Solved: Greens Theorem as a Fundamental Theorem of

Explain why the two forms of Greens Theorem are analogs of the Fundamental Theorem of Calculus.

Referring to both forms of Greens Theorem, match each idea in Column 1 to an idea in Column 2: Line integral for flux Double integral of the curl Line integral for circulation Double integral of the divergence

Compute the two-dimensional curl of F = 84x3y, xy2 + x49.

Compute the two-dimensional divergence of F = 84x3y, xy2 + x49.

How do you use a line integral to compute the area of a plane region?

Why does a two-dimensional vector field with zero curl on a region have zero circulation on a closed curve that bounds the region?

Why does a two-dimensional vector field with zero divergence on a region have zero outward flux across a closed curve that bounds the region?

Sketch a two-dimensional vector field that has zero curl everywhere in the plane.

Sketch a two-dimensional vector field that has zero divergence everywhere in the plane.

Discuss one of the parallels between a conservative vector field and a source-free vector field.

regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. Is the vector field conservative? F = 8x, y9; R = 51x, y2: x2 + y2 26

regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. Is the vector field conservative? F = 8y, x9; R is the square with vertices 10, 02, 11, 02, 11, 12, and 10, 12.

regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. Is the vector field conservative? F = 82y, -2x9; R is the region bounded by y = sin x and y = 0, for 0 x p.

regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. Is the vector field conservative? F = 8 -3y, 3x9; R is the triangle with vertices 10, 02, 11, 02, and 10, 22.

regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. Is the vector field conservative? F = 82xy, x2 - y29; R is the region bounded by y = x12 - x2 and y = 0.

regions R and vector fields F. a. Compute the two-dimensional curl of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. Is the vector field conservative? F = 80, x2 + y29; R = 51x, y2: x2 + y2 16

1722. Area of regions Use a line integral on the boundary to find the area of the following regions. A disk of radius 5

1722. Area of regions Use a line integral on the boundary to find the area of the following regions. A region bounded by an ellipse with major and minor axes of length 12 and 8, respectively.

1722. Area of regions Use a line integral on the boundary to find the area of the following regions. 51x, y2: x2 + y2 166

1722. Area of regions Use a line integral on the boundary to find the area of the following regions. The region shown in the figure

1722. Area of regions Use a line integral on the boundary to find the area of the following regions. The region bounded by the parabolas r1t2 = 8t, 2t29 and r1t2 = 8t, 12 - t29, for -2 t 2

1722. Area of regions Use a line integral on the boundary to find the area of the following regions. The region bounded by the curve r1t2 = 8t11 - t22, 1 - t29, for -1 t 1 (Hint: Plot the curve.)

2328. Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. State whether the vector field is source free. F = 8x, y9; R = 51x, y2: x2 + y2 46

2328. Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. State whether the vector field is source free. F = 8y, -x9; R is the square with vertices 10, 02, 11, 02, 11, 12, and 10, 12.

2328. Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. State whether the vector field is source free. F = 8y, -3x9; R is the region bounded by y = 4 - x2 and y = 0.

2328. Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. State whether the vector field is source free. F = 8 -3y, 3x9; R is the triangle with vertices 10, 02, 13, 02, and 10, 12.

2328. Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. State whether the vector field is source free. F = 82xy, x2 - y29; R is the region bounded by y = x12 - x2 and y = 0.

2328. Greens Theorem, flux form Consider the following regions R and vector fields F. a. Compute the two-dimensional divergence of the vector field. b. Evaluate both integrals in Greens Theorem and check for consistency. c. State whether the vector field is source free. F = 8x2 + y2, 09; R = 51x, y2: x2 + y2 16

2934. Line integrals Use Greens Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. C C 12x + ey22 dy - 14y2 + ex22 dx, where C is the boundary of the square with vertices 10, 02, 11, 02, 11, 12, and 10, 12

2934. Line integrals Use Greens Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. C C 12x - 3y2 dy - 13x + 4y2 dx, where C is the unit circle

2934. Line integrals Use Greens Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. C C f dy - g dx, where 8 f, g9 = 80, xy9 and C is the triangle with vertices 10, 02, 12, 02, and 10, 42

2934. Line integrals Use Greens Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. C C f dy - g dx, where 8 f, g9 = 8x2, 2y29 and C is the upper half of the unit circle and the line segment -1 x 1 oriented clockwise

2934. Line integrals Use Greens Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. The circulation line integral of F = 8x2 + y2, 4x + y39, where C is the boundary of 51x, y2: 0 y sin x, 0 x p6

2934. Line integrals Use Greens Theorem to evaluate the following line integrals. Unless stated otherwise, assume all curves are oriented counterclockwise. The flux line integral of F = 8ex - y, ey - x9, where C is the boundary of 51x, y2: 0 y x, 0 x 16

3538. General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = 8x, y9; R is the half-annulus 51r, u2: 1 r 2, 0 u p6.

