<Text-field style="Heading 1" layout="Heading 1">Problem 8-5</Text-field>

Integral Equivalent of Divergence

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Show that the divergence of the vector field

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is the limiting ratio of the surface flux to the enclosed volume.

(For the surface, use the sphere with center LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNigtSSNtaUdGJDYlUSJ1RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVEiLEYnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GMTYlUSJ2RidGNEY3RjotRjE2JVEid0YnRjRGN0Y+Rj5GPg== and radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEickYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.)

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<Text-field style="Heading 1" layout="Heading 1">Solution (Maple 14)</Text-field>

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Tools\342\211\273Load Package: Vector Calculus

JSFH

<Text-field style="Heading 2" layout="Heading 2">Divergence by Differential Operator</Text-field>

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Set coordinate system

Tools\342\211\273Tasks\342\211\273Browse:

Calculus - Vector\342\211\273Vector Algebra and Settings\342\211\273Set Coordinates\342\211\273The More Common Systems

The More Common Coordinate Systems

Cartesian [x\134,y],Cartesian [x\134,y\134,z],Cartesian - other,polar,cylindrical,sphericaluse DocumentTools in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local q; Do(q = %ComboBox1): if q=Cartesian [x,y] then Do(%MathContainer2=MathML[Export]([x,y])); elif q=Cartesian [x,y,z] then Do(%MathContainer2=MathML[Export]([x,y,z])); elif q=Cartesian - other then Do(%MathContainer2=NULL); elif q=polar then Do(%MathContainer2=NULL); elif q=cylindrical then Do(%MathContainer2=NULL); elif q=spherical then Do(%MathContainer2=NULL); end if: Do(%MathContainer3 = NULL); end module; end use;

use DocumentTools,VectorCalculus in # Enter Maple commands to be executed when the specified # action is carried out on the component. # Use: # Do( %component_name ); # and # Do( %component_name = value ); # to set and get properties of the component. # You can also use arbitrary expressions # involving components, e.g.: # Do( %target = %input1 + 2*%input2 ); # Note the %-prefix to each component name. # See ?CustomizingComponents for more information. module() local q,q1,q2,q3; Do(q = %ComboBox1); if q<>Cartesian [x,y] and q<> Cartesian [x,y,z] then Do(q1 = %MathContainer2); if type([q1],'list'('list')) then q2:=q1; elif type([q1],'list') then q2:=[q1];end if; if q2=NULL then return;end if; end if; if q=Cartesian [x,y] then SetCoordinates(cartesian[x,y]); elif q=Cartesian [x,y,z] then SetCoordinates(cartesian[x,y,z]); elif q=Cartesian - other then if nops(q2)=3 then SetCoordinates(parse(StringTools:-Join(["cartesian",convert(q2,string)],""))) else return;end if; elif q=polar then if nops(q2)=2 then SetCoordinates(parse(StringTools:-Join(["polar",convert(q2,string)],""))) else return;end if; elif q=cylindrical then if nops(q2)=3 then SetCoordinates(parse(StringTools:-Join(["cylindrical",convert(q2,string)],""))) else return;end if; elif q=spherical then if nops(q2)=3 then SetCoordinates(parse(StringTools:-Join(["spherical",convert(q2,string)],""))) else return;end if; end if; q3 := GetCoordinates(); Do(%MathContainer3 = MathML[Export]([op(0,q3),[op(q3)]])); end module; end use;

The field F

Using angle brackets, enter the field F as a vector.

