Vector Calculus with Maple - Math5720, Fall 2011

Linear Algebra package: vector algebra operations

with(LinearAlgebra);

a:=Vector(<1,2,3>); b:=Vector(<-3,4,5>); a+3*b; Norm(a,2);

DotProduct(a,b); a.b;

VectorAngle(a,b);

evalf(%);

CrossProduct(a,b);

VectorCalculus package

with(VectorCalculus); BasisFormat(true);

? BasisFormat;

SetCoordinates('cartesian'[x,y,z]);

Differentiation of vector-functions, curves

r:=<cos(2*t),-sin(2*t),t>;

r1:=diff(r,t); r2:=diff(r,t,t);

To use standard Vector notation, set BasisFormat to false.

BasisFormat(false);

Differentiation of vector-functions, curves

r:=<cos(2*t),-sin(2*t),t>;

r1:=diff(r,t); r2:=diff(r,t,t);

The arc length computation

evalf(Int(Norm(r1,2),t=0..2*Pi));

Evaluation of the derivative at the given point

v:=(subs(t=Pi, r1));

simplify(v);

with(plots);

Plotting plane curve Example (Cycloid, see Section 9.5 Problem 21, page 398) (See : plot[parametric] in Maple Help)

plot([sin(t)+t,cos(t)+1,t=0..6*Pi],scaling=constrained);

Ploting space curve

with(plots):

spacecurve([r[1],r[2],r[3], t=0..2*Pi], axes=normal, thickness=3,numpoints=400);

Gradient, directional derivatives, potential, divergence, curl

f:=x^3*y^2*z;

F:=Gradient(f,[x,y,z]);

dirder:= DirectionalDiff(f,a,[x,y,z]);

subs(x=-1,y=4,z=3,dirder); v:='v';

p:=ScalarPotential(F);

p;

Divergence(F);

Curl(F);

rr:=sqrt(x^2+y^2+z^2);

Laplacian(1/rr,[x,y,z]);

simplify(%);

G:=VectorField(<-x*y^2,x,x-y>); Divergence(G);Curl(G);

Evaluating the value of vector field at the point

evalVF(Curl(G),<1,2,3>);

Visualization of lines and surfaces

Few spacecurves in plotted jointly (helix and a tangent line at t=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2J1EjcGlGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3L0Y2USdub3JtYWxGJw==)

ri:=simplify(subs(t=Pi,r)); di:=simplify(subs(t=Pi,r1)); tl:=evalm(ri+t*di);

p1:=spacecurve([r[1],r[2],r[3],t=0..2*Pi,color=blue], axes=normal): p2:=spacecurve(tl,t=-3..3,color=red):display(p1,p2,thickness=4);

Surface (helixoid and torus)

plot3d( [ v*cos(u), v*sin(u), v*u ], u=0..4*Pi,v=0..1,numpoints=3000);

plot3d([ (2+cos(y))*cos(x), (2+cos(y))*sin(x), sin(y) ], x=0..2*Pi, y=0..2*Pi,scaling=constrained,style=patchnogrid);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkjbWlHRiQ2JlEjYzFGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjQtSSNtb0dGJDYuUSomY29sb25lcTtGJ0YyRjUvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZLLUkobWZlbmNlZEdGJDYnLUYjNjgtRiw2JlEkY29zRidGL0YyRjUtRk82JS1GIzYoLUYsNiZRInhGJ0YvRjJGNS8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdRicvJStleGVjdXRhYmxlR0YxLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0YyRjVGMkY1LUY4Ni5RKCZtaW51cztGJ0YyRjVGO0Y9Rj9GQUZDRkVGRy9GSlEsMC4yMjIyMjIyZW1GJy9GTUZjby1JI21uR0YkNiVRIjJGJ0YyRjUtRjg2LlExJkludmlzaWJsZVRpbWVzO0YnLyUlYm9sZEdGMS9GM1Enbm9ybWFsRidGO0Y9Rj9GQUZDRkVGRy9GSlEmMC4wZW1GJy9GTUZhcEZTLUZPNiUtRiM2Ki1GLDYjUSFGJy1GIzYpLUZmbzYlUSQwLjRGJ0YyRjVGaW8tRiw2JlEieUYnRi9GMkY1RmduRlxvRjJGNUZncEZnbkZqbkZcb0YyRjVGMkY1LUY4Ni5RIixGJ0YyRjVGOy9GPlEldHJ1ZUYnRj9GQUZDRkVGR0ZgcC9GTVEsMC4zMzMzMzMzZW1GJy1GLDYmUSRzaW5GJ0YvRjJGNUZWRl9vRmVvRmlvRmlxRmNwRmJxRl9xRmduRmpuRlxvRjJGNUYyRjUvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRictRjg2LlEiO0YnRjJGNUY7RmVxRj9GQUZDRkVGR0ZgcEZMLyUrZm9yZWdyb3VuZEdRKFswLDAsMF1GJ0ZnbkZqbkZcb0YyRjU=

