Chapter 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 17.1 Green's Theorem (1) Green's Theorem: Let R be a domain whose boundary C is a simple closed curve, oriented counterclockwise. If M(x, y) and N(x, y) are differentiable and have continuous first partial derivatives on R, then . (2) The area formula of line integral: The area enclosed in a closed curve C can be calculated be the line integral A = . (3) The divergence (flux density) of a two-dimensional vector field F = Mi + Nj at the point (x, y) is div F = (4) The k-component of the curl (also called a scalar curl or circulation density) of a two- dimensional vector field F = Mi + Nj at the point (x, y) is curl (F) = (curl F) k = More general forms of Green's Theorem are the following: (5) Green's Theorem (circulation-curl form or tangential form): curl (F) (6) Green's Theorem (flux-divergence form or normal form): div (F) 17.1.1 Find divergence Example 1. Find the divergence of F(x, y) = ( )i + ( )j. divF:=diff(x^2-y,x)+diff(x*y-y^2,y); (1.1.1) Example 2. Find the divergence of F(x, y) = ( )i + ( )j. divF:=diff(x+exp(x)*sin(y),x)+diff(x+exp(x)*cos(y),y); (1.1.2) You can use the command "diverge" in the "linalg" package to calculate the divergence of a vector field as follows. Example 3. Use the command "diverge" in the "linalg" package to find the divergence of F(x, y) = ( )i + ( )j. with(linalg): Chapte 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 333 F:=[x^2-y,x*y-y^2]; (1.1.3) diverge(F,[x,y]); (1.1.4) 17.1.2 Use Green's Theorem to evaluate the line integrals Example 4. Evaluate the counterclockwise circulation of the vector field F(x, y) = + j over C, where C consists of the line segments from (0,0) to (1, 0), from (1, 0) to (0, 1), and from (0, 1) to (0, 0). Step1: Get the graph of the curve C. plot({[0,0],[1,0],[0,1],[0,0]}, color='blue',style=LINE); 1 0 0 1 Step 2: Change the line integral to double integral using Green's Theorem. P:=x^4; Q:=x*y; (1.2.1) func:=diff(Q,x)-diff(P,y); (1.2.2) int(int(func, y=0..1-x),x=0..1); 1 (1.2.3) 6 Example 5. Evaluate the counterclockwise circulation of over C, where C is the circle M:=3*y-exp(sin(x)); N:=7*x+sqrt(y^4+1); (1.2.4) func:=diff(N,x)-diff(M,y); (1.2.5) int(int(func,y=-sqrt(9-x^2)..sqrt(9-x^2)),x=-3..3); (1.2.6) Example 6. Evaluate the counterclockwise circulation of the vector field F(x, y) = i + j over C, where C is the boundary of the semi-annual region in the upper half-plane between the circles and the circle . (1.1.3) 334 Chapte 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS plot({sqrt(1-x^2), sqrt(4-x^2)}, x=-2..2, scaling= constrained); 2 1 0 1 2 x M:=y^2; N:=3*x*y; (1.2.7) func:=diff(N,x)-diff(M,y); (1.2.8) Change to the polar coordinates. x:=r*cos(theta): y:=r*sin(theta): int(int(func*r,r=1..2),theta=0..Pi); 14 (1.2.9) 3 x:='x': y:='y': You can also use the rectangular coordinates to evaluate the integral above as follows. int(int(y, y=0..sqrt(4-x^2)), x=-2..2)-int(int(y, y=0..sqrt (1-x^2)), x=-1..1); 14 (1.2.10) 3 Example 7. Evaluate the line integral where C is the square cut from the first quadrant by the lines x = 1 and y =1. plot({[0,0],[1,0],[1,1],[0,1],[0,0]},style=LINE,color= 'black'); (1.1.3) Chapte 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 335 1 0 0 1 M:=y^2; N:=x*y; (1.2.11) func:=diverge([M,N], [x,y]); (1.2.12) int(int(func, x=0..1),y=0..1); 1 (1.2.13) 2 Example 8. Calculate the outward flux of the field F(x, y) = i + j across the square bounded by the lines x y func:=diverge([x, y^2], [x,y]); (1.2.14) Flux:=int(int(func, x=-1..1),y=-1..1); (1.2.15) 17.1.3 Find the area of a region using the formula of line integral Example 9. Find the area enclosed by the ellipse by the line integral. Step1: Get the parametric equation of the