Context Menu: Assign to a Name\342\211\273F

JSFH

Divergence by differential operator

Common Symbols or Operators palette: Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEmJkRlbDtGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMTExMTExMWVtRictRiw2LVEpJm1pZGRvdDtGJ0YvRjJGNUY3RjlGO0Y9Rj8vRkJRLDAuMTY2NjY2N2VtRicvRkVGSy1JI21pR0YkNiVRIkZGJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRidGLw==

Context Menu: Evaluate and Display Inline

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<Text-field style="Heading 2" layout="Heading 2">Flux through Surface of Sphere</Text-field>

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Compute surface flux, that is, 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the surface of a sphere of radius LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5NjE7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy

Tools\342\211\273Tasks\342\211\273Browse:

Calculus - Vector\342\211\273Integration\342\211\273Flux\342\211\2733-D\342\211\273Through a Sphere

Flux through a Sphere

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For the Vector Field:

use DocumentTools in Do(%ComboBox2(value) = "Select Coordinate System"): Do(%MathContainer4 = NULL); Do(%MathContainer5 = NULL); Do(%MathContainer6 = NULL); Do(%MathContainer7 = NULL); Do(%MathContainer8 = NULL); Do(%MathContainer13 = NULL); Do(%MathContainer14 = NULL); Do(%MathContainer9 = NULL); Do(%MathContainer10 = NULL); Do(%MathContainer11 = NULL); Do(%MathContainer12 = NULL); Do(%MathContainer4(visible) = false); Do(%MathContainer5(visible) = false); Do(%MathContainer6(visible) = false); Do(%MathContainer7(visible) = false); Do(%MathContainer8(visible) = false); Do(%MathContainer13(visible) = false); Do(%MathContainer14(visible) = false); Do(%MathContainer9(visible) = false); Do(%MathContainer10(visible) = false); Do(%MathContainer11(visible) = false); Do(%MathContainer12(visible) = false); Do(%Label4(visible) = false); Do(%Label5(visible) = false); Do(%Label6(visible) = false); Do(%Label7(visible) = false); Do(%Label8(visible) = false); Do(%Label9(visible) = false); Do(%Label10(visible) = false); Do(%Label11(visible) = false); Do(%Label12(visible) = false); Do(%Button3(visible) = false); Do(%Button4(visible) = false); Do(%Button6(visible) = false); Do(%Button7(visible) = false); Do(%Button5(visible) = false); Do(%Button11(enabled) = false); Do(%Button12(enabled) = false); Do(%Button13(enabled) = false); Do(%Button14(enabled) = false); Do(%Button15(enabled) = false); Do(%Button8(visible) = false); Do(%Button9(visible) = false); Do(%Button10(visible) = false); Do(%RadioButton0(visible) = false); Do(%RadioButton1(visible) = false); end use;

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Select Coordinate System,Cartesian [x\134,y\134,z],Cartesian - other,cylindrical,spherical,bipolarcylindrical,bispherical,cardioidal,cardioidcylindrical,casscylindrical,conical,ellcylindrical,hypercylindrical,invcasscylindrical,logcylindrical,logcoshcylindrical,oblatespheroidal,paraboloidal,paracylindrical,prolatespheroidal,rosecylindrical,sixsphere,tangentcylindrical,tangentsphere,toroidaluse DocumentTools in module() local q; if GetProperty(ComboBox2,value) = "Select Coordinate System" then Do(%Label4(Visible) = false); Do(%MathContainer4 =NULL); Do(%MathContainer4(visible) = false); Do(%Button3(visible) = false); Do(%MathContainer5 = NULL); Do(%MathContainer5(visible) = false); return;end if; Do(q = %ComboBox2); if q=Cartesian [x,y,z] then Do(%MathContainer4=MathML[Export]([x,y,z])); elif q=Cartesian - other then Do(%MathContainer4 = NULL); else Do(%MathContainer4 = NULL); end if: Do(%MathContainer5 = NULL); Do(%Label4(visible) = true); Do(%MathContainer4(visible) = true); Do(%Button3(visible) = true); Do(%MathContainer5(visible) = true); #Do(%MathContainer7 = NULL); #Do(%MathContainer8 = NULL); end module; end use;

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JSFH

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use DocumentTools, StringTools in xxxVFconstruct := module() local q,q1,q2; export q3; Do(q = %ComboBox2); Do(q1 = %MathContainer4); if type([q1],'list'('list')) then q2:=q1; elif type([q1],'list') then q2:=[q1];end if; if q2=NULL then return;end if; if q=Cartesian [x,y,z] then q3 := parse(Join(["cartesian",convert(q2,string)],"")); elif q=Cartesian - other then if nops(q2)=3 then q3:=(parse(Join(["cartesian",convert(q2,string)],""))) else return;end if; else if nops(q2)=2 then q3:=(parse(Join([convert(q,string),convert(q2,string)],""))) else return;end if; end if; Do(%MathContainer5 = MathML[Export]([op(0,q3),[op(q3)]])); Do(%Label5(visible) = true): Do(%Label6(visible) = true): Do(%Label7(visible) = true): Do(%MathContainer6(visible) = true); Do(%MathContainer7(visible) = true); Do(%MathContainer8(visible) = true); Do(%Button4(visible) = true); Do(%Button6(visible) = true); Do(%Button7(visible) = true); Do(%Button5(visible) = true); end module; end use;

JSFH

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use DocumentTools in Do(%MathContainer6 = NULL); end use;

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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use DocumentTools in module() local q1,q2,q3; Do(q1 = %MathContainer6); Do(q2 = %MathContainer7); Do(q3 = %MathContainer8); if nops([q1])*nops([q2])*nops([q3])=0 then return;end if; Do(%Label8(visible) = true); Do(%Label9(visible) = true); Do(%Label10(visible) = true); Do(%Label11(visible) = true); Do(%Label12(visible) = true); Do(%MathContainer9(visible) = true); Do(%MathContainer10(visible) = true); Do(%MathContainer11(visible) = true); Do(%MathContainer12(visible) = true); Do(%Button8(visible) = true); Do(%Button9(visible) = true); Do(%Button10(visible) = true); Do(%RadioButton0(visible) = true); Do(%RadioButton1(visible) = true); Do(%RadioButton0(value) = true); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

use DocumentTools in Do(%MathContainer7 = NULL); end use;

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use DocumentTools in Do(%MathContainer8 = NULL); end use;

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use DocumentTools in Do(%MathContainer9 = NULL); Do(%MathContainer10 = NULL); Do(%MathContainer11 = NULL); end use;

use DocumentTools in Do(%MathContainer12 = NULL); end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8=

use DocumentTools in Do(%MathContainer13 = NULL); Do(%MathContainer14 = NULL); end use;

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use DocumentTools in module() local q1,q2,q3,q4; Do(q1 = %MathContainer9); Do(q2 = %MathContainer10); Do(q3 = %MathContainer11); Do(q4 = %MathContainer12); if nops([q1])*nops([q2])*nops([q3])*nops([q4])=0 then return;end if; Do(%Button11(enabled) = true); Do(%MathContainer13(visible) = true); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

JSFH

use DocumentTools, VectorCalculus in module() local q, F1, F2, F3, F, a,b,c,rad, DIR; Do(F1 = %MathContainer6); Do(F2 = %MathContainer7); Do(F3 = %MathContainer8); if nops([F1])*nops([F2])*nops([F3])=0 then return;end if; F := VectorField([F1,F2,F3],coords=xxxVFconstruct:-q3); Do(a = %MathContainer9); Do(b = %MathContainer10); Do(c = %MathContainer11); Do(rad = %MathContainer12); if nops([a])*nops([b])*nops([c])*nops([rad])=0 then return;end if; Cent := Vector([a,b,c],coords=cartesian[x,y,z]); if Do(%RadioButton0) then DIR:=outward else DIR:=inward;end if; q := Flux(F,Sphere(Cent,rad,DIR),inert); Do(%MathContainer13 = q); Do(%Button12(enabled) = true); Do(%Button13(enabled) = true); Do(%MathContainer14(visible) = true); end module; end use;

use DocumentTools in module() local q; Do(q = %MathContainer13); Do(%MathContainer13 = subsop([1,1] = simplify(op([1,1],q)),q)); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

use DocumentTools in module() local q; Do(q = %MathContainer13); Do(%MathContainer14 = value(q)); Do(%Button14(enabled) = true); Do(%Button15(enabled) = true); end module; end use;

use DocumentTools in module() local q; Do(q = %MathContainer14); Do(%MathContainer14 = simplify(q)); end module; end use;

use DocumentTools in module() local q; Do(q = %MathContainer14); Do(%MathContainer14 = evalf(q)); end module; end use;

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<Text-field style="Heading 2" layout="Heading 2">Ratio of Flux to Volume</Text-field>

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Divide by the volume of a sphere, etc.

Copy/paste the flux and press the Enter key.

Context Menu: Factor

Divide by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRRIjRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2JS1GLzYkUSIzRidGMi8lJ2l0YWxpY0dRJXRydWVGJy9GM1EnaXRhbGljRicvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRkQvJSliZXZlbGxlZEdRJmZhbHNlRictSSNtaUdGJDYlUScmIzk2MDtGJy9GO0ZJRjItSSNtb0dGJDYtUSJ+RidGMi8lJmZlbmNlR0ZJLyUqc2VwYXJhdG9yR0ZJLyUpc3RyZXRjaHlHRkkvJSpzeW1tZXRyaWNHRkkvJShsYXJnZW9wR0ZJLyUubW92YWJsZWxpbWl0c0dGSS8lJ2FjY2VudEdGSS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRl1vLUklbXN1cEdGJDYlLUZLNiVRJyYjOTYxO0YnRk5GMkY1LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0Yy

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 1" layout="Heading 1">Solution (Maple 15)</Text-field>

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Tools\342\211\273Load Package: Vector Calculus

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

<Text-field style="Heading 2" layout="Heading 2">Divergence by Differential Operator</Text-field>

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Set coordinate system

Tools\342\211\273Tasks\342\211\273Browse:

Calculus - Vector\342\211\273Vector Algebra and Settings\342\211\273Set Coordinates

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Set Coordinates

use DocumentTools in Do(%MathContainer0 = NULL); Do(%MathContainer1 = NULL); Do(%ComboBox0(enabled) = true); Do(%ComboBox5(enabled) = false); Do(%ComboBox6(enabled) = false); Do(%Label0(enabled) = true); Do(%MathContainer0(visible) = true); SetProperty(Button0,enabled,true); Do(%MathContainer1(visible) = true); if Do(%ComboBox0) = Cartesian [x,y] then Do(%MathContainer0 = MathML[Export]([x,y]));end if; end use;

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use DocumentTools in Do(%ComboBox0(enabled) = false); Do(%ComboBox5(enabled) = true); Do(%ComboBox6(enabled) = false); Do(%Label0(enabled) = true); Do(%MathContainer0(visible) = true); SetProperty(Button0,enabled,true); Do(%MathContainer1(visible) = true); Do(%MathContainer0 = NULL); Do(%MathContainer1 = NULL); end use;

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use DocumentTools in Do(%ComboBox0(enabled) = false); Do(%ComboBox5(enabled) = false); Do(%ComboBox6(enabled) = true); Do(%Label0(enabled) = true); Do(%MathContainer0(visible) = true); SetProperty(Button0,enabled,true); Do(%MathContainer1(visible) = true); Do(%MathContainer0 = NULL); Do(%MathContainer1 = NULL); end use;

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Cartesian [x\134,y],Cartesian [x\134,y\134,z],Cartesian - other,polar,cylindrical,sphericaluse DocumentTools in module() local q; Do(q = %ComboBox0): if q=Cartesian [x,y] then Do(%MathContainer0=MathML[Export]([x,y])); elif q=Cartesian [x,y,z] then Do(%MathContainer0=MathML[Export]([x,y,z])); elif q=Cartesian - other then Do(%MathContainer0=NULL); elif q=polar then Do(%MathContainer0=NULL); elif q=cylindrical then Do(%MathContainer0=NULL); elif q=spherical then Do(%MathContainer0=NULL); end if: Do(%MathContainer1 = NULL); end module; end use;

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bipolar,cardioid,cassinian,elliptic,hyperbolic,invcassinian,logarithmic,logcosh,parabolic,rose,tangentuse DocumentTools in Do(%MathContainer0 = NULL); Do(%MathContainer1 = NULL); end use;

