEdGJTYkNyU3IywkKihJInJHRiUiIiQtSSRzaW5HNiRJKnByb3RlY3RlZEdGNEkoX3N5c2xpYkdGJTYjSSZ0aGV0YUdGJSIiIiwmKiQtSSRjb3NHRjNGNiIiI0YwISIiRjhGOEY9NyMsJCooRjFGPUYvRj1GO0Y4ISInNyMiIiE3JVEickYlUSZ0aGV0YUYlUSRwaGlGJSZJJ1ZlY3RvckdGMzYjSSdjb2x1bW5HRiU=

dTwoForm(%);

NiMsKCooSSJyRzYiIiIjLUkkc2luRzYkSSpwcm90ZWN0ZWRHRitJKF9zeXNsaWJHRiY2I0kmdGhldGFHRiYiIiIsJiokLUkkY29zR0YqRi1GJyIiJCEiIkYvRi8iIicqKEYoRi9GJUYnRjJGJyEjNyomRihGNEYlRidGNg==

StarThreeForm(%);

NiMsJCooLCgqKEkickc2IiIiIy1JJHNpbkc2JEkqcHJvdGVjdGVkR0YtSShfc3lzbGliR0YoNiNJJnRoZXRhR0YoIiIiLCYqJC1JJGNvc0dGLEYvRikiIiQhIiJGMUYxIiInKihGKkYxRidGKUY0RikhIzcqJkYqRjZGJ0YpRjhGMUYnISIjRipGN0Y3

simplify(%);

NiMiIiE=

dR:=dZeroForm(r*cos(theta));

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StarOneForm(%);

NiMtSSdSVEFCTEVHNiI2JSIpZ3NUPy1JJ01BVFJJWEdGJTYkNyU3IyooSSJyR0YlIiIjLUkkY29zRzYkSSpwcm90ZWN0ZWRHRjNJKF9zeXNsaWJHRiU2I0kmdGhldGFHRiUiIiItSSRzaW5HRjJGNUY3NyMsJComRjhGL0YuRjchIiI3IyIiITclUSJyRiVRJnRoZXRhRiVRJHBoaUYlJkknVmVjdG9yR0YyNiNJJ2NvbHVtbkdGJQ==

dTwoForm(%);

NiMiIiE=

StarThreeForm(%);

NiMiIiE=

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