1. Use Green's Theorem to find the counterclockwise circulation of the field F around the curve C as given below: F = (xy + y²)î + (x – y)j y %3D (1, 1) x = y? y = x2
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- See imageConsider the vector field F and the curve C below. F(x, y) (3 + 8xy2)i + 8x²yj, C is the arc of the hyperbola y = 1/x from (1, 1) to (4, = (a) Find a potential function f such that F = Vf. f(x, y) = (b) Use part (a) to evaluate Sc Vf. dr along the given curve C.(a) We consider the vector field F(x, y, 2) = (x², xy, 1) = x²ï+xyj+ 1k. (i) Calculate curl F = ▼ × F. (ii) Let the curve C be defined by the parameterization = {r(t) = (t, t², 1) , te (0,1)} . Calculate F - dr F - dr along this curve starting at the point A = (0,0, 1) and ending at B = (1, 1, 1).
- Consider the field F = (x + y)i - (x² + y2), and the counterclockwise curve C whose boundary is defined by the lines y = 0, x = 2, and y = x. (a) Sketch a graph of the curve C described above. Indicate the direction on the graph. (b) Express the circulation of the field F as a double integral using Green's theorem over C. Evaluate the integral to find the circulation. (c) Express the flux of the field F as a double integral using Green's theorem over C. Evaluate the integral to find the flux.Calculate the circulation of the field F around the closed curve C. Circulation means line integralF = x 3y 2 i + x 3y 2 j; curve C is the counterclockwise path around the rectangle with vertices at (0,0),(3,0).(3,2) and (0.2)3. Let D be the upper half-plane (y > 0) and consider the vector field F: D+R² given by F(x, y) = (7, =7). Test whether F is a gradient field, and if so, find a potential P.
- Consider the vector field F = Evaluate F dr along the curve C: F(t) = t²7 + t³ j, 02. Find the circulation and flux of the field F(r.y) = (-y,r) along and across the curve (cost, 4 sin t), tE [0, 2]Consider the vector field F = Evaluate r(t) = Pi+tj, 0Find the outward flux for the field F(x,y) = 〈x,y2〉over the rectangle bounded by the curve C shown in the pictureConsider the vector field i = P(x, y) = -3y+x(x² + y? – 1), ý = Q(x, y) = 3æ + y(a² + y? – 1) .2 Rewrite this ODE in polar coordinates and solve it. Use the result of the previous part to find an explicit formula for the Poincaré map.Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,