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bipolarcylindrical,bispherical,cardioidal,cardioidcylindrical,casscylindrical,conical,ellcylindrical,hypercylindrical,invcasscylindrical,logcylindrical,logcoshcylindrical,oblatespheroidal,paraboloidal,paracylindrical,prolatespheroidal,rosecylindrical,sixsphere,tangentcylindrical,tangentsphere,toroidaluse DocumentTools in Do(%MathContainer0 = NULL); Do(%MathContainer1 = NULL); end use;

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use DocumentTools, VectorCalculus in module() local q,q1,q2,q3; if Do(%RadioButton4) then Do(q = %ComboBox0); Do(q1 = %MathContainer0); if type([q1],'list'('list')) then q2:=q1; elif type([q1],'list') then q2:=[q1];end if; if q2=NULL then return;end if; if q=Cartesian [x,y] then q3 := parse(StringTools:-Join(["cartesian",convert(q2,string)],"")); elif q=Cartesian [x,y,z] then q3 := parse(StringTools:-Join(["cartesian",convert(q2,string)],"")); elif q=Cartesian - other then if nops(q2)=2 or nops(q2)=3 then q3:=(parse(StringTools:-Join(["cartesian",convert(q2,string)],""))) else return;end if; elif q=polar then if nops(q2)=2 then q3:=(parse(StringTools:-Join(["polar",convert(q2,string)],""))) else return;end if; elif q=cylindrical then if nops(q2)=3 then q3:=(parse(StringTools:-Join(["cylindrical",convert(q2,string)],""))) else return;end if; elif q=spherical then if nops(q2)=3 then q3:=(parse(StringTools:-Join(["spherical",convert(q2,string)],""))) else return;end if; end if; Do(%MathContainer1 = MathML[Export]([op(0,q3),[op(q3)]])); SetCoordinates(q3); elif Do(%RadioButton5) then Do(q = %ComboBox5); Do(q1 = %MathContainer0); if type([q1],'list'('list')) then q2:=q1; elif type([q1],'list') then q2:=[q1];end if; if q2=NULL then return;end if; if nops(q2)<>2 then return; end if; q3 := parse(StringTools:-Join([convert(q,string),convert(q2,string)],"")); Do(%MathContainer1 = MathML[Export]([op(0,q3),[op(q3)]])); SetCoordinates(q3); elif Do(%RadioButton6) then Do(q = %ComboBox6); Do(q1 = %MathContainer0); if type([q1],'list'('list')) then q2:=q1; elif type([q1],'list') then q2:=[q1];end if; if q2=NULL then return;end if; if nops(q2)<>3 then return; end if; q3 := parse(StringTools:-Join([convert(q,string),convert(q2,string)],"")); Do(%MathContainer1 = MathML[Export]([op(0,q3),[op(q3)]])); SetCoordinates(q3); else return; end if; end module; end use;

use DocumentTools, VectorCalculus in Do(%RadioButton7 = true): Do(%MathContainer0 = NULL): Do(%MathContainer1 = NULL): Do(%ComboBox0 = "Cartesian [x,y]"): SetProperty(ComboBox0,enabled,false): Do(%ComboBox5 = bipolar): SetProperty(ComboBox5,enabled,false): Do(%ComboBox6 = bipolarcylindrical): SetProperty(ComboBox6,enabled,false): SetProperty(Label0,enabled,false): SetProperty(MathContainer0,visible,false): SetProperty(MathContainer1,visible,false): SetProperty(Button0,enabled,false): SetCoordinates(cartesian); end use;

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The field F

Using angle brackets, enter the field F as a vector.

Context Menu: Assign to a Name\342\211\273F

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Divergence by differential operator

Context Menu: Evaluate and Display Inline

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<Text-field style="Heading 2" layout="Heading 2">Flux through Surface of Sphere</Text-field>

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Tools\342\211\273Tasks\342\211\273Browse:

Calculus - Vector\342\211\273Integration\342\211\273Flux\342\211\2733-D\342\211\273Through a Sphere

Flux through a Sphere

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For the Vector Field:

use DocumentTools in Do(%ComboBox4(value) = "Select Coordinate System"): Do(%MathContainer17 = NULL); Do(%MathContainer18 = NULL); Do(%MathContainer19 = NULL); Do(%MathContainer20 = NULL); Do(%MathContainer21 = NULL); Do(%MathContainer26 = NULL); Do(%MathContainer27 = NULL); Do(%MathContainer22 = NULL); Do(%MathContainer23 = NULL); Do(%MathContainer24 = NULL); Do(%MathContainer25 = NULL); Do(%MathContainer17(visible) = false); Do(%MathContainer18(visible) = false); Do(%MathContainer19(visible) = false); Do(%MathContainer20(visible) = false); Do(%MathContainer21(visible) = false); Do(%MathContainer26(visible) = false); Do(%MathContainer27(visible) = false); Do(%MathContainer22(visible) = false); Do(%MathContainer23(visible) = false); Do(%MathContainer24(visible) = false); Do(%MathContainer25(visible) = false); Do(%Label15(visible) = false); Do(%Label16(visible) = false); Do(%Label17(visible) = false); Do(%Label18(visible) = false); Do(%Label19(visible) = false); Do(%Label20(visible) = false); Do(%Label21(visible) = false); Do(%Label22(visible) = false); Do(%Label23(visible) = false); Do(%Button18(visible) = false); Do(%Button19(visible) = false); Do(%Button21(visible) = false); Do(%Button22(visible) = false); Do(%Button20(visible) = false); Do(%Button26(enabled) = false); Do(%Button27(enabled) = false); Do(%Button28(enabled) = false); Do(%Button29(enabled) = false); Do(%Button30(enabled) = false); Do(%Button23(visible) = false); Do(%Button24(visible) = false); Do(%Button25(visible) = false); Do(%RadioButton2(visible) = false); Do(%RadioButton3(visible) = false); end use;

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Select Coordinate System,Cartesian [x\134,y\134,z],Cartesian - other,cylindrical,spherical,bipolarcylindrical,bispherical,cardioidal,cardioidcylindrical,casscylindrical,conical,ellcylindrical,hypercylindrical,invcasscylindrical,logcylindrical,logcoshcylindrical,oblatespheroidal,paraboloidal,paracylindrical,prolatespheroidal,rosecylindrical,sixsphere,tangentcylindrical,tangentsphere,toroidaluse DocumentTools in module() local q; if GetProperty(ComboBox4,value) = "Select Coordinate System" then Do(%Label15(Visible) = false); Do(%MathContainer17 =NULL); Do(%MathContainer17(visible) = false); Do(%Button18(visible) = false); Do(%MathContainer18 = NULL); Do(%MathContainer18(visible) = false); return;end if; Do(q = %ComboBox4); if q=Cartesian [x,y,z] then Do(%MathContainer17=MathML[Export]([x,y,z])); elif q=Cartesian - other then Do(%MathContainer17 = NULL); else Do(%MathContainer17 = NULL); end if: Do(%MathContainer18 = NULL); Do(%Label15(visible) = true); Do(%MathContainer17(visible) = true); Do(%Button18(visible) = true); Do(%MathContainer18(visible) = true); #Do(%MathContainer20 = NULL); #Do(%MathContainer21 = NULL); end module; end use;

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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use DocumentTools, StringTools in xxxVFconstruct := module() local q,q1,q2; export q3; Do(q = %ComboBox4); Do(q1 = %MathContainer17); if type([q1],'list'('list')) then q2:=q1; elif type([q1],'list') then q2:=[q1];end if; if q2=NULL then return;end if; if q=Cartesian [x,y,z] then q3 := parse(Join(["cartesian",convert(q2,string)],"")); elif q=Cartesian - other then if nops(q2)=3 then q3:=(parse(Join(["cartesian",convert(q2,string)],""))) else return;end if; else if nops(q2)=2 then q3:=(parse(Join([convert(q,string),convert(q2,string)],""))) else return;end if; end if; Do(%MathContainer18 = MathML[Export]([op(0,q3),[op(q3)]])); Do(%Label16(visible) = true): Do(%Label17(visible) = true): Do(%Label18(visible) = true): Do(%MathContainer19(visible) = true); Do(%MathContainer20(visible) = true); Do(%MathContainer21(visible) = true); Do(%Button19(visible) = true); Do(%Button21(visible) = true); Do(%Button22(visible) = true); Do(%Button20(visible) = true); end module; end use;

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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use DocumentTools in Do(%MathContainer19 = NULL); end use;

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use DocumentTools in module() local q1,q2,q3; Do(q1 = %MathContainer19); Do(q2 = %MathContainer20); Do(q3 = %MathContainer21); if nops([q1])*nops([q2])*nops([q3])=0 then return;end if; Do(%Label19(visible) = true); Do(%Label20(visible) = true); Do(%Label21(visible) = true); Do(%Label22(visible) = true); Do(%Label23(visible) = true); Do(%MathContainer22(visible) = true); Do(%MathContainer23(visible) = true); Do(%MathContainer24(visible) = true); Do(%MathContainer25(visible) = true); Do(%Button23(visible) = true); Do(%Button24(visible) = true); Do(%Button25(visible) = true); Do(%RadioButton2(visible) = true); Do(%RadioButton3(visible) = true); Do(%RadioButton2(value) = true); end module; end use;

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use DocumentTools in Do(%MathContainer20 = NULL); end use;

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use DocumentTools in Do(%MathContainer21 = NULL); end use;

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use DocumentTools in Do(%MathContainer22 = NULL); Do(%MathContainer23 = NULL); Do(%MathContainer24 = NULL); end use;

use DocumentTools in Do(%MathContainer25 = NULL); end use;

use DocumentTools in Do(%MathContainer26 = NULL); Do(%MathContainer27 = NULL); end use;

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use DocumentTools in module() local q1,q2,q3,q4; Do(q1 = %MathContainer22); Do(q2 = %MathContainer23); Do(q3 = %MathContainer24); Do(q4 = %MathContainer25); if nops([q1])*nops([q2])*nops([q3])*nops([q4])=0 then return;end if; Do(%Button26(enabled) = true); Do(%MathContainer26(visible) = true); end module; end use;

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use DocumentTools, VectorCalculus in module() local q, F1, F2, F3, F, a,b,c,rad, DIR; Do(F1 = %MathContainer19); Do(F2 = %MathContainer20); Do(F3 = %MathContainer21); if nops([F1])*nops([F2])*nops([F3])=0 then return;end if; F := VectorField([F1,F2,F3],coords=xxxVFconstruct:-q3); Do(a = %MathContainer22); Do(b = %MathContainer23); Do(c = %MathContainer24); Do(rad = %MathContainer25); if nops([a])*nops([b])*nops([c])*nops([rad])=0 then return;end if; Cent := Vector([a,b,c],coords=cartesian[x,y,z]); if Do(%RadioButton2) then DIR:=outward else DIR:=inward;end if; q := Flux(F,Sphere(Cent,rad,DIR),inert); Do(%MathContainer26 = q); Do(%Button27(enabled) = true); Do(%Button28(enabled) = true); Do(%MathContainer27(visible) = true); end module; end use;

use DocumentTools in module() local q; Do(q = %MathContainer26); Do(%MathContainer26 = subsop([1,1] = simplify(op([1,1],q)),q)); end module; end use;

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use DocumentTools in module() local q; Do(q = %MathContainer26); Do(%MathContainer27 = value(q)); Do(%Button29(enabled) = true); Do(%Button30(enabled) = true); end module; end use;

use DocumentTools in module() local q; Do(q = %MathContainer27); Do(%MathContainer27 = simplify(q)); end module; end use;

use DocumentTools in module() local q; Do(q = %MathContainer27); Do(%MathContainer27 = evalf(q)); end module; end use;

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<Text-field style="Heading 2" layout="Heading 2">Ratio of Flux to Volume</Text-field>

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Divide by the volume of a sphere, etc.

Copy/paste the flux and press the Enter key.

Context Menu: Factor

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

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