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Line integralse (few examples with LineInt command)

with(VectorCalculus): SetCoordinates( cartesian[x,y] );

Don't forget to indicate that coordinate system is two-dimensional now

LineInt( VectorField( <x^3,y*cos(x)> ), LineSegments( <0,0>, <1,1>, <1,-1> ) );

LineInt( VectorField( <x^2,y^2> ), Path( <t,t^2>, t=0..2 ) );

LineInt( VectorField( <y,-x> ), Circle( <0,0>, 4 ) );

In 3-dimensional space

SetCoordinates( cartesian[x,y,z] );

LineInt(VectorField(<x^2,x*y*z,z^2*y>),Path(<cos(t),-sin(t),t^2>,t=0..Pi));

Double Integrals as iterated integrals

evalf(int(int(ln(x*y)/x,x=y+1..y^2+2),y=1..2));

int(int((x^2*y),y=1-x..1-x^2),x=0..1);

int( x*y, [x,y] = Triangle( <0,0>, <1,0>, <0,1> ) );

int( sin(x)*cos(x*y)*tan(z), [x,y,z] = Parallelepiped( 0..Pi, 0..Pi/2, 0..Pi/4 ) );

evalf(%);

Flux integrals

Flux( VectorField( <y,-x,x*y>, cartesian[x,y,z] ), Surface( <u,t,u^2+v^2>, [u,v] = Rectangle( 0..1, 2..3 ) ) );

Flux( VectorField( <x,y,z>, cartesian[x,y,z] ), Sphere( <0,0,0>, 4 ) );

evalf(Flux( VectorField( <cos(x*y),-x,x*y>, cartesian[x,y,z] ), Surface( <s,t,s^2+t^2>, [s,t] = Region( 0..1, s^2-2..s+2 ) ) ) );

Surface Integrals

SurfaceInt( x+y+z, [x,y,z] = Surface( <s,t,4-2*s-t>, [s,t] = Triangle(<0,0>,<1,0>,<0,1>) ) );

SurfaceInt( x*y*z, [x,y,z] = Box( 1..2, 3..5, 5..6 ) );

Plotting parallelepiped

with(plottools);with(plots);

p1:=parallelepiped(<1,0,0>,<0,2,0>,<0,0,1>,[1,3,5]):

display(p1);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=

Computer Project #1 (due to Tuesday, October 18)

Use Maple to solve the following problems (URL address of the file with the scanned pages of the 9th edition of the textbook http://homepages.wmich.edu/~ledyaev/m572cp.pdf) 1. (9th edition) Section 9.5: Problems 22 , 25 (page 399) . Display simultaneously vectors of tangential and normal acceleration at the indicated points . 2. (9th edition) Section 10.1 Problems 11, 12 (page 425) :line integrals

3. (9th edition) Section 10.6 Problems 11, 12 page 456), sketch surfaces using Maple (9th edition)

4. (9th edition) Section 10.9 9, 10 (page 473), sketch surfaces using Maple