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- Vipularrow_forwarda) Show that F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j is the gradient of some function f. Find f b) Evaluate the line integral ʃC F dr where the vector field is given by F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j and C is the curve on the circle x 2 + y 2 = 9 from (3, 0) to (0, 3) in a counterclockwise direction.arrow_forwardfellas is it controversial to existarrow_forward
- Find the Curl of the vector field. F(x,y, z) = 3e* sin(y)i+ 7e* cos(y j+ &z k OS %3D (70* cos(y)- 30" cos(y)) k а. (7e" sin(y) + 3e" sin(y) k OC (7e* cos(y) + 3e* cos() O No correct answer e. cos karrow_forwardUse Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)arrow_forwardConsider the function p(r, y) = x +y. Graph at least 4 level curves for p • Compute the gradient field of • Show that the vector field is orthogonal to the level curve at the point (1,1) • Show that the vector field is orthogonal to the level curves at all points (x, y)arrow_forward
- Find a tangent vector of the curve7(t) = (t², 2 sin(t), 2 cos(t)) at (0,0, 2) (0,0,1) (1,0,0) None of the above or below (0,1,0) O (1,1,1) ) (1/2,0, 1/2)arrow_forwardWe are given a vector field and a parametric curvearrow_forwardThe gradient vector field of f(x,y)=y(2x2 -y3 ) is given by: O1. (2xy)i +(x2 -3y² )i O II. (4xy)i -(2x2 -3y? )i O II (4xy)i +(4x2 -3y² )i OV. (4xy)i +(2x2 -3y² )iarrow_forward
- Evaluate the line integral (3ry² + 6y) dr, where C is the path traced by first moving from the point (-3, 1) to the point (2, 1) along a straight line, then moving from the point (2, 1) to the point (5,2) along the parabola x = y² + 1.arrow_forward3. Let f(x, y) = sin x + sin y. (NOTE: You may use software for any part of this problem.) (a) Plot a contour map of f. (b) Find the gradient Vf. (c) Plot the gradient vector field Vf. (d) Explain how the contour map and the gradient vector field are related. (e) Plot the flow lines of Vf. (f) Explain how the flow lines and the vector field are related. (g) Explain how the flow lines of Vf and the contour map are related.arrow_forwardfind the curl of U = exy ax +sin(xy)ay + cos2(xz) azarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage