Green s Theorem. Fundamental Theorem for Conservative Vector Fields

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1 Assignment - Mathematics 4(Model Answer) onservative vector field and Green theorem onservative Vector Fields If F = φ, for some differentiable function φ in a domaind, then we say that F is conservative field and φ is its potential function. NOTE: ) If F F is not conservative. ) If F = and F is defined on a region D, where the component functions of F are continuous and have continuous first partial derivatives, then F is conservative on D. Fundamental Theorem for onservative Vector Fields Assume that F = φ on a domain D. ) If is any smooth curve from A(x, y, z ) to B(x, y, z ) in D, then F. dr = φ. dr = φ(x, y, z ) φ(x, y, z ) (i.e.) The line integral is independent of the path joining the two points A and B. ) The circulation around a closed path (that is A = B ) is zero. F. dr = Green s Theorem Let be a piecewise smooth, simple closed curve enclosing a region in the plane. Let F = F i + F j be a vector field with F and F having continuous first partial derivatives in an open region containing. Then, F dr = F dx + F dy ounterclockwise circulation = ( F url integral F y ) da ollege of Engineering & Technology

2 Problem : Evaluate the line integral, ( 3y dx dy) Where is the boundary of the region. o x, Verify the result using Green theorem. y sin x SOLUTION: F dr = F dx + F dy = ( F F y ) da L.H.S.: FO. r (t) =< t, > < t <. r (t) =<, > 3. F (r(t)) =<, > 4. F (r(t)) r (t) = + = 5. F dr = F (r(t)) r (t)dt = dt = FO. r (t) =< t, sint > < t <. r (t) =<, cost > 3. F (r(t)) =< 3sint, > ollege of Engineering & Technology

3 4. F (r(t)) r (t) = 3sint + cost. 5. F dr = F (r(t)) r sint = 6. (t)dt = 3sint + cost. dt = 3cost.H.S: F =< 3y, > F = F y = 3 F dr = ( F = 3 cos(x) = 6 F sin (x) y ) da = 3 dy dx = 3 sin(x) dx Since L.H.S=.H.S Green theorem is satisfied. Problem : Show that the vector field F (x, y, z ) = y cos x i +( y sin x + e z ) j + y e z Hence, evaluate the line integral /,3, ) ( (,, ) Solution: F. dr F =< y cos x, y sin x + e z, y e z >. k, is conservative. F = = y z y z F F F 3 y cos x y sin x + e z y e z =< y y ez z y sin x + ez, ( y ez z y cos x), y sin x + e z y y cos x =< e z e z, ( ), ycosx ycosx >= <,, > ollege of Engineering & Technology

4 Since F =.Then F is conservative (path independent) and F = φ. To get potential field φ F = φ F =< F, F, F 3 > =< φ, φ, φ > y z φ = y cos x φ = y cos(x) dx = y sin(x) + A(y, z) φ y = y sin x + e z φ = ysin(x) + e z dy = y sin(x) + ye z + B(x, z) φ z = y ez φ 3 = ye z dz = ye z + c(x, y). φ = φ U φ U φ 3 = y sin(x) + ye z I = F. dr = φ. dr = φ(x, y, z ) φ(x, y, z ) = φ (, 3,) φ (,, ) = 9 + 3e 4 e. Problem 3: a) If F is a vector field with components have continuous nd derivatives prove that:. ( F) =. b) If f is a scalar function with components have continuous nd derivatives prove that: ( F) =. SOLUTION: a). ( f) =. y z = y z y z F F F 3 F F F 3 b) ( f) = = F 3 y F z F 3 y + F yz + F z F zy = f y f y z f z = ( f yz f zy ) i ( f z f z ) j ( f y f y ) k = ollege of Engineering & Technology

5 Problem 4: show that the vector field F(x, y, z ) = ( 8 x 3 + z ) i - 3 z j + ( x z - 3 y) k is conservative, and hence evaluate the line integral Solution: (,,3) (,,) F.dr. F =< 8 x 3 + z, 3 z, x z 3 y > F = = y z y z F F F 3 8 x 3 + z 3 z x z 3 y =< ( x z 3 y) y z ( 3 z), ( ( x z 3 y) z (8x3 + z )), ( 3 z) x z 3 y =< 3 + 3, (z z), ( ) >= <,, > y Since F =.Then F is conservative. To get potential field φ F = φ F =< F, F, F 3 > =< φ, φ, φ > y z φ = 8 x3 + z φ = 8x 3 + z dx = x 4 + z x + A(y, z) φ y = 3 z φ = 3z dy = 3yz + B(x, z). φ z = x z 3 y φ 3 = xz 3ydz = z x 3yz + c(x, y). φ = φ U φ U φ 3 = φ = z x 3yz + x 4 I = F. dr = φ. dr = φ(x, y, z ) φ(x, y, z ) = φ(,,3) φ(,,) = = 4. ollege of Engineering & Technology

6 Problem 5: verify Green s theorem for the vector field, F(x, y) = x y i + y j and the curve, which is the boundary of the region bounded by, y = x and y = x. Solution: : y = x & F(x, y) = x y i + y j.. r (t) =< t, t > < t <. r (t) =<, t > 3. F (r(t)) =< t 4, t 4 > 4. F (r(t)) r (t) = t 4 + t F dr = F (r(t)) r (t)dt = t 4 + t 5 dt = 8 5 : y = x & F(x, y) = x y i + y j.. r (t) =< t, t > < t <. r (t) =<, > 3. F (r(t)) =< ( t) 3, ( t) > 4. F (r(t)) r (t) = ( t) 3 ( t). ollege of Engineering & Technology

7 5. F dr = F (r(t)) r 7. (t)dt = ( t) 3 ( t) dt = so F dr = F dr + F dr = c c c =..H.S: F =< x y, y > F = F y = x F dr = ( F F x y ) da = x dy dx = x 3 + x 4 dx x = x4 4 + x5 5 = = Since L.H.S=.H.S. Then Green theorem is satisfied. Problem 6: Use Green s theorem to evaluate the line integral F dr where: F ( 4 e x ) ˆi (sin y 3x ) ˆj,and the curve c is shown in figure. y G Solution: a b x F dr = F dx + F dy = ( F F ( 4 e x ) ˆi (sin y 3x ) ˆj F y ) da F = 6x F y = ollege of Engineering & Technology

8 F dx + F dy = 6x da G Use polar coordinates: a < r < b & < θ <. b 6rcos(θ)rdrdθ = a b 3 cos(θ) a 3 cos(θ) dθ = ( b 3 a 3 ) sin(θ) = b 3 a 3. Problem 7: Given the potential function, f( x, y, z)= x + y + z, using it construct a conservative vector field F ( x, y, z ) and hence evaluate the line integrals : (,,3) (i) F. (ii). dr (,,) F. d r, where : r ( t ) = ( + cos t, 3 + sin t, 4 ), t. x f x = (x + y + z ) 3 y f y = (x + y + z ) 3 z f z = (x + y + z ) 3 x F(x, y, z) = f = (x + y + z ), y 3 (x + y + z ), z 3 (x + y + z ) 3 i) F = f(,,3) f(,,) = 4 (,,3). dr (,,) ii) F dr =. ollege of Engineering & Technology