ellipse. x:=a*cos(t); y:=b*sin(t); Step 2: Evaluate the line integral. lowlmt:=0: uplmt:=2*Pi: ifunc:=1/2*(x*diff(y,t)-y*diff(x,t)); A:=int(ifunc,t=lowlmt..uplmt); You can verify the result using the area formula in double integral as follows. x:='x': y:='y': assume(a>0): assume(b>0): (1.1.3) 336 Chapte 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS A:=int(int(1, y=-b*sqrt(1-x^2/a^2)..b*sqrt(1-x^2/a^2)), x=- a..a); (1.3.1) 17.1.4 Use Green's Theorem to change the path of the line integrals Example 10. If F(x, y yi + xj)/( ) and C is the square bounded by the lines x y , find the integral of F over C. x:='x': y:='y': P:=-y/(x^2+y^2): Q:=x/(x^2+y^2): testf:=diff(Q,x)-diff(P,y); simplify(testf); 0 Hence, = 0 except at the origin (0, 0), where the first partial derivatives do not exist. Then the path can be changed to any simple closed path containing the origin, for example, the unit circle. Hence, we can evaluate the integral using the new path , oriented counterclockwise. x:=cos(t); y:=sin(t); rvec:=[x,y]; fieldf:=simplify([-y/(x^2+y^2),x/(x^2+y^2)]); dr:=diff(rvec,t); assume(t>0): ifunc:=simplify(dotprod(fieldf,dr)); int(ifunc,t=0..2*Pi); x:='x': y:='y': Exercises 1. Verify Green's Theorem for the line integral , where C is the unit circle, oriented counterclockwise by (a) evaluating the integral directly, (b) applying Green's Theorem. 2. Use Green's Theorem to evaluate the line integral , where C is the boundary of the unit square , oriented counterclockwise. (1.1.3) (1.3.1) Chapte 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS 337 3. Use Green's Theorem to evaluate the line integral , where C is the triangle with vertices , oriented counterclockwise. 4. Use Green's Theorem to evaluate the work done by the force F = [x + y, y], over the close loop C, where C is the boundary of the region enclosed by 1, oriented counterclockwise. 5. Use Green's Theorem to evaluate the line integral r, where F = [ x2, 2] and C consists of the arcs 1, oriented counterclockwise. 17.2 Stokes' Theorem (1) The curl vector of a vector field F = Mi + j + Pk is curl (F) = (2) Stokes' Theorem: The circulation of a vector field F = Mi + j + Pk around the boundary C of an oriented surface S in the direction counterclockwise with respect to the surface's unit vector n is equal to the integral of curl F over S. curl (F (3) The identity curl grad( f ) = 0, i.e., V# Vf = 0 is important. (4) If = 0 at every point of a simply connected open region D in space, then F is a conservative vector field on D. Note. In Maple, the command "curl" in the "linalg" package can be used to calculate the curl vector of a vector field. 17.2.1 Find the curl and divergence of a vector field Example 1. Find the curl of the vector field F(x, y, z) = i + j + k. F:=[x^2*y,y*z^2,z*x^2]; v:=[x,y,z]; (2.1.1) curl(F,v); (2.1.2) Example 2. Find the curl of the vector field F(x, y, z) = )i + j + k. F:=[x^2-y,4*z,x^2]; v:=[x,y,z]; (2.1.3) curl(F,v); (2.1.4) (1.1.3) (1.3.1) 338 Chapte 17 FUNDAMENTAL THEOREMS OF VECTOR ANALYSIS (2.1.4) 17.2.2 Determine by its curl whether or not a vector field is conservative Example 3. Determine whether or not the vector field F(x, y, z) = i j + k is conservative. F:=[y*z,z^2,x^2]: v:=[x,y,z]: curl(F,v); (2.2.1) Hence, F is not a conservative field. Example 4. Determine whether or not the vector field F(x, y, z) = i + ( )j + k is conservative. F:=[y*z,y^2+x*z,x*y]: curl(F,v); (2.2.2) Hence, F is a conservative field. Example 5. Let r = [x, y, z], r = ||r||. To verify (1) grad (r) = r/r; (2) curl( r ) = 0; and (3) diverge (rr) = 4r. (1) Verify grad (r) = r/r. r:=[x,y,z]: v=[x,y,z]: absr:=sqrt(x^2+y^2+z^2); (2.2.3) grad(absr,v); (2.2.4) Hence, grad (r) = r/r. (2) Verify curl( r ) = 0. curl(r,v); (2.2.5) (3) Verify diverge (rr) = 4r.