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  • The Divergence TheoremMATH 311, Calculus III

    J. Robert Buchanan

    Department of Mathematics

    Summer 2011

    J. Robert Buchanan The Divergence Theorem

  • Greens Theorem Revisited

    Greens Theorem:C

    M(x , y) dx + N(x , y) dy =

    R

    (Nx My

    )dA

    R

    T

    C

    n

    x

    y

    J. Robert Buchanan The Divergence Theorem

  • Greens Theorem Vector Form (1 of 3)

    Simple closed curve C is described by the vector-valuedfunction

    r(t) = x(t), y(t) for a t b.

    The unit tangent vector and unit (outward) normal vector to Care respectively

    T(t) =1

    r(t)x (t), y (t) and n(t) = 1

    r(t)y (t),x (t).

    J. Robert Buchanan The Divergence Theorem

  • Greens Theorem Vector Form (2 of 3)

    If the vector field F(x , y) = M(x , y)i + N(x , y)j, then along thesimple closed curve C:

    F n = M(x(t), y(t)),N(x(t), y(t)) 1r(t)

    y (t),x (t)

    =(M(x(t), y(t))y (t) N(x(t), y(t))x (t)

    ) 1r(t)

    .

    Now consider the line integralC

    F n ds.

    Note: this is a line integral with respect to arc length.

    J. Robert Buchanan The Divergence Theorem

  • Greens Theorem Vector Form (2 of 3)

    If the vector field F(x , y) = M(x , y)i + N(x , y)j, then along thesimple closed curve C:

    F n = M(x(t), y(t)),N(x(t), y(t)) 1r(t)

    y (t),x (t)

    =(M(x(t), y(t))y (t) N(x(t), y(t))x (t)

    ) 1r(t)

    .

    Now consider the line integralC

    F n ds.

    Note: this is a line integral with respect to arc length.

    J. Robert Buchanan The Divergence Theorem

  • Greens Theorem Vector Form (2 of 3)

    If the vector field F(x , y) = M(x , y)i + N(x , y)j, then along thesimple closed curve C:

    F n = M(x(t), y(t)),N(x(t), y(t)) 1r(t)

    y (t),x (t)

    =(M(x(t), y(t))y (t) N(x(t), y(t))x (t)

    ) 1r(t)

    .

    Now consider the line integralC

    F n ds.

    Note: this is a line integral with respect to arc length.

    J. Robert Buchanan The Divergence Theorem

  • Greens Theorem Vector Form (3 of 3)

    C

    F n ds = b

    a(F n)(t) r(t)dt

    =

    ba

    (M(x(t), y(t))y (t) N(x(t), y(t))x (t)

    ) r(t)r(t)

    dt

    =

    ba

    (M(x(t), y(t))y (t) N(x(t), y(t))x (t)

    )dt

    =

    C

    M(x , y) dy N(x , y) dx

    =

    R

    (Mx

    +Ny

    )dA (by Greens Theorem)

    =

    R F dA

    J. Robert Buchanan The Divergence Theorem

  • Summary and Objective

    Greens Theorem in vector form statesC

    F n ds =

    R F(x , y) dA.

    A double integral of the divergence of a two-dimensional vectorfield over a region R equals a line integral around the closedboundary C of R.

    The Divergence Theorem (also called Gausss Theorem) willextend this result to three-dimensional vector fields.

    J. Robert Buchanan The Divergence Theorem

  • Summary and Objective

    Greens Theorem in vector form statesC

    F n ds =

    R F(x , y) dA.

    A double integral of the divergence of a two-dimensional vectorfield over a region R equals a line integral around the closedboundary C of R.

    The Divergence Theorem (also called Gausss Theorem) willextend this result to three-dimensional vector fields.

    J. Robert Buchanan The Divergence Theorem

  • Divergence Theorem

    Remark: the Divergence Theorem equates surface integralsand volume integrals.