3538. General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = 8 -y, x9; R is the annulus 51r, u2: 1 r 3, 0 u 2p6.

3538. General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = 82x + y, x - 4y9; R is the quarter-annulus 51r, u2: 1 r 4, 0 u p>26.

3538. General regions For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves are oriented counterclockwise. F = 8x - y, -x + 2y9; R is the parallelogram 51x, y2: 1 - x y 3 - x, 0 x 16.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a. The work required to move an object around a closed curve C in the presence of a vector force field is the circulation of the force field on the curve. b. If a vector field has zero divergence throughout a region (on which the conditions of Greens Theorem are met), then the circulation on the boundary of that region is zero. c. If the two-dimensional curl of a vector field is positive throughout a region (on which the conditions of Greens Theorem are met), then the circulation on the boundary of that region is positive (assuming counterclockwise orientation).

4043. Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F = h ln 1x2 + y22, tan -1 y x i, where R is the annulus 51r, u2: 1 r 2, 0 u 2p6

4043. Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F = _12x2 + y22, where R is the half annulus 51r, u2: 1 r 3, 0 u p6

4043. Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F = 8y cos x, -sin x9, where R is the square 51x, y2: 0 x p>2, 0 y p>26

4043. Circulation and flux For the following vector fields, compute (a) the circulation on and (b) the outward flux across the boundary of the given region. Assume boundary curves have counterclockwise orientation. F = 8x + y2, x2 - y9, where R = 51x, y2: 3y2 x 36 - y26

4445. Special line integrals Prove the following identities, where C is a simple closed smooth oriented curve. C C dx = C C dy = 0

4445. Special line integrals Prove the following identities, where C is a simple closed smooth oriented curve. C C f 1x2 dx + g1y2 dy = 0, where f and g have continuous derivatives on the region enclosed by C

Double integral to line integral Use the flux form of Greens Theorem to evaluate 4R 12xy + 4y32 dA, where R is the triangle with vertices 10, 02, 11, 02, and 10, 12.

Area line integral Show that the value of C C xy2 dx + 1x2y + 2x2 dy depends only on the area of the region enclosed by C.

Area line integral In terms of the parameters a and b, how is the value of AC ay dx + bx dy related to the area of the region enclosed by C, assuming counterclockwise orientation of C?

4952. Stream function Recall that if the vector field F = 8f, g9 is source free (zero divergence), then a stream function c exists such that f = cy and g = -cx. a. Verify that the given vector field has zero divergence. b. Integrate the relations f = cy and g = -cx to find a stream function for the field. F = 84, 29

4952. Stream function Recall that if the vector field F = 8f, g9 is source free (zero divergence), then a stream function c exists such that f = cy and g = -cx. a. Verify that the given vector field has zero divergence. b. Integrate the relations f = cy and g = -cx to find a stream function for the field. F = 8y2, x29

4952. Stream function Recall that if the vector field F = 8f, g9 is source free (zero divergence), then a stream function c exists such that f = cy and g = -cx. a. Verify that the given vector field has zero divergence. b. Integrate the relations f = cy and g = -cx to find a stream function for the field. F = 8 -e-x sin y, e-x cos y9

4952. Stream function Recall that if the vector field F = 8f, g9 is source free (zero divergence), then a stream function c exists such that f = cy and g = -cx. a. Verify that the given vector field has zero divergence. b. Integrate the relations f = cy and g = -cx to find a stream function for the field. F = 8x2, -2xy9

5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a. Verify that the curl and divergence of the given field is zero. b. Find a potential function w and a stream function c for the field. c. Verify that w and c satisfy Laplaces equation wxx + wyy = cxx + cyy = 0. F = 8ex cos y, -ex sin y9

5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a. Verify that the curl and divergence of the given field is zero. b. Find a potential function w and a stream function c for the field. c. Verify that w and c satisfy Laplaces equation wxx + wyy = cxx + cyy = 0. F = 8x3 - 3xy2, y3 - 3x2y9

5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a. Verify that the curl and divergence of the given field is zero. b. Find a potential function w and a stream function c for the field. c. Verify that w and c satisfy Laplaces equation wxx + wyy = cxx + cyy = 0. F = h tan-1 y x , 1 2 ln 1x2 + y22 i

5356. Ideal flow A two-dimensional vector field describes ideal flow if it has both zero curl and zero divergence on a simply connected region (excluding the origin if necessary). a. Verify that the curl and divergence of the given field is zero. b. Find a potential function w and a stream function c for the field. c. Verify that w and c satisfy Laplaces equation wxx + wyy = cxx + cyy = 0. F = 8x, y9 x2 + y2

Flow in an ocean basin An idealized two- dimensional ocean is modeled by the square region R = 3-p>2, p>24 * 3-p>2, p>24 with boundary C. Consider the stream function c1x, y2 = 4 cos x cos y defined on R (see figure). a. The horizontal (east-west) component of the velocity is u = cy and the vertical (north-south) component of the velocity is v = -cx. Sketch a few representative velocity vectors and show that the flow is counterclockwise around the region. b. Is the velocity field source free? Explain. c. Is the velocity field irrotational? Explain. d. Let C be the boundary of R. Find the total outward flux across C. e. Find the circulation on C assuming counterclockwise orientation.

Greens Theorem as a Fundamental Theorem of Calculus Show that if the circulation form of Greens Theorem is applied to the vector field h 0, f 1x2 c i and R = 51x, y2: a x b, 0 y c6, then the result is the Fundamental Theorem of Calculus, L b a df dx dx = f 1b2 - f 1a2.

Greens Theorem as a Fundamental Theorem of Calculus Show that if the flux form of Greens Theorem is applied to the vector field h f 1x2 c , 0 i and R = 51x, y2: a x b, 0 y c6, then the result is the Fundamental Theorem of Calculus, L b a df dx dx = f 1b2 - f 1a2.

Whats wrong? Consider the rotation field F = 8 -y, x9 x2 + y2 . a. Verify that the two-dimensional curl of F is zero, which suggests that the double integral in the circulation form of Greens Theorem is zero. b. Use a line integral to verify that the circulation on the unit circle of the vector field is 2p. c. Explain why the results of parts (a) and (b) do not agree.

Whats wrong? Consider the radial field F = 8x, y9 x2 + y2. a. Verify that the divergence of F is zero, which suggests that the double integral in the flux form of Greens Theorem is zero. b. Use a line integral to verify that the outward flux across the unit circle of the vector field is 2p. c. Explain why the results of parts (a) and (b) do not agree.

Conditions for Greens Theorem Consider the radial field F = 8f, g9 = 8x, y9 2x2 + y2 = r _ r _ . a. Explain why the conditions of Greens Theorem do not apply to F on a region that includes the origin. b. Let R be the unit disk centered at the origin and compute OR a 0f 0x + 0g 0y b dA. c. Evaluate the line integral in the flux form of Greens Theorem on the boundary of R. d. Do the results of parts (b) and (c) agree? Explain.

Flux integrals Assume the vector field F = 8 f, g9 is source free (zero divergence) with stream function c. Let C be any smooth simple curve from A to the distinct point B. Show that the flux integral 1C F # n ds is independent of path; that is, 1C F # n ds = c1B2 - c1A2.

Streamlines are tangent to the vector field Assume that the vector field F = 8 f, g9 is related to the stream function c by cy = f and cx = -g on a region R. Prove that at all points of R, the vector field is tangent to the streamlines (the level curves of the stream function).

Streamlines and equipotential lines Assume that on _2, the vector field F = 8 f, g9 has a potential function w such that f = wx and g = wy, and it has a stream function c such that f = cy and g = -cx. Show that the equipotential curves (level curves of w) and the streamlines (level curves of c) are everywhere orthogonal.

Channel flow The flow in a long shallow channel is modeled by the velocity field F = 80, 1 - x29, where R = 51x, y2: _ x _ 1 and _ y _ 56. a. Sketch R and several streamlines of F. b. Evaluate the curl of F on the lines x = 0, x = 14 , x = 12 , and x = 1. c. Compute the circulation on the boundary of R. d. How do you explain the fact that the curl of F is nonzero at points of R, but the circulation is zero?