    Theorem (Divergence Theorem)

    Let Q R3 be a region bounded by a closed surface Q andlet n be the unit outward normal to Q. If F is a vector functionthat has continuous first partial derivatives in Q, then

    QF n dS =

    Q F dV .

    J. Robert Buchanan The Divergence Theorem

  • Proof (1 of 7)

    Suppose F(x , y , z) = M(x , y , z)i + N(x , y , z)j + P(x , y , z)k,then the Divergence Theorem can be stated as

    QF n dS

    =

    Q

    M(x , y , z)i n dS +

    QN(x , y , z)j n dS

    +

    Q

    P(x , y , z)k n dS

    =

    Q

    Mx

    dV +

    Q

    Ny

    dV +

    Q

    Pz

    dV

    =

    Q F(x , y , z) dV .

    J. Robert Buchanan The Divergence Theorem

  • Proof (2 of 7)

    Thus the theorem will be proved if we can show thatQ

    M(x , y , z)i n dS =

    Q

    Mx

    dVQ

    N(x , y , z)j n dS =

    Q

    Ny

    dVQ

    P(x , y , z)k n dS =

    Q

    Pz

    dV .

    All of the proofs are similar so we will focus only on the third.

    J. Robert Buchanan The Divergence Theorem

  • Proof (3 of 7)

    Suppose region Q can be described as

    Q = {(x , y , z) |g(x , y) z h(x , y), for (x , y) R}

    where R is a region in the xy -plane.

    Think of Q as being bounded by three surfaces S1 (top), S2(bottom), and S3 (side).

    J. Robert Buchanan The Divergence Theorem

  • Proof (4 of 7)

    S1: z=hHx,yL

    S2: z=gHx,yL S3

    x

    y

    z

    On surface S3 the the unit outward normal is parallel to thexy -plane and thus

    QP(x , y , z) k n

    =0

    dS =

    Q0 dS = 0.

    J. Robert Buchanan The Divergence Theorem

  • Proof (5 of 7)

    Now we calculate the surface integral over S1.

    S1 = {(x , y , z) | z h(x , y) = 0, for (x , y) R}

    Unit outward normal:

    n =(z h(x , y))(z h(x , y))

    =hx(x , y)i hy (x , y)j + k

    [hx(x , y)]2 + [hy (x , y)]2 + 1

    andk n = 1

    [hx(x , y)]2 + [hy (x , y)]2 + 1

    J. Robert Buchanan The Divergence Theorem

  • Proof (6 of 7)

    S1

    P(x , y , z)k n dS =

    S1

    P(x , y , z)[hx(x , y)]2 + [hy (x , y)]2 + 1

    dS

    =

    R

    P(x , y ,h(x , y)) dA

    In a similar way we can show the surface integral over S2 isS2

    P(x , y , z)k n dS =

    RP(x , y ,g(x , y)) dA.

    J. Robert Buchanan The Divergence Theorem

  • Proof (7 of 7)

    Finally,Q

    P(x , y , z)k n dS

    =

    S1

    P(x , y , z)k n dS +

    S2P(x , y , z)k n dS

    +

    S3

    P(x , y , z)k n dS

    =

    R

    P(x , y ,h(x , y)) dA

    RP(x , y ,g(x , y)) dA

    =

    R[P(x , y ,h(x , y)) P(x , y ,g(x , y))] dA

    =

    R

    P(x , y , z)z=h(x ,y)z=g(x ,y)

    dA

    =

    R

    h(x ,y)g(x ,y)

    Pz

    dz dA =

    Q

    Pz

    dV .

    J. Robert Buchanan The Divergence Theorem

  • Example (1 of 2)

    Let Q be the solid unit sphere centered at the origin. Use theDivergence Theorem to calculate the flux of the vector fieldF(x , y , z) = z, y , x over the surface of the unit sphere.

    J. Robert Buchanan The Divergence Theorem

  • Example (2 of 2)

    F(x , y , z) = z, y , x F = 1

    S = {(x , y , z) | x2 + y2 + z2 = 1}Q = {(x , y , z) | x2 + y2 + z2 1}

    According to the Divergence Theorem,S

    F n dS =

    Q F dV =

    Q

    1 dV =43.

    J. Robert Buchanan The Divergence Theorem

  • Example (1 of 3)

    Let Q be the solid region bounded by the parabolic cylinderz = 1 x2 and the planes z = 0, y = 0, and y + z = 2.Calculate the flux of the vector field

    F(x , y , z) = xy i + (y2 + exz2)j + sin(xy)k

    over the boundary of Q.

    J. Robert Buchanan The Divergence Theorem

  • Example (2 of 3)

    Region Q:

    1 x 10 y 2 z0 z 1 x2

    -1.0

    -0.5

    0.0

    0.5

    1.0

    x

    0.0

    0.5

    1.0

    1.5

    2.0y

    0.0

    0.5

    1.0

    z

    J. Robert Buchanan The Divergence Theorem

  • Example (3 of 3)

    F(x , y , z) = xy , y2 + exz2 , sin(xy) F = 3y

    S = {(x , y , z) | z = 1 x2, z = 0, y = 0, y + z = 2}Q = {(x , y , z) |0 z 1 x2, 0 y 2 z}

    According to the Divergence Theorem,S

    F n dS =

    Q F dV =

    Q

    3y dV

    =

    11

    1x20

    2z0

    3y dy dz dx

    =18435

    J. Robert Buchanan The Divergence Theorem

  • Example (3 of 3)

    F(x , y , z) = xy , y2 + exz2 , sin(xy) F = 3y

    S = {(x , y , z) | z = 1 x2, z = 0, y = 0, y + z = 2}Q = {(x , y , z) |0 z 1 x2, 0 y 2 z}

    According to the Divergence Theorem,S

    F n dS =

    Q F dV =

    Q

    3y dV

    =

    11

    1x20

    2z0

    3y dy dz dx

    =18435

    J. Robert Buchanan The Divergence Theorem

  • Example (3 of 3)

    F(x , y , z) = xy , y2 + exz2 , sin(xy) F = 3y

    S = {(x , y , z) | z = 1 x2, z = 0, y = 0, y + z = 2}Q = {(x , y , z) |0 z 1 x2, 0 y 2 z}

    According to the Divergence Theorem,S

    F n dS =

    Q F dV =

    Q

    3y dV

    =

    11

    1x20

    2z0

    3y dy dz dx

    =18435

    J. Robert Buchanan The Divergence Theorem

  • Identities (1 of 2)

    Show that

    S( F) n dS = 0.

    By the Divergence TheoremS

    ( F) n dS =

    Q ( F) dV

    =

    Q

    0 dV

    = 0

    J. Robert Buchanan The Divergence Theorem

  • Identities (1 of 2)

    Show that

    S( F) n dS = 0.

    By the Divergence TheoremS

    ( F) n dS =

    Q ( F) dV

    =

    Q

    0 dV

    = 0

    J. Robert Buchanan The Divergence Theorem

  • Identities (2 of 2)

    Show that

    SDnf (x , y , z) dS =

    Q2f (x , y , z) dV .

    S

    Dnf (x , y , z) dS =

    Sf (x , y , z) n dS

    =

    Q f (x , y , z) dV (Divergence Th.)

    =

    Q2f (x , y , z) dV

    J. Robert Buchanan The Divergence Theorem

  • Identities (2 of 2)

    Show that

    SDnf (x , y , z) dS =

    Q2f (x , y , z) dV .

    S

    Dnf (x , y , z) dS =

    Sf (x , y , z) n dS

    =

    Q f (x , y , z) dV (Divergence Th.)

    =

    Q2f (x , y , z) dV

    J. Robert Buchanan The Divergence Theorem

  • Average Value of a